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Abstract

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. Previously, no results were known for spanners with low spanner diameter and for maintaining spanners under insertions and deletions.