# Bounding the locality of distributed routing algorithms

## Abstract

We examine bounds on the locality of routing. A local routing algorithm makes a sequence of distributed forwarding decisions, each of which is made using only local information. Specifically, in addition to knowing the node for which a message is destined, an intermediate node might also know (1) its local neighbourhood (the subgraph corresponding to all network nodes within \(k\) hops of itself, for some fixed \(k\)), (2) the node from which the message originated, and (3) the incoming port (which of its neighbours last forwarded the message). Our objective is to determine, as \(k\) varies, which of these parameters are necessary and/or sufficient to permit local routing on a network modelled by a connected undirected graph. In particular, we establish tight bounds on \(k\) for the feasibility of deterministic \(k\)-local routing for various combinations of these parameters, as well as corresponding bounds on dilation (the worst-case ratio of actual route length to shortest path length).

## Keywords

Distributed algorithms Local routing Dilation## Notes

### Acknowledgments

The authors wish to thank Therese Biedl who observed that our origin-oblivious and predecessor-oblivious \((n/2)\)-local routing algorithm (Algorithm 3) identifies a shortest path. Also, the authors thank the anonymous reviewers for their helpful suggestions. Some of these results appeared in preliminary form at the 28th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC 2009) [2]. This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

## References

- 1.Bose, P., Brodnik, A., Carlsson, S., Demaine, E.D., Fleischer, R., López-Ortiz, A., Morin, P., Munro, I.: Online routing in convex subdivisions. Int. J. Comput. Geom. Appl.
**12**(4), 283–295 (2002)MATHCrossRefGoogle Scholar - 2.Bose, P., Carmi, P., Durocher, S.: Bounding the locality of distributed routing algorithms. In: Proceedings of the ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC), vol. 28, pp. 250–259. ACM (2009)Google Scholar
- 3.Bose, P., Morin, P.: Competitive online routing in geometric graphs. Theor. Comput. Sci.
**324**, 273–288 (2004)MathSciNetMATHCrossRefGoogle Scholar - 4.Bose, P., Morin, P.: Online routing in triangulations. SIAM J. Comput.
**33**(4), 937–951 (2004)MathSciNetMATHCrossRefGoogle Scholar - 5.Bose, P., Morin, P., Stojmenović, I., Urrutia, J.: Routing with guaranteed delivery in ad hoc wireless networks. Wirel. Netw.
**7**(6), 609–616 (2001)MATHCrossRefGoogle Scholar - 6.Braverman, M.: On ad hoc routing with guaranteed delivery. In: Proceedings of the ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC), vol 27, p. 418. ACM (2008)Google Scholar
- 7.Chávez, E., Dobrev, S., Kranakis, E., Opatrny, J., Stacho, L., Urrutia, J.: Local construction of planar spanners with irregular transmission ranges. In: Proceedings of the Latin American Symposium on Theoretical Informatics (LATIN), volume 3887 of Lecture Notes in Computer Science, pp. 286–297. Springer, Berlin (2006)Google Scholar
- 8.Chávez, E., Dobrev, S., Kranakis, E., Opatrny, J., Stacho, L., Urrutia, J.: Route discovery with constant memory in oriented planar geometric networks. Networks
**48**(1), 7–15 (2006)Google Scholar - 9.Chen, D., Devroye, L., Dujmović, V., Morin, P.: Memoryless routing in convex subdivisions: Random walks are optimal. In: Proceedings of the European Workshop on Computational Geometry (EuroCG), pp. 109–112 (2010) Google Scholar
- 10.Durocher, S., Kirkpatrick, D., Narayanan, L.: On routing with guaranteed delivery in three-dimensional ad hoc wireless networks. Wirel. Netw.
**16**, 227–235 (2010)Google Scholar - 11.Durocher, S., Kranakis, E., Krizanc, D., Narayanan, L.: Balancing traffic load using one-turn rectilinear routing. J. Interconnect. Netw.
**10**(1–2), 93–120 (2009)Google Scholar - 12.Finn, G.G.: Routing and addressing problems in large metropolitan-scale internetworks. Technical Report ISI/RR-87-180. Information Sciences Institute (1987)Google Scholar
- 13.Flury, R., Wattenhofer, R.: Randomized 3D geographic routing. In: Proceedings of the IEEE Conference on Computer Communications (INFOCOM), pp. 834–842. IEEE (2008)Google Scholar
- 14.Fraigniaud, P., Gavoille, C.: Local memory requirement of universal routing schemes. In: Proceedings of the ACM SIGACT-SIGOPS Symposium on Parallel Algorithms and Architecture (SPAA), pp. 183–188. ACM (1996)Google Scholar
- 15.Fraigniaud, P., Gavoille, C.: Universal routing schemes. Distrib. Comput.
**10**, 65–78 (1997)CrossRefGoogle Scholar - 16.Fraigniaud, P., Gavoille, C.: Interval routing schemes. Algorithmica
**21**(2), 155–182 (1998)MathSciNetMATHCrossRefGoogle Scholar - 17.Fraigniaud, P., Gavoille, C.: A theoretical model for routing complexity. In: Proceedings of the International Colloquium on Structural Information and Communication Complexity (SIROCCO), pp. 98–113. Carleton Scientific (1998)Google Scholar
- 18.Fraser, M.: Local routing on tori. Adhoc Sens. Wirel. Netw.
**6**, 179–196 (2008)Google Scholar - 19.Fraser, M., Kranakis, E., Urrutia, J.: Memory requirements for local geometric routing and traversal in digraphs. In: Proceedings of the Canadian Conference on, Computational Geometry (CCCG). vol. 20 (2008)Google Scholar
- 20.Guan, X.: Face routing in wireless ad-hoc networks. PhD thesis, University of Toronto (2009)Google Scholar
- 21.Kranakis, E., Singh, H., Urrutia, J.: Compass routing on geometric networks. In: Proceedings of the Canadian Conference on Computational Geometry (CCCG), vol. 11, pp. 51–54 (1999)Google Scholar
- 22.Kuhn, F., Wattenhofer, R., Zollinger, A.: Ad-hoc networks beyond unit disk graphs. In: Joint Workshop on Foundations of Mobile Computing, pp. 69–78. ACM (2003)Google Scholar
- 23.Li, X.-Y., Wang, Y., Song, W.-Z.: Applications of \(k\)-local MST for topology control and broadcasting in wireless ad hoc networks. IEEE Trans. Parallel Distrib. Syst.
**15**(12), 1057–1069 (2004)CrossRefGoogle Scholar - 24.Peleg, D., Upfal, E.: A trade-off between space and efficiency for routing tables. J. ACM
**36**(3), 510–530 (1989)MathSciNetMATHCrossRefGoogle Scholar - 25.Santoro, N., Khatib, R.: Labelling and implicit routing in networks. Comput. J.
**29**(1), 5–8 (1985)MathSciNetCrossRefGoogle Scholar - 26.Stojmenović, I.: Position based routing in ad hoc networks. IEEE Commun. Mag.
**40**(7), 128–134 (2002)CrossRefGoogle Scholar