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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 n) time. The second algorithm computes a (3+ϵ)-spanner of K with O(n n) edges in O(n n) time. The latter result is optimal: We show that for any 2 ≤ k ≤ n - Θ(√(n n)), spanners witharXiv:0712.0554v1 fatcat:safhlubecjaarcfymjylbendj4