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SIAM J. Comput. 38, pp. 1803-1820 (18 pages)
Spanners of Complete $k$-Partite Geometric Graphs
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Add to FacebookWe 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.
© 2009 Society for Industrial and Applied Mathematics
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