Advertisement

Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

  • Martin Nöllenburg
  • Roman Prutkin
  • Ignaz Rutter
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9472)

Abstract

A greedily routable region (GRR) is a closed subset of \(\mathbb R^2\), in which each destination point can be reached from each starting point by choosing the direction with maximum reduction of the distance to the destination in each point of the path. Recently, Tan and Kermarrec proposed a geographic routing protocol for dense wireless sensor networks based on decomposing the network area into a small number of interior-disjoint GRRs. They showed that minimum decomposition is NP-hard for polygons with holes.

We consider minimum GRR decomposition for plane straight-line drawings of graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing style which has become a popular research topic in graph drawing. We show that minimum decomposition is still NP-hard for graphs with cycles, but can be solved optimally for trees in polynomial time. Additionally, we give a 2-approximation for simple polygons, if a given triangulation has to be respected.

Notes

Acknowledgements

The second author thanks Jie Gao for pointing him to the topic of GRR decompositions.

References

  1. 1.
    Alamdari, S., Chan, T.M., Grant, E., Lubiw, A., Pathak, V.: Self-approaching graphs. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 260–271. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  2. 2.
    Bose, P., Morin, P., Stojmenović, I., Urrutia, J.: Routing with guaranteed delivery in ad hoc wireless networks. Wireless Netw. 7(6), 609–616 (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    Calinescu, G., Fernandes, C.G., Reed, B.: Multicuts in unweighted graphs and digraphs with bounded degree and bounded tree-width. J. Algorithms 48(2), 333–359 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chazelle, B., Dobkin, D.: Optimal convex decompositions. In: Computational Geometry, pp. 63–133 (1985)Google Scholar
  5. 5.
    Chen, D., Varshney, P.K.: A survey of void handling techniques for geographic routing in wireless networks. Commun. Surv. Tutor. 9(1), 50–67 (2007)CrossRefGoogle Scholar
  6. 6.
    Costa, M., Létocart, L., Roupin, F.: A greedy algorithm for multicut and integral multiflow in rooted trees. Oper. Res. Lett. 31(1), 21–27 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Costa, M.C., Létocart, L., Roupin, F.: Minimal multicut and maximal integer multiflow: A survey. Eur. J. Oper. Res. 162(1), 55–69 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dehkordi, H.R., Frati, F., Gudmundsson, J.: Increasing-chord graphs on point sets. J. Graph Algorithms Appl. (2015, to appear). doi: 10.7155/jgaa.00348
  9. 9.
    Fang, Q., Gao, J., Guibas, L., de Silva, V., Zhang, L.: Glider: gradient landmark-based distributed routing for sensor networks. In: INFOCOM 2005, pp. 339–350. IEEE (2005)Google Scholar
  10. 10.
    Garg, N., Vazirani, V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18(1), 3–20 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Icking, C., Klein, R., Langetepe, E.: Self-approaching curves. Math. Proc. Camb. Phil. Soc. 125, 441–453 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Keil, J.M.: Decomposing a polygon into simpler components. SIAM J. Comput. 14(4), 799–817 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Knuth, D.E., Raghunathan, A.: The problem of compatible representatives. SIAM J. Discrete Math. 5(3), 422–427 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mauve, M., Widmer, J., Hartenstein, H.: A survey on position-based routing in mobile ad hoc networks. IEEE Netw. 15(6), 30–39 (2001)CrossRefGoogle Scholar
  16. 16.
    Nöllenburg, M., Prutkin, R., Rutter, I.: Partitioning graph drawings and triangulated simple polygons into greedily routable regions (2015). CoRR arXiv:1509.05635
  17. 17.
    Nöllenburg, M., Prutkin, R., Rutter, I.: On self-approaching and increasing-chord drawings of 3-connected planar graphs. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 476–487. Springer, Heidelberg (2014) Google Scholar
  18. 18.
    Tan, G., Bertier, M., Kermarrec, A.M.: Convex partition of sensor networks and its use in virtual coordinate geographic routing. In: INFOCOM 2009, pp. 1746–1754. IEEE (2009)Google Scholar
  19. 19.
    Tan, G., Kermarrec, A.M.: Greedy geographic routing in large-scale sensor networks: a minimum network decomposition approach. IEEE/ACM Trans. Netw. 20(3), 864–877 (2012)CrossRefGoogle Scholar
  20. 20.
    Zhu, X., Sarkar, R., Gao, J.: Shape segmentation and applications in sensor networks. In: INFOCOM 2007, pp. 1838–1846. IEEE (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Martin Nöllenburg
    • 1
  • Roman Prutkin
    • 2
  • Ignaz Rutter
    • 2
  1. 1.Algorithms and Complexity GroupTU WienViennaAustria
  2. 2.Institute of Theoretical InformaticsKarlsruhe Institute of TechnologyKarlsruheGermany

Personalised recommendations