Partitioned Neighborhood Spanners of Minimal Outdegree

Matthias Fischer, Tamas Lukovszki, Martin Ziegler


A geometric spanner with vertex set P in R^d 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 and the stretch factor f>1 describing the "quality" of approximation. PNGs have been throughly investigated with respect to small values of f. We, on the other hand, present in this work results about small values of k. The aim of minimizing this parameter rather than the first one arises from two observations: 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 constructive and yield efficient algorithms for actually computing the corresponding graphs even in degenerate cases.