Dynamic algorithms for geometric spanners of small diameter: Randomized solutions

Let S be a set of n points in IR 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 log log n) expected amortized time per insertion and deletion, in the model of random updates, as introduced by Mulmuley. Key words: Computational geometry, proximity problems, skip lists, randomization, dynamic data structures