Proceedings of the international symposium on Optimal algorithms
Optimally sparse spanners in 3-dimensional Euclidean space
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Euclidean spanners: short, thin, and lanky
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
On approximating arbitrary metrices by tree metrics
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
Journal of the ACM (JACM)
Approximating a Finite Metric by a Small Number of Tree Metrics
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Cost-distance: two metric network design
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Some NP-complete geometric problems
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
| Hi-index | 0.00 |
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 non-negative 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 by giving a counter-example. Thus, we show that for this instance that every tree solution differs from the optimal network bya factor Ω(log n).