Skip to main content
SpringerLink
Log in

Geodesic Fault-Tolerant Additive Weighted Spanners

  • Conference paper
  • First Online:
Computing and Combinatorics (COCOON 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11653))

Included in the following conference series:

Abstract

Let S be a set of n points and let w be a function that assigns non-negative weights to points in S. The additive weighted distance \(d_w(p, q)\) between two points \(p,q \in S\) is defined as \(w(p) + d(p, q) + w(q)\) if \(p \ne q\) and it is zero if \(p = q\). Here, d(p, q) is the geodesic Euclidean distance between p and q. For a real number \(t > 1\), a graph G(S, E) is called a t-spanner for the weighted set S of points if for any two points p and q in S the distance between p and q in graph G is at most t.\(d_w(p, q)\) for a real number \(t > 1\). For some integer \(k \ge 1\), a t-spanner G for the set S is a (k, t)-vertex fault-tolerant additive weighted spanner, denoted with (k, t)-VFTAWS, if for any set \(S' \subset S\) with cardinality at most k, the graph \(G \setminus S'\) is a t-spanner for the points in \(S \setminus S'\). For any given real number \(\epsilon > 0\), we present algorithms to compute a \((k, 4+\epsilon )\)-VFTAWS for the metric space \((S, d_w)\) resulting from the following: (i) points in S are in the free space of the polygonal domain, (ii) points in S lying on a terrain.

This research is supported in part by SERB MATRICS grant MTR/2017/000474.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. 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)

    Google Scholar 

  2. Abam, M.A., de Berg, M.: Kinetic spanners in \(\mathbb{R}^d\). Discrete Comput. Geom. 45(4), 723–736 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Abam, M.A., de Berg, M., Farshi, M., Gudmundsson, J.: Region-fault tolerant geometric spanners. Discrete Comput. Geom. 41(4), 556–582 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Abam, M.A., de Berg, M., Gudmundsson, J.: A simple and efficient kinetic spanner. Comput. Geom. 43(3), 251–256 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Google Scholar 

  7. Abu-Affash, A.K., Aschner, R., Carmi, P., Katz, M.J.: Minimum power energy spanners in wireless ad hoc networks. Wirel. Netw. 17(5), 1251–1258 (2011)

    Article  Google Scholar 

  8. Alon, N., Seymour, P.D., Thomas, R.: Planar separators. SIAM J. Discrete Math. 7(2), 184–193 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Althöfer, I., Das, G., Dobkins, D., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete Comput. Geom. 9(1), 81–100 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Arikati, S., Chen, D.Z., Chew, L.P., Das, G., Smid, M., Zaroliagis, C.D.: Planar spanners and approximate shortest path queries among obstacles in the plane. In: Diaz, J., Serna, M. (eds.) ESA 1996. LNCS, vol. 1136, pp. 514–528. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61680-2_79

    Chapter  Google Scholar 

  11. Aronov, B., et al.: Sparse geometric graphs with small dilation. Comput. Geom. 40(3), 207–219 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Google Scholar 

  13. Arya, S., Mount, D.M., Smid, M.: Dynamic algorithms for geometric spanners of small diameter: randomized solutions. Comput. Geom. 13(2), 91–107 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    Google Scholar 

  15. Arya, S., Smid, M.H.M.: Efficient construction of a bounded-degree spanner with low weight. Algorithmica 17(1), 33–54 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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

    Chapter  Google Scholar 

  17. Bose, P., Carmi, P., Couture, M., Maheshwari, A., Smid, M., Zeh, N.: Geometric spanners with small chromatic number. Comput. Geom. 42(2), 134–146 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bose, P., Carmi, P., Farshi, M., Maheshwari, A., Smid, M.: Computing the greedy spanner in near-quadratic time. Algorithmica 58(3), 711–729 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bose, P., Carmi, P., Chaitman, L., Collette, S., Katz, M.J., Langerman, S.: Stable roommates spanner. Comput. Geom. 46(2), 120–130 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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

    Chapter  Google Scholar 

  21. Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S.: On plane constrained bounded-degree spanners. Algorithmica 81, 1392 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bose, P., Gudmundsson, J., Smid, M.: Constructing plane spanners of bounded degree and low weight. Algorithmica 42(3), 249–264 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bose, P., Smid, M., Xu, D.: Delaunay and diamond triangulations contain spanners of bounded degree. Int. J. Comput. Geom. Appl. 19(02), 119–140 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Carmi, P., Chaitman, L.: Bounded degree planar geometric spanners. CoRR, arXiv:1003.4963 (2010)

  25. Carmi, P., Chaitman, L.: Stable roommates and geometric spanners. In: Proceedings of the 22nd Annual Canadian Conference on Computational Geometry, pp. 31–34 (2010)

    Google Scholar 

  26. Carmi, P., Smid, M.H.M.: An optimal algorithm for computing angle-constrained spanners. J. Comput. Geom. 3(1), 196–221 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Chew, L.P.: There are planar graphs almost as good as the complete graph. J. Comput. Syst. Sci. 39(2), 205–219 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  28. Czumaj, A., Zhao, H.: Fault-tolerant geometric spanners. Discrete Comput. Geom. 32(2), 207–230 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Das, G., Joseph, D.: Which triangulations approximate the complete graph? In: Djidjev, H. (ed.) Optimal Algorithms. LNCS, vol. 401, pp. 168–192. Springer, Heidelberg (1989). https://doi.org/10.1007/3-540-51859-2_15

    Chapter  Google Scholar 

  30. Das, G., Narasimhan, G.: A fast algorithm for constructing sparse Euclidean spanners. Int. J. Comput. Geom. Appl. 7(4), 297–315 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Eppstein, D.: Spanning trees and spanners. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 425–461. Elsevier, Amsterdam (1999)

    Google Scholar 

  32. Gudmundsson, J., Knauer, C.: Dilation and detour in geometric networks. In: Gonzalez, T. (ed.) Handbook of Approximation Algorithms and Metaheuristics. Chapman & Hall, Boca Raton (2007)

    Google Scholar 

  33. Gudmundsson, J., Levcopoulos, C., Narasimhan, G.: Fast greedy algorithms for constructing sparse geometric spanners. SIAM J. Comput. 31(5), 1479–1500 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  34. Har-Peled, S., Mendel, M.: Fast construction of nets in low-dimensional metrics and their applications. SIAM J. Comput. 35(5), 1148–1184 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kapoor, S., Li, X.-Y.: Efficient construction of spanners in d-dimensions. CoRR, arXiv:1303.7217 (2013)

  36. Keil, J.M., Gutwin, C.A.: The Delaunay triangulation closely approximates the complete Euclidean graph. In: Dehne, F., Sack, J.-R., Santoro, N. (eds.) WADS 1989. LNCS, vol. 382, pp. 47–56. Springer, Heidelberg (1989). https://doi.org/10.1007/3-540-51542-9_6

    Chapter  Google Scholar 

  37. Levcopoulos, C., Narasimhan, G., Smid, M.H.M.: Improved algorithms for constructing fault-tolerant spanners. Algorithmica 32(1), 144–156 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lukovszki, T.: New results of fault tolerant geometric spanners. In: Proceedings of Workshop on Algorithms and Data Structures, pp. 193–204 (1999)

    Google Scholar 

  39. Narasimhan, G., Smid, M.H.M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  40. Peleg, D., Schäffer, A.: Graph spanners. J. Graph Theory 13(1), 99–116 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  41. Segal, M., Shpungin, H.: Improved multi-criteria spanners for ad-hoc networks under energy and distance metrics. In: IEEE INFOCOM, pp. 6–10 (2010)

    Google Scholar 

  42. Smid, M.: Geometric spanners with few edges and degree five. In: Gudmundsson, J., Jay, C.B. (eds.) CATS 2006, pp. 7–9. Australian Computer Society (2006)

    Google Scholar 

  43. 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)

    Google Scholar 

  44. Talwar, K.: Bypassing the embedding: algorithms for low dimensional metrics. In: Proceedings of ACM Symposium on Theory of Computing, pp. 281–290 (2004)

    Google Scholar 

  45. Wang, Y., Li, X.-Y.: Minimum power assignment in wireless ad hoc networks with spanner property. J. Comb. Optim. 11(1), 99–112 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, IIT Guwahati, Guwahati, India

    Sukanya Bhattacharjee & R. Inkulu

Authors
  1. Sukanya Bhattacharjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Inkulu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Inkulu .

Editor information

Editors and Affiliations

  1. The University of Texas at Dallas, Richardson, TX, USA

    Ding-Zhu Du

  2. Xidian University, Xi'an, China

    Zhenhua Duan

  3. Xidian University, Xi'an, China

    Cong Tian

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Bhattacharjee, S., Inkulu, R. (2019). Geodesic Fault-Tolerant Additive Weighted Spanners. In: Du, DZ., Duan, Z., Tian, C. (eds) Computing and Combinatorics. COCOON 2019. Lecture Notes in Computer Science(), vol 11653. Springer, Cham. https://doi.org/10.1007/978-3-030-26176-4_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-26176-4_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26175-7

  • Online ISBN: 978-3-030-26176-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation