Abstract
Given a set S of n points, a weight function w to associate a non-negative weight to each point in S, a positive integer \(k \ge 1\), and a real number \(\epsilon > 0\), we devise the following algorithms to compute a k-vertex fault-tolerant spanner network G(S, E) for the metric space induced by the weighted points in S: (1) When the points in S are located in a simple polygon, we present an algorithm to compute G with multiplicative stretch \(\sqrt{10}+\epsilon \), and the number of edges in G is \(O(k n (\lg {n})^2)\). (2) When the points in S are located in the free space of a polygonal domain \(\mathcal{P}\), we present an algorithm to compute G of size \(O(\sqrt{h} k n(\lg {n})^2)\) and its multiplicative stretch is \(6+\epsilon \). Here, h is the number of simple polygonal holes in \(\mathcal{P}\).
This research is supported in part by SERB MATRICS grant MTR/2017/000474.
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References
Abam, M.A., Adeli, M., Homapour, H., Asadollahpoor, P.Z.: Geometric spanners for points inside a polygonal domain. In: Proceedings of Symposium on Computational Geometry, pp. 186–197 (2015)
Abam, M.A., de Berg, M., Farshi, M., Gudmundsson, J.: Region-fault tolerant geometric spanners. Discret. Comput. Geom. 41(4), 556–582 (2009)
Abam, M.A., de Berg, M., Farshi, M., Gudmundsson, J., Smid, M.H.M.: Geometric spanners for weighted point sets. Algorithmica 61(1), 207–225 (2011)
Abam, M.A., de Berg, M., Seraji, M.J.R.: Geodesic spanners for points on a polyhedral terrain. In: Proceedings of Symposium on Discrete Algorithms, pp. 2434–2442 (2017)
Abam, M.A., Har-Peled, S.: New constructions of SSPDs and their applications. Comput. Geom. 45(5), 200–214 (2012)
Alon, N., Seymour, P.D., Thomas, R.: Planar separators. SIAM J. Discret. Math. 7(2), 184–193 (1994)
Althöfer, I., Das, G., Dobkin, D., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discret. Comput. Geom. 9(1), 81–100 (1993). https://doi.org/10.1007/BF02189308
Aronov, B., et al.: Sparse geometric graphs with small dilation. Comput. Geom. 40(3), 207–219 (2008)
Arya, S., Das, G., Mount, D.M., Salowe, J.S., Smid, M.H.M.: Euclidean spanners: short, thin, and lanky. In: Proceedings of Annual ACM Symposium on Theory of Computing, pp. 489–498 (1995)
Arya, S., Mount, D.M., Smid, M.: Dynamic algorithms for geometric spanners of small diameter: randomized solutions. Comput. Geom. 13(2), 91–107 (1999)
Arya, S., Mount, D.M., Smid, M.H.M.: Randomized and deterministic algorithms for geometric spanners of small diameter. In: Proceedings of Annual Symposium on Foundations of Computer Science, pp. 703–712 (1994)
Arya, S., Smid, M.H.M.: Efficient construction of a bounded-degree spanner with low weight. Algorithmica 17(1), 33–54 (1997)
Bhattacharjee, S., Inkulu, R.: Fault-tolerant additive weighted geometric spanners. In: Pal, S.P., Vijayakumar, A. (eds.) CALDAM 2019. LNCS, vol. 11394, pp. 29–41. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11509-8_3
Bhattacharjee, S., Inkulu, R.: Geodesic fault-tolerant additive weighted spanners. In: Du, D.-Z., Duan, Z., Tian, C. (eds.) COCOON 2019. LNCS, vol. 11653, pp. 38–51. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26176-4_4
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., Carmi, P., Chaitman, L., Collette, S., Katz, M.J., Langerman, S.: Stable roommates spanner. Comput. Geom. 46(2), 120–130 (2013)
Bose, P., Carmi, P., Couture, M.: Spanners of additively weighted point sets. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 367–377. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-69903-3_33
Bose, P., Czyzowicz, J., Kranakis, E., Krizanc, D., Maheshwari, A.: Polygon cutting: revisited. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 1998. LNCS, vol. 1763, pp. 81–92. Springer, Heidelberg (2000). https://doi.org/10.1007/978-3-540-46515-7_7
Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S.: On plane constrained bounded-degree spanners. Algorithmica 81(4), 1392–1415 (2018). https://doi.org/10.1007/s00453-018-0476-8
Carmi, P., Chaitman, L.: Stable roommates and geometric spanners. In: Proceedings of the 22nd Annual Canadian Conference on Computational Geometry, pp. 31–34 (2010)
Chew, L.P.: There are planar graphs almost as good as the complete graph. J. Comput. Syst. Sci. 39(2), 205–219 (1989)
Czumaj, A., Zhao, H.: Fault-tolerant geometric spanners. Discret. Comput. Geom. 32(2), 207–230 (2004)
Das, G., Narasimhan, G.: A fast algorithm for constructing sparse Euclidean spanners. Int. J. Comput. Geom. Appl. 7(4), 297–315 (1997)
Eppstein, D.: Spanning trees and spanners. In: Sack, J.-R., Urrutia, J. (ed.) Handbook of Computational Geometry, pp. 425–461. Elsevier (1999)
Gudmundsson, J., Knauer, C.: Dilation and detour in geometric networks. In: Gonzalez, T. (ed.) Handbook of Approximation Algorithms and Metaheuristics. Chapman & Hall (2007)
Gudmundsson, J., Levcopoulos, C., Narasimhan, G.: Fast greedy algorithms for constructing sparse geometric spanners. SIAM J. Comput. 31(5), 1479–1500 (2002)
Kapoor, S., Li, X.-Y.: Efficient construction of spanners in d-dimensions. CoRR, abs/1303.7217 (2013)
Levcopoulos, C., Narasimhan, G., Smid, M.H.M.: Improved algorithms for constructing fault-tolerant spanners. Algorithmica 32(1), 144–156 (2002)
Lukovszki, T.: New results on fault tolerant geometric spanners. In: Dehne, F., Sack, J.-R., Gupta, A., Tamassia, R. (eds.) WADS 1999. LNCS, vol. 1663, pp. 193–204. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48447-7_20
Narasimhan, G., Smid, M.H.M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (2007)
Peleg, D., Schäffer, A.: Graph spanners. J. Graph Theory 13(1), 99–116 (1989)
Solomon, S.: From hierarchical partitions to hierarchical covers: optimal fault-tolerant spanners for doubling metrics. In: Proceedings of Symposium on Theory of Computing, pp. 363–372 (2014)
Talwar, K.: Bypassing the embedding: algorithms for low dimensional metrics. In: Proceedings of ACM Symposium on Theory of Computing, pp. 281–290 (2004)
Varadarajan, K.R.: A divide-and-conquer algorithm for min-cost perfect matching in the plane. In: Proceedings of Annual Symposium on Foundations of Computer Science, pp. 320–329 (1998)
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Inkulu, R., Singh, A. (2020). Vertex Fault-Tolerant Spanners for Weighted Points in Polygonal Domains. In: Wu, W., Zhang, Z. (eds) Combinatorial Optimization and Applications. COCOA 2020. Lecture Notes in Computer Science(), vol 12577. Springer, Cham. https://doi.org/10.1007/978-3-030-64843-5_32
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