Tree-Approximations for the Weighted Cost-Distance Problem
We generalize the Cost-Distance problem: Given a set of n sites in k-dimensional Euclidean space and a weighting over pairs of sites, construct a network that minimizes the cost (i.e. weight) of the network and the weighted distances between all pairs of sites. It turns out that the optimal solution can contain Steiner points as well as cycles. Furthermore, there are instances where crossings optimize the network. We then investigate how trees can approximate the weighted Cost-Distance problem. We show that for anyg iven set of n sites and a nonnegative weighting of pairs, provided the sum of the weights is polynomial, one can construct in polynomial time a tree that approximates the optimal network within a factor of O(log n). Finally, we show that better approximation rates are not possible for trees. We prove this bygiv ing a counter-example. Thus, we show that for this instance that everytre e solution differs from the optimal network bya factor ω(log n).
KeywordsPolynomial Time Span Tree Minimum Span Tree Steiner Tree Optimal Network
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- 2.S. Arya, G. Das, D. M. Mount, J. S. Slowe, and M. Smid. Euclidean spanners: Short, thin, and lanky. In Proc. 27th ACM Symp. Theory of Computing. ACM, 1995.Google Scholar
- 3.Bartal. On approximating arbitrarymet rics by tree metrics. In ACM Symposium on Theory of Computing, 1998.Google Scholar
- 4.M. Charikar, C. Chekuri, A. Goel, S. Guha, and S. Plotkin. Approximating a finite metric bya small number of tree metrics. In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, 1998.Google Scholar
- 5.Gautam Das, Paul J. Heffernan, and Giri Narasimhan. Optimally sparse spanners in 3-dimensional euclidean space. In Symposium on Computational Geometry, pages 53–62, 1993.Google Scholar
- 6.David Eppstein. Spanning trees and spanners. In Jörg-Rudiger Sack and Jorge Urrutia, editors, Handbook of Computational Geometry, chapter 9, pages 425–461. Elsevier Science Publishing, 2000.Google Scholar
- 7.M. R. Garey, R. L. Graham, and David S. Johnson. Some NP-complete geometric problems. In ACM Symposium on Theory of Computing, pages 10–22, 1976.Google Scholar
- 10.A. Meyerson, K. Munagala, and S. Plotkin. Cost-Distance: Two metric network design. In Proc. 41st Symp. Foundations of Computer Science. IEEE, 2000.Google Scholar
- 11.Birgitta Weber. Netzwerke optimiert für lokale Kosten-Distanzen. Lübeck University, Diploma Thesis, 2001.Google Scholar