Abstract
We introduce the weak gap property for directed graphs whose vertex set S is a metric space of size n. We prove that, if the doubling dimension of S is a constant, any directed graph satisfying the weak gap property has O(n) edges and total weight \(O( \log n ) \cdot wt({\mathord{\it MST}}(S))\), where \(wt({\mathord{\it MST}}(S))\) denotes the weight of a minimum spanning tree of S. We show that 2-optimal TSP tours and greedy spanners satisfy the weak gap property.
This work was supported by NSERC.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Althöfer, I., Das, G., Dobkin, D.P., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete & Computational Geometry 9, 81–100 (1993)
Assouad, P.: Plongements lipschitziens dans ℝN. Bulletin de la Société Mathématique de France 111, 429–448 (1983)
Bose, P., Carmi, P., Farshi, M., Maheshwari, A., Smid, M.: Computing the greedy spanner in near-quadratic time. To appear in Algorithmica
Callahan, P.B., Kosaraju, S.R.: A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. Journal of the ACM 42, 67–90 (1995)
Chandra, B., Das, G., Narasimhan, G., Soares, J.: New sparseness results on graph spanners. International Journal of Computational Geometry & Applications 5, 125–144 (1995)
Chandra, B., Karloff, H., Tovey, C.: New results on the old k-opt algorithm for the traveling salesman problem. SIAM Journal on Computing 28, 1998–2029 (1999)
Das, G., Heffernan, P., Narasimhan, G.: Optimally sparse spanners in 3-dimensional Euclidean space. In: Proceedings of the 9th ACM Symposium on Computational Geometry, pp. 53–62 (1993)
Das, G., Narasimhan, G., Salowe, J.: A new way to weigh malnourished Euclidean graphs. In: Proceedings of the 6th ACM-SIAM Symposium on Discrete Algorithms, pp. 215–222 (1995)
Gudmundsson, J., Levcopoulos, C., Narasimhan, G.: Fast greedy algorithms for constructing sparse geometric spanners. SIAM Journal on Computing 31, 1479–1500 (2002)
Har-Peled, S., Mendel, M.: Fast construction of nets in low-dimensional metrics and their applications. SIAM Journal on Computing 35, 1148–1184 (2006)
Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, Berlin (2001)
Lin, S.: Computer solutions of the traveling salesman problem. Bell Systems Technical Journal 44, 2245–2269 (1965)
Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (2007)
Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, Berlin (1988)
Soares, J.: Approximating Euclidean distances by small degree graphs. Discrete & Computational Geometry 11, 213–233 (1994)
Vaidya, P.M.: An O(n logn) algorithm for the all-nearest-neighbors problem. Discrete & Computational Geometry 4, 101–115 (1989)
Yao, A.C.: On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM Journal on Computing 11, 721–736 (1982)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Smid, M. (2009). The Weak Gap Property in Metric Spaces of Bounded Doubling Dimension. In: Albers, S., Alt, H., Näher, S. (eds) Efficient Algorithms. Lecture Notes in Computer Science, vol 5760. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03456-5_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-03456-5_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03455-8
Online ISBN: 978-3-642-03456-5
eBook Packages: Computer ScienceComputer Science (R0)