WADS 2017: Algorithms and Data Structures pp 85-96

# $$\delta$$-Greedy t-spanner

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

## Abstract

We introduce a new geometric spanner, $$\delta$$-Greedy, whose construction is based on a generalization of the known Path-Greedy and Gap-Greedy spanners. The $$\delta$$-Greedy spanner combines the most desirable properties of geometric spanners both in theory and in practice. More specifically, it has the same theoretical and practical properties as the Path-Greedy spanner: a natural definition, small degree, linear number of edges, low weight, and strong $$(1+\varepsilon )$$-spanner for every $$\varepsilon >0$$. The $$\delta$$-Greedy algorithm is an improvement over the Path-Greedy algorithm with respect to the number of shortest path queries and hence with respect to its construction time. We show how to construct such a spanner for a set of n points in the plane in $$O(n^2 \log n)$$ time.

The $$\delta$$-Greedy spanner has an additional parameter, $$\delta$$, which indicates how close it is to the Path-Greedy spanner on the account of the number of shortest path queries. For $$\delta = t$$ the output spanner is identical to the Path-Greedy spanner, while the number of shortest path queries is, in practice, linear.

Finally, we show that for a set of n points placed independently at random in a unit square the expected construction time of the $$\delta$$-Greedy algorithm is $$O(n \log n)$$. Our analysis indicates that the $$\delta$$-Greedy spanner gives the best results among the known spanners of expected $$O(n \log n)$$ time for random point sets. Moreover, analysis implies that by setting $$\delta = t$$, the $$\delta$$-Greedy algorithm provides a spanner identical to the Path-Greedy spanner in expected $$O(n \log n)$$ time.

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### References

1. 1.
Alewijnse, S.P.A., Bouts, Q.W., ten Brink, A.P., Buchin, K.: Computing the greedy spanner in linear space. Algorithmica 73(3), 589–606 (2015)
2. 2.
Alewijnse, S.P.A., Bouts, Q.W., ten Brink, A.P., Buchin, K.: Distribution-sensitive construction of the greedy spanner. Algorithmica, 1–23 (2016)Google Scholar
3. 3.
Arya, S., Mount, D.M., Smid, M.H.M.: Randomized and deterministic algorithms for geometric spanners of small diameter. In FOCS, pp. 703–712 (1994)Google Scholar
4. 4.
Bar-On, G., Carmi, P.: $$\delta$$-greedy t-spanner. CoRR, abs/1702.05900 (2017)Google Scholar
5. 5.
Bose, P., Carmi, P., Farshi, M., Maheshwari, A., Smid, M.: Computing the greedy spanner in near-quadratic time. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 390–401. Springer, Heidelberg (2008). doi:10.1007/978-3-540-69903-3_35
6. 6.
Callahan, P.B.: Optimal parallel all-nearest-neighbors using the well-separated pair decomposition. In: FOCS, pp. 332–340 (1993)Google Scholar
7. 7.
Callahan, P.B., Kosaraju, S.R.: A decomposition of multi-dimensional point-sets with applications to k-nearest-neighbors and n-body potential fields. In: STOC, pp. 546–556 (1992)Google Scholar
8. 8.
Chandra, B.: Constructing sparse spanners for most graphs in higher dimensions. Inf. Process. Lett. 51(6), pp. 289–294 (1994)Google Scholar
9. 9.
Chandra, B., Das, G., Narasimhan, G., Soares, J.: New sparseness results on graph spanners. Int. J. Comp. Geom. and Applic. 5, 125–144 (1995)
10. 10.
Clarkson, K.L.: Approximation algorithms for shortest path motion planning. In STOC, pp. 56–65 (1987)Google Scholar
11. 11.
Das, G., Heffernan, P.J., Narasimhan, G.: Optimally sparse spanners in 3-dimensional Euclidean space. In SoCG, pp. 53–62 (1993)Google Scholar
12. 12.
Das, G., Narasimhan, G.: A fast algorithm for constructing sparse Euclidean spanners. Int. J. Comp. Geom. and Applic. 7(4), 297–315 (1997)
13. 13.
Farshi, M., Gudmundsson, J.: Experimental study of geometric t-spanners. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 556–567. Springer, Heidelberg (2005). doi:10.1007/11561071_50
14. 14.
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). doi:10.1007/978-3-540-72845-0_21
15. 15.
Farshi, M., Gudmundsson, J.: Experimental study of geometric t-spanners. ACM Journal of Experimental Algorithmics 14 (2009)Google Scholar
16. 16.
Gudmundsson, J., Levcopoulos, C., Narasimhan, G.: Fast greedy algorithms for constructing sparse geometric spanners. SIAM J. Comput. 31(5), 1479–1500 (2002)
17. 17.
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). doi:10.1007/3-540-19487-8_23
18. 18.
Levcopoulos, C., Narasimhan, G., Smid, M.H.M.: Improved algorithms for constructing fault-tolerant spanners. Algorithmica 32(1), 144–156 (2002)
19. 19.
Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, New York (2007)
20. 20.
Soares, J.: Approximating Euclidean distances by small degree graphs. Discrete & Computational Geometry 11(2), 213–233 (1994)Google Scholar