Spanners of complete k-partite geometric graphs

Authors:
Prosenjit Bose;Paz Carmi;Mathieu Couture;Anil Maheshwari;Pat Morin;Michiel Smid
Affiliations:
School of Computer Science, Carleton University, Ottawa, Ontario, Canada;School of Computer Science, Carleton University, Ottawa, Ontario, Canada;School of Computer Science, Carleton University, Ottawa, Ontario, Canada;School of Computer Science, Carleton University, Ottawa, Ontario, Canada;School of Computer Science, Carleton University, Ottawa, Ontario, Canada;School of Computer Science, Carleton University, Ottawa, Ontario, Canada
Venue:
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Year:
2008

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Abstract

We address the following problem: Given a complete k-partite geometric graph K whose vertex set is a set of n points in Rd, 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 + Ɛ)- spanner of K with O(n) edges in O(n log n) time. The second algorithm computes a (3 + Ɛ)-spanner of K with O(n log n) edges in O(n log n) 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.