ISAAC 2015: Algorithms and Computation pp 637-649

# Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

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.

### 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)
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)
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)
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)
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)
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)
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)
11. 11.
Icking, C., Klein, R., Langetepe, E.: Self-approaching curves. Math. Proc. Camb. Phil. Soc. 125, 441–453 (1999)
12. 12.
Keil, J.M.: Decomposing a polygon into simpler components. SIAM J. Comput. 14(4), 799–817 (1985)
13. 13.
Knuth, D.E., Raghunathan, A.: The problem of compatible representatives. SIAM J. Discrete Math. 5(3), 422–427 (1992)
14. 14.
Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982)
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)
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)
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