Spanners of Complete k -Partite Geometric Graphs

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.