Dynamic algorithms for geometric spanners of small diameter: Randomized solutions

Abstract Let S be a set of n points in R d and let t >1 be a real number. A t-spanner for S is a directed graph having the points of S as its vertices, such that for any pair p and q of points there is a path from p to q of length at most t times the Euclidean distance between p and q. Such a path is called a t-spanner path. The spanner diameter of such a spanner is defined as the smallest integer D such that for any pair p and q of points there is a t-spanner path from p to q containing at most D edges. A randomized algorithm is given for constructing a t-spanner that, with high probability, contains O( n ) edges and has spanner diameter O(log n ). A data structure of size O( n log d n ) is given that maintains this t-spanner in O(log d n loglog n ) expected amortized time per insertion and deletion, in the model of random updates, as introduced by Mulmuley.

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