Skip to main content
SpringerLink
Log in

Permutation Routing and (ℓ, k)-Routing on Plane Grids

  • Chapter
  • First Online:
Book cover Graphs and Algorithms in Communication Networks

Part of the book series: Texts in Theoretical Computer Science. An EATCS Series ((TTCS))

  • 1420 Accesses

Abstract

The packet routing problem plays an essential role in communication networks. It consists in transferring data from some origins to some destinations within a reasonable amount of time. In the (ℓ, k)-routing problem, each node can send at most ℓ packets and receive at most k packets. Permutation routing is the particular case ℓ = k = 1. In the r-central routing problem, all nodes at distance at most r from a fixed node v want to send a packet to v. Here, we survey the results on permutation routing, the r-central routing, and the general (ℓ, k)-routing problems on regular plane grids, that is, square grids, triangular grids, and hexagonal grids. We assume the store-and-forward Δ-port model with synchronous transmission, and we consider both full and half-duplex networks.

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amini, O., Huc, F., Sau, I., Žerovnik, J.: (ℓ, k)-Routing on Plane Grids. Rapport de Recherche 6480 INRIA (2008)

    Google Scholar 

  2. Aspnes, J., Azar, Y., Fiat, A., Plotkin, S., Waarts, O.: On-line routing of virtual circuits with applications to load balancing and machine scheduling. Journal of the ACM (JACM) 44(3), 486–504 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Aspnes, J., Busch, C., Dolev, S., Fatourou, P., Georgiou, C., Shvartsman, A.: Eight Open Problems in Distributed Computing. Bulletin of the EATCS 90, 109–126 (2006)

    MATH  MathSciNet  Google Scholar 

  4. Awerbuch, B., Azar, Y.: Local optimization of global objectives: competitive distributed deadlock resolution and resource allocation. In: 35th Annual Symposium on Foundations of Computer Science (FOCS), pp. 240–249 (1994)

    Google Scholar 

  5. Azar, Y., Cohen, E., Fiat, A., Kaplan, H., Räcke, H.: Optimal oblivious routing in polynomial time. Journal of Computer and System Sciences 69(3), 383–394 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Borodin, A., Hopcroft, J.: Routing, merging and sorting on parallel models of computation. Proceedings of the fourteenth annual ACM symposium on Theory of computing pp. 338–344 (1982)

    Google Scholar 

  7. Busch, C., Magdon-Ismail, M., Xi, J.: Oblivious routing on geometric networks. Proceedings of the 17th annual ACM symposium on Parallelism in algorithms and architectures pp. 316–324 (2005)

    Google Scholar 

  8. Busch, C., Magdon-Ismail, M., Xi, J.: Optimal oblivious path selection on the mesh. Proceedings of the International Parallel and Distributed Processing Symposium (IPDPS 2005), Denver, Colorado, USA, April (2005)

    Google Scholar 

  9. Dobravec, T., Robič, B., Žerovnik, J.: Permutation routing in double-loop networks: design and empirical evaluation. Journal of Systems Architecture 48, 387–402 (2003)

    Article  Google Scholar 

  10. Dobravec, T., Žerovnik, J., Robič, B.: An optimal message routing algorithm for circulant networks. Journal of Systems Architecture 52, 298–306 (2006)

    Article  Google Scholar 

  11. Fraigniaud, P., Lazard, E.: Methods and problems of communication in usual networks. Discrete Applied Mathematics 53, 79–134 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  12. Havill, J. T.: Online Packet Routing on Linear Arrays and Rings. Lecture Notes in Computer Science 2076, 773–784 (2001)

    Article  MathSciNet  Google Scholar 

  13. Hwang, F., Lin, T., Jan, R.: A Permutation Routing Algorithm for Double Loop Network. Parallel Processing Letters 7(3), 259–265 (1997)

    Article  Google Scholar 

  14. Hwang, F., Yao, Y., Dasgupta, B.: Some permutation routing algorithms for low-dimensional hypercubes. Theoretical Computer Science 270, 111–124 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Iwama, K., Kambayashi, Y., Miyano, E.: New Bounds for Oblivious Mesh Routing. Journal of Graph Algorithms and Applications 5(5), 17–38 (2001)

    MATH  MathSciNet  Google Scholar 

  16. Iwama, K., Miyano, E.: Oblivious Routing Algorithms on the Mesh of Buses. Journal of Parallel and Distributed Computing 60, 137–149 (2000)

    Article  MATH  Google Scholar 

  17. Iwama, K., Miyano, E.: An O(\(\sqrt{N}\)) Oblivious Routing Algorithm for Two-Dimensional Meshes of Constant Queue-Size. Journal of Algorithms 41, 262–279 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Jan, G. E., Lin, M. B.: Concentration, load balancing, partial permutation routing, and superconcentration on cube-connected cycles parallel computers. Journal of Parallel and Distributed Compututing 65, 1471–1482 (2005)

    Article  MATH  Google Scholar 

  19. Kaklamanis, C., Krizanc, D., Tsantilas, T.: Tight bounds for oblivious routing in the hypercube. Theory of Computing Systems 24(1), 223–232 (1991)

    MATH  MathSciNet  Google Scholar 

  20. Klavžar, S., Vesel, A., Žigert, P.: On resonance graphs of catacondensed hexagonal graphs: structure, coding, and Hamiltonian path algorithm. MATCH Communications in Mathematical and in Computer Chemistry 49(49), 99–116 (2003)

    MATH  Google Scholar 

  21. Kranakis, E., Sing, H., Urrutia, J.: Compas Routing in Geometric Graphs. In: 11th Canadian Conference of Computational Geometry, pp. 51–54 (1999)

    Google Scholar 

  22. Kunde, M., Niedermeier, R., Rossmanith, P.: Faster sorting and routing on grids with diagonals. In: 11th Symposium of Theoretical Computer Science, 775, pp. 225–236. Lecture Notes on Computer Science (1994)

    Google Scholar 

  23. Kunde, M., Tensi, T.: (k–k) Routing on Multidimensional Mesh-Connected Arrays. Journal of Parallel and Distributed Computing 11, 146–155 (1991)

    Article  Google Scholar 

  24. Leighton, F., Maggs, B. M., Richa, A. W.: Fast Algorithms for Finding O(Congestion + Dilation) Packet Routing Schedules. In: 28th Annual Hawaii International Conference on System Sciences, pp. 555–563 (1995)

    Google Scholar 

  25. Leighton, F. T., Maggs, B. M., Rao, S. B.: Packet Routing and Job-Shop Scheduling in O(congestion + dilation) Steps. Combinatorica 14(2), 167–186 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  26. Leighton, T.: Introduction to Parallel Algorithms and Architectures: Arrays-Trees-Hypercubes. Morgan-Kaufman, San Mateo, California (1992)

    Google Scholar 

  27. Leighton, T., Maggs, B., Rao, S.: Universal packet routing algorithms. In: 29th Annual Symposium on Foundations of Computer Science (FOCS), pp. 256–269 (1988)

    Google Scholar 

  28. Leighton, T., Makedon, F., Tollis, I. G.: A 2n – 2 Step Algorithm for Routing in an n × n Array with Constant-Size Queues. Algorithmica 14, 291–304 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  29. Litman, A., Moran-Schein, S.: Fast, minimal, and oblivious routing algorithms on the mesh with bounded queues. Journal of Interconnection Networks 2, 445–469 (2001)

    Article  Google Scholar 

  30. Maggs, B.: A Survey of Congestion + Dilation Results for Packet Scheduling. 40th Annual Conference on Information Sciences and Systems 22(24), 1505–1510 (2006)

    Article  Google Scholar 

  31. Makedon, F., Symvonis, A.: Optimal algorithms for the many-to-one routing problem on 2-dimensional meshes. Microprocessors and Microsystems 17, 361–367 (1993)

    Article  Google Scholar 

  32. Nocetti, F. G., Stojmenović, I., Zhang, J.: Addressing and Routing in Hexagonal Networks with Applications for Tracking Mobile Users and Connection Rerouting in Cellular Networks. IEEE Transactions on Parallel and Distributed Systems 13(9), 963–971 (2002)

    Article  Google Scholar 

  33. Osterloh, A.: Optimal oblivious routing on d-dimensional Meshes. Theoretical Computer Science 333, 331–346 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  34. Ostrovsky, R., Rabani, Y.: Universal O(congestion + dilation + log1+ε N) Local Control Packet Switching Algorithms. In: 29th Annual ACM Symposium on the Theory of Computing, pp. 644–653. New York (1997)

    Google Scholar 

  35. Pietracaprina, A., Pucci, G.: Optimal Many-to-One Routing on the Mesh with Constant Queues. Lecture Notes in Computer Science 2150, 645–649 (2001)

    Article  Google Scholar 

  36. Räcke, H.: Exploiting locality for data management in systems of limited bandwidth. In: 38th Annual Symposium on Foundations of Computer Science (FOCS), pp. 284–293 (1997)

    Google Scholar 

  37. Räcke, H.: Minimizing congestion in general networks. In: 43rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 43–52 (2002)

    Google Scholar 

  38. Rajasekaran, S., Overholt, R.: Constant queue routing on a mesh. Journal of Parallel and Distributed Computing 15, 160–166 (1992)

    Article  MATH  Google Scholar 

  39. Robič, B., Žerovnik, J.: Minimum 2-terminal routing in 2-jump circulant graphs. Computers and Artificial Intelligence 19(1), 37–46 (2000)

    MATH  MathSciNet  Google Scholar 

  40. Sau, I., Žerovnik, J.: An Optimal Permutation Routing Algorithm on Full-Duplex Hexagonal Networks. Discrete Mathematics and Theoretical Computer Science 10(3), 49–62 (2008)

    MATH  MathSciNet  Google Scholar 

  41. Scheideler, C.: Universal Routing Strategies for Interconnection Networks. Springer (1998)

    Google Scholar 

  42. Sibeyn, J.: Routing on Triangles, Tori and Honeycombs. International Journal of Foundations of Computer Science 8(3), 269–287 (1997)

    Article  MATH  Google Scholar 

  43. Sibeyn, J., Kaufman, M.: Deterministic 1-k routing on meshes (with applications to worm-hole routing). In: LNCS (ed.) 11th Symposium on Theoretical Aspects of Computer Science, vol. 775, pp. 237–248 (1994)

    Google Scholar 

  44. Sibeyn, J. F., Chlebus, B. S., Kaufmann, M.: Deterministic Permutation Routing on Meshes. Journal of Algorithms 22(1), 111–141 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  45. Srinivasan, A., Teo, C. P.: A constant-factor approximation algorithm for packet routing, and balancing local vs. global criteria. In: STOC ’97: Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pp. 636–643. ACM, New York, NY, USA (1997). DOI http://doi.acm.org/10.1145/258533.258658

  46. Stojmenović, I.: Honeycomb Networks: Topological Properties and Communication Algorithms. IEEE Transactions on Parallel and Distributed Systems 8(10), 1036–1042 (1997)

    Article  Google Scholar 

  47. Suel, T.: Routing and Sorting on Meshes with Row and Column Buses. In: Parallel Processing Symposium, vol. Eighth International Volume, pp. 411–417 (1994)

    Google Scholar 

  48. Tošić, R., Masulović, D., Stojmenović, I., Brunvoll, J., Cyvin, B., Cyvin, S.: Enumeration of Polyhex Hydrocarbons up to h=17. Journal of Chemical Information and Computer Sciences 35, 181–187 (1995)

    Google Scholar 

  49. Trobec, R.: Two-dimensional regular d-meshes. Parallel Computing 26, 1945–1953 (2000)

    Article  MATH  Google Scholar 

  50. Valiant, L., Brebner, G.: Universal schemes for parallel communication. Proceedings of the thirteenth annual ACM symposium on Theory of computing pp. 263–277 (1981)

    Google Scholar 

Download references

Acknowledgements

We want to thank Omid Amini, Frédéric Giroire, Florian Huc, and Rastislav Královič for insightful remarks and discussions.

Author information

Authors and Affiliations

  1. MASCOTTE, INRIA, I3S, CNRS UMR6070, University of Nice-Sophia Antipolis, Nice, France

    Ignasi Sau

  2. Graph Theory and Combinatorics Group, Department of Applied Mathematics IV, Universitat Politécnica de Catalunya, Barcelona, Spain

    Ignasi Sau

  3. University of Maribor, Smetanova 17, 2000, Maribor, Slovenia

    Janez Žerovnik

  4. Institute of Mathematics, Physics and Mechanics, Jadranska 19, Ljubljana, Slovenia

    Janez Žerovnik

Authors
  1. Ignasi Sau
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Janez Žerovnik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Ignasi Sau or Janez Žerovnik .

Editor information

Editors and Affiliations

  1. LS II für Mathematik, RWTH Aachen, Templergraben 55, Aachen, 52056, Germany

    Arie Koster

  2. Depto. Matematica Aplicada IV, Universitat Politècnica de Catalunya, C. Jordi Girona 1-3, Barcelona, 08034, Spain

    Xavier Muñoz

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Sau, I., Žerovnik, J. (2009). Permutation Routing and (ℓ, k)-Routing on Plane Grids. In: Koster, A., Muñoz, X. (eds) Graphs and Algorithms in Communication Networks. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02250-0_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-02250-0_10

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02249-4

  • Online ISBN: 978-3-642-02250-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation