A Framework for Computing the Greedy Spanner

ABSTRACT

The highest quality geometric spanner (e.g. in terms of edge count, both in theory and in practice) known to be computable in polynomial time is the greedy spanner. The state-of-the-art in computing this spanner are a O(n2 log n) time, O(n2) space algorithm and a O(n2 log2 n) time, O(n) space algorithm, as well as the 'improved greedy' algorithm, taking O(n3 log n) time in the worst case and O(n2) space but being faster in practice thanks to a caching strategy.

We identify why this caching strategy gives speedups in practice. We formalize this into a framework and give a general efficiency lemma. From this we obtain many new time bounds, both on old algorithms and on new algorithms we introduce in this paper. Interestingly, our bounds are in terms of the well-separated pair decomposition, a data structure not actually computed by the caching algorithms.

Specifically, we show that the 'improved greedy' algorithm has a O(n2 log n log Φ) running time (where Φ is the spread of the point set) and a variation has a O(n2 log2 n) running time. We give a variation of the linear space state-of-the-art algorithm and an entirely new algorithm with a O(n2 log n log Φ) running time, both of which improve its space usage by a factor O(1/(t−1)), where t is the dilation of the spanner.

We present experimental results comparing all the above algorithms. The experiments show that our new algorithm is much more space efficient than the existing linear space algorithm - up to 200 times when using low t - while being comparable in running time and much easier to implement.

References

1. S. P. A. Alewijnse, Q. W. Bouts, A. P. ten Brink, and K. Buchin. Computing the greedy spanner in linear space. CoRR, arXiv:1306.4919, 2013.
2. P. Bose, P. Carmi, M. Farshi, A. Maheshwari, and M. Smid. Computing the greedy spanner in near-quadratic time. Algorithmica, 58(3):711--729, 2010.
3. P. B. Callahan. Dealing with Higher Dimensions: The Well-Separated Pair Decomposition and Its Applications. PhD thesis, Johns Hopkins University, Baltimore, Maryland, 1995.
4. P. B. Callahan and S. R. Kosaraju. A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. ACM, 42(1):67--90, 1995.
5. L. P. Chew. There are planar graphs almost as good as the complete graph. J. Comput. System Sci., 39(2):205--219, 1989.
6. M. Farshi and J. Gudmundsson. Experimental study of geometric t-spanners. ACM J. Experimental Algorithmics, 14, 2009.
7. J. Gao, L. J. Guibas, J. Hershberger, L. Zhang, and A. Zhu. Geometric spanners for routing in mobile networks. IEEE J. Selected Areas in Communications, 23(1):174--185, 2005.
8. J. Gudmundsson and C. Knauer. Dilation and detours in geometric networks. In T. Gonzales, editor, Handbook on Approximation Algorithms and Metaheuristics, pages 52--1--52--16. Chapman & Hall/CRC, Boca Raton, 2006.
9. J. M. Keil. Approximating the complete euclidean graph. In 1st Scandinavian Workshop on Algorithm Theory (SWAT), volume 318 of LNCS, pages 208--213. Springer, 1988.
10. G. Narasimhan and M. Smid. Geometric Spanner Networks. Cambridge University Press, New York, NY, USA, 2007.
11. D. Peleg and A. A. Schäffer. Graph spanners. Journal of Graph Theory, 13(1):99--116, 1989.

Index Terms

• Theory of computation

  • Design and analysis of algorithms