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 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∈+∈ε)-spanner of K with O(n) edges in O(n logn) time. The second algorithm computes a (3∈+∈ε)-spanner of K with O(n logn) edges in O(n logn) time. Finally, we show that there exist complete k-partite geometric graphs K such that every subgraph of K with a subquadratic number of edges has stretch factor at least 3