Abstract
The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it on n points take Ω(n 2) time, limiting its use on large data sets.
We observe that for many point sets, the greedy spanner has many ‘short’ edges that can be determined locally and usually quickly, and few or no ‘long’ edges that can usually be determined quickly using local information and the well-separated pair decomposition. We give experimental results showing large to massive performance increases over the state-of-the-art on nearly all tests and real-life data sets. On the theoretical side we prove a near-linear expected time bound on uniform point sets and a near-quadratic worst-case bound.
Our bound for point sets drawn uniformly and independently at random in a square follows from a local characterization of t-spanners we give on such point sets: we give a geometric property that holds with high probability on such point sets. This property implies that if an edge set on these points has t-paths between pairs of points ‘close’ to each other, then it has t-paths between all pairs of points.
This characterization gives a O(n log2 n log2 logn) expected time bound on our greedy spanner algorithm, making it the first subquadratic time algorithm for this problem on any interesting class of points. We also use this characterization to give a O((n + |E|) log2 n loglogn) expected time algorithm on uniformly distributed points that determines if E is a t-spanner, making it the first subquadratic time algorithm for this problem that does not make assumptions on E.
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
Abam, M.A., de Berg, M., Farshi, M., Gudmundsson, J.: Region-fault tolerant geometric spanners. Discr. Comp. Geom. 41(4), 556–582 (2009)
Agarwal, P.K., Klein, R., Knauer, C., Langerman, S., Morin, P., Sharir, M., Soss, M.: Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D. Discrete Comput. Geom. 39(1), 17–37 (2008)
Alewijnse, S.P.A., Bouts, Q.W., ten Brink, A.P., Buchin, K.: Computing the greedy spanner in linear space. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 37–48. Springer, Heidelberg (2013)
Alewijnse, S.P.A., Bouts, Q.W., ten Brink, A.P., Buchin, K.: Distribution-sensitive construction of the greedy spanner. CoRR, arXiv:1401.1085 (2014)
Atallah, M.: Some dynamic computational geometry problems. Computers and Mathematics with Applications 11, 1171–1181 (1985)
Bose, P., Carmi, P., Farshi, M., Maheshwari, A., Smid, M.: Computing the greedy spanner in near-quadratic time. Algorithmica 58(3), 711–729 (2010)
Bose, P., Devroye, L., Evans, W., Kirkpatrick, D.: On the spanning ratio of Gabriel graphs and beta-skeletons. SIAM Journal on Discrete Mathematics 20(2), 412–427 (2006)
Buchin, K.: Constructing Delaunay triangulations along space-filling curves. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 119–130. Springer, Heidelberg (2009)
Callahan, P.B.: Dealing with Higher Dimensions: The Well-Separated Pair Decomposition and Its Applications. PhD thesis, Johns Hopkins University, Baltimore, Maryland (1995)
Chew, L.P.: There are planar graphs almost as good as the complete graph. J. Comput. System Sci. 39(2), 205–219 (1989)
Devroye, L.: On the expected size of some graphs in computational geometry. Comput. Math. Appl. 15, 53–64 (1988)
Eppstein, D., Wortman, K.A.: Minimum dilation stars. Comput. Geom. 37(1), 27–37 (2007)
Gao, J., Guibas, L.J., Hershberger, J., Zhang, L., Zhu, A.: Geometric spanners for routing in mobile networks. IEEE J. Selected Areas in Communications 23(1), 174–185 (2005)
Gudmundsson, J., Knauer, C.: Dilation and detours in geometric networks. In: Gonzales, T. (ed.) Handbook on Approximation Algorithms and Metaheuristics, pp. 52-1– 52-16. Chapman and Hall/CRC, Boca Raton (2006)
Keil, J.M.: Approximating the complete Euclidean graph. In: Karlsson, R., Lingas, A. (eds.) SWAT 1988. LNCS, vol. 318, pp. 208–213. Springer, Heidelberg (1988)
Narasimhan, G., Smid, M.: Approximating the stretch factor of Euclidean graphs. SIAM J. Comput. 30(3), 978–989 (2000)
Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, New York (2007)
Peleg, D., Schäffer, A.A.: Graph spanners. Journal of Graph Theory 13(1), 99–116 (1989)
Santi, P.: Topology control in wireless ad hoc and sensor networks. ACM Computing Surveys (CSUR) 37(2), 164–194 (2005)
Sharir, M., Agarwal, P.: Davenport-Schinzel Sequences and their Geometric Applications. Cambridge University Press (1995)
Steele, J.M.: Probability Theory and Combinatorial Optimization. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 69. SIAM (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Alewijnse, S.P.A., Bouts, Q.W., Ten Brink, A.P. (2014). Distribution-Sensitive Construction of the Greedy Spanner. In: Schulz, A.S., Wagner, D. (eds) Algorithms - ESA 2014. ESA 2014. Lecture Notes in Computer Science, vol 8737. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44777-2_6
Download citation
DOI: https://doi.org/10.1007/978-3-662-44777-2_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-44776-5
Online ISBN: 978-3-662-44777-2
eBook Packages: Computer ScienceComputer Science (R0)