Dynamic algorithms for geometric spanners of small diameter: Randomized solutions - Arya, Sunil; Mount, David M.; Smid, Michiel (Computational geometry : theory and applications, 1999)
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( nlog d n) is given that maintains this t-spanner in O(log d nloglog n) expected amortized time per insertion and deletion, in the model of random updates, as introduced by Mulmuley.