Abstract
We show that for every set \(\mathcal S\) of n points in the plane and a designated point \(rt \in \mathcal S\), there exists a tree T that has small maximum degree, depth and weight. Moreover, for every point \(v \in \mathcal S\), the distance between rt and v in T is within a factor of (1 + ε) close to their Euclidean distance ||rt,v||. We call these trees narrow-shallow-low-light (NSLLTs). We demonstrate that our construction achieves optimal (up to constant factors) tradeoffs between all parameters of NSLLTs. Our construction extends to point sets in \(\mathbb R^d\), for an arbitrarily large constant d. The running time of our construction is O(n ·logn).
We also study this problem in general metric spaces, and show that NSLLTs with small maximum degree, depth and weight can always be constructed if one is willing to compromise the root-distortion. On the other hand, we show that the increased root-distortion is inevitable, even if the point set \(\mathcal S\) resides in a Euclidean space of dimension Θ(logn).
On the bright side, we show that if one is allowed to use Steiner points then it is possible to achieve root-distortion (1 + ε) together with small maximum degree, depth and weight for general metric spaces.
Finally, we establish some lower bounds on the power of Steiner points in the context of Euclidean spanning trees and spanners.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Alpert, C.J., Hu, T.C., Huang, J.H., Kahng, A.B., Karger, D.: Prim-Dijkstra tradeoffs for improved performance-driven routing tree design. IEEE Trans. on CAD of Integrated Circuits and Systems 14(7), 890–896 (1995)
Arya, S., Das, G., Mount, D.M., Salowe, J.S., Smid, M.H.M.: Euclidean spanners: short, thin, and lanky. In: 27th ACM Symposium on Theory of Computing, pp. 489–498. ACM Press, New York (1995)
Awerbuch, B., Baratz, A., Peleg, D.: Cost-sensitive analysis of communication protocols. In: 9th ACM Symposium on Principles of Distributed Computing, pp. 177–187. ACM Press, New York (1990)
Awerbuch, B., Baratz, A., Peleg, D.: Efficient Broadcast and Light-Weight Spanners (manuscript) (1991)
Chan, T.M.: Euclidean Bounded-Degree Spanning Tree Ratios. Discrete & Computational Geometry 32(2), 177–194 (2004)
Cong, J., Kahng, A.B., Robins, G., Sarrafzadeh, M., Wong, C.K.: Performance-Driven Global Routing for Cell Based ICs. In: 9th IEEE International Conference on Computer Design: VLSI in Computer & Processors, pp. 170–173. IEEE press, New York (1991)
Cong, J., Kahng, A.B., Robins, G., Sarrafzadeh, M., Wong, C.K.: Provably good performance-driven global routing. IEEE Trans. on CAD of Integrated Circuits and Sys. 11(6), 739–752 (1992)
Dinitz, Y., Elkin, M., Solomon, S.: Shallow-Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners. In: 49th IEEE Symposium on Foundations of Computer Science, pp. 519–528. EEE Press, New York (2008)
Eppstein, D.: Spanning trees and spanners. Technical report 96–16, Dept. of Information and Computer-Science, University of California, Irvine (1996)
Farshi, M., Gudmundsson, J.: Experimental Study of Geometric t-Spanners: A Running Time Comparison. In: Demetrescu, C. (ed.) WEA 2007. LNCS, vol. 4525, pp. 270–284. Springer, Heidelberg (2007)
Fekete, S.P., Khuller, S., Klemmstein, M., Raghavachari, B., Young, N.E.: A Network-Flow Technique for Finding Low-Weight Bounded-Degree Spanning Trees. J. Algorithms 24(2), 310–324 (1997)
Grünewald, M., Lukovszki, T., Schindelhauer, C., Volbert, K.: Distributed Maintenance of Resource Efficient Wireless Network Topologies. In: Monien, B., Feldmann, R.L. (eds.) Euro-Par 2002. LNCS, vol. 2400, pp. 935–946. Springer, Heidelberg (2002)
Gupta, A.: Steiner points in tree metrics don’t (really) help. In: 12th ACM-SIAM Symposium on Discrete Algorithms, pp. 220–227. SIAM Press, Philadelphia (2001)
Khuller, S., Raghavachari, B., Young, N.E.: Balancing Minimum Spanning and Shortest Path Trees. In: 4th ACM-SIAM Symposium on Discrete Algorithms, pp. 243–250. ACM Press, New York (1993)
Lukovszki, T.: New Results on Fault Tolerant Geometric Spanners. In: Dehne, F., Gupta, A., Sack, J.-R., Tamassia, R. (eds.) WADS 1999. LNCS, vol. 1663, pp. 193–204. Springer, Heidelberg (1999)
Lukovszki, T.: New Results on Geometric Spanners and Their Applications. Ph.D thesis, Dept. of Computer-Science, University of Paderborn, Paderborn, Germany (1999)
Lukovszki, T., Schindelhauer, C., Volbert, K.: Resource Efficient Maintenance of Wireless Network Topologies. J. UCS 12(9), 1292–1311 (2006)
Monma, C.L., Suri, S.: Transitions in Geometric Minimum Spanning Trees. Discrete & Computational Geometry 8, 265–293 (1992)
Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (2007)
Papadimitriou, C.H., Vazirani, U.V.: On Two Geometric Problems Related to the Traveling Salesman Problem. J. Algorithms 5(2), 231–246 (1984)
Ruppert, J., Seidel, R.: Approximating the d-dimensional complete Euclidean graph. In: 3rd Canadian Conference on Computational Geometry, pp. 207–210 (1991)
Salowe, J.S., Richards, D.S., Wrege, D.E.: Mixed spanning trees: a technique for performance-driven routing. In: 3rd ACM Great Lakes symposium on VLSI, pp. 62–66. ACM Press, New York (1993)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Elkin, M., Solomon, S. (2009). Narrow-Shallow-Low-Light Trees with and without Steiner Points. In: Fiat, A., Sanders, P. (eds) Algorithms - ESA 2009. ESA 2009. Lecture Notes in Computer Science, vol 5757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04128-0_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-04128-0_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04127-3
Online ISBN: 978-3-642-04128-0
eBook Packages: Computer ScienceComputer Science (R0)