TR-07-22: Spanners of complete k-Partite Geometric Graphs

Abstract

We address the following problem: Given a complete $k$-partite geometric graph $K$ whose vertex set is a set of $n$ points in $\mathbb{R}^d$, compute a spanner of $K$ that has a “small” stretch factor and “few” edges. We present two algorithms for this problem. The first algorithm computes a $(5+\epsilon)$-spanner of $K$ with $O(n)$ edges in $O(n \log n)$ time. The second algorithm computes a $(3+\epsilon)$-spanner of $K$ with $O(n \log n)$ edges in $O(n \log n)$ time. The latter result is optimal: We show that for any $2 \leq k \leq n – \Theta(\sqrt{n \log n} )$, spanners with $O(n \log n)$ edges and stretch factor less than $3$ do not exist for all complete $k$-partite geometric graphs.

TR-07-22.pdf