Partitioned Neighborhood Spanners of Minimal Outdegree1

A geometric spanner with vertex set P IRD is a sparse approximation of the complete Euclidean graph determined by P. We introduce the notion of partitioned neighborhood graphs (PNGs), unifying and generalizing most constructions of spanners treated in literature. Two important parameters characterizing their properties are the outdegree k 2 IN and the stretch factor f > 1 describing the ‘quality’ of the approximation. PNGs have been throughly investigated with respect to small values of f . We present in this work results about small values of k. The aim of minimizing k rather than f arises from two observations: a) k determines the amount of space required for storing PNGs. b) Many algorithms employing a (previously constructed) spanner have running times depending on its outdegree. Our results include, for fixed dimensions D as well as asymptotically, upper and lower bounds on this optimal value of k. The upper bounds are shown constructively and yield efficient algorithms for actually computing the corresponding PNGs even in degenerate cases. 1Partially supported by EU ESPRIT Long Term Research Project 20244 (ALCOM-IT) and DFG Grants Me872/7-1 and Me872/4-1