Multi-dimensional Interval Routing Schemes

  • Michele Flammini
  • Giorgio Gambosi
  • Umberto Nanni
  • Richard B. Tan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 972)

Abstract

Interval Routing Scheme (k-IRS) is a compact routing scheme on general networks. It has been studied extensively and recently been implemented on the latest generation INMOS Transputer Router chip. In this paper we introduce an extension of the Interval Routing Scheme k-IRS to the multi-dimensional case (k, d)-MIRS, where k is the number of intervals and d is the number of dimensions. Whereas k-IRS only represents compactly a single shortest path between any two nodes, with this new extension we are able to represent all shortest paths compactly. This is useful for fault-tolerance and traffic distribution in a network. We study efficient representations of all shortest paths between any pair of nodes for general network topologies and for specific interconnection networks such as rings, grids, tori and hypercubes. For these interconnection networks we show that for about the same space complexity as k-IRS we can represent all shortest paths in (k, d)-MIRS (as compared to only a single shortest path in k-IRS). Moreover, tradeoffs are derived between the dimension d and the number of intervals k in multi-dimensional interval routing schemes on hypercubes, grids and tori.

Keywords

Compact Routing Methods Interval Routing Schemes Interconnection Networks Shortest Paths Dimensions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [ABLP89]
    B. Awerbuch, A. Bar-Noy, N. Linial, and D. Peleg, Compact Distributed Data Structures for Adaptive Routing, Proc. 21 st ACM Symp. on Theory of Computing (1989), pp. 479–489.Google Scholar
  2. [ABLP90]
    B. Awerbuch, A. Bar-Noy, N. Linial, and D. Peleg, Improved Routing Strategies with Succinct Tables. Journal of Algorithms, 11 (1990), pp. 307–341.CrossRefGoogle Scholar
  3. [BLT94]
    E. M. Bakker, J. van Leeuwen and R. B. Tan, Some Characterization Results in Compact Routing Schemes, Manuscript (1994).Google Scholar
  4. [BLT91]
    E. M. Bakker, J. van Leeuwen and R. B. Tan, Linear Interval Routing Schemes, Tech. Rep. RUU-CS-91-7, Dept. of Computer Science, Utrecht University (1991). Also in: Algorithms Review 2 (2) (1991), pp. 45–61.Google Scholar
  5. [F95]
    M. Flammini, Compact Routing Models: Some Complexity Results and Extensions, Ph. D. Thesis, Dept. of System and Computer Science, University of Rome “La Sapienza”, 1995.Google Scholar
  6. [FGS93]
    M. Flammini, G. Gambosi and S. Salomone, Boolean Routing, Proc. 7 th International Workshop on Distributed Algorithms (WDAG'93), Springer-Verlag LNCS 725 (1993), pp. 219–233.Google Scholar
  7. [FGS94]
    M. Flammini, G. Gambosi and S. Salomone, On Devising Boolean Routing Schemes, extended abstract, in: M. Nagl (Ed.), Graph-Theoretic Concepts in Computer Science (WG'95), Proceedings 21st International Workshop, Springer-Verlag LNCS (1995).Google Scholar
  8. [FGS95]
    M. Flammini, G. Gambosi and S. Salomone, Interval Routing Schemes, Proc. 12 th Symp. on Theoretical Aspects of Computer Science (STACS'95), Springer-Verlag LNCS 900 (1995), pp. 279–290.Google Scholar
  9. [FLM95]
    M. Flammini, J. van Leeuwen and A. Marchetti Spaccamela, The Complexity of Interval Routing on Random Graphs, to appear in Proc. 20th Symposium on Mathematical Foundation of Computer Science (MFCS'95) (1995).Google Scholar
  10. [FJ86]
    G. N. Frederickson and R. Janardan, Optimal Message Routing Without Complete Routing Tables, Proc. 5th Annual ACM Symposium on Principles of Distributed Computing (1986), pp. 88–97. Also as: Designing Networks with Compact Routing Tables, Algorithmica 3 (1988), pp. 171–190.Google Scholar
  11. [FJ89]
    G. N. Frederickson and R. Janardan, Efficient Message Routing in Planar Networks, SIAM Journal on Computing 18 (1989), pp. 843–857.CrossRefGoogle Scholar
  12. [FJ90]
    G. N. Frederickson and R. Janardan, Space Efficient Message Routing in c-Decomposable Networks, SIAM Journal on Computing 19 (1990), pp. 164–181.CrossRefGoogle Scholar
  13. [HKR91]
    H. Hofestädt, A. Klein and E. Reyzl, Performance Benefits from Locally Adaptive Interval Routing in Dynamically Switched Interconnection Networks, Proc. 2nd European Distributed Memory Computing Conference (1991), pp. 193–202.Google Scholar
  14. [I91]
    The T9000 Transputer Products Overview Manual, Inmos (1991).Google Scholar
  15. [KKR93]
    E. Kranakis, D. Krizanc and S. S. Ravi, On Multi-Label Linear Interval Routing Schemes, in: J. van Leeuwen (Ed.), Graph-Theoretic Concepts in Computer Science (WG'93), Proceedings 19th International Workshop, Springer-Verlag LNCS 790 (1993), pp. 338–349.Google Scholar
  16. [LT83]
    J. van Leeuwen and R. B. Tan, Routing with Compact Routing Tables, Tech. Rep. RUU-CS-83-16, Dept. of Computer Science, Utrecht University (1983). Also as: Computer Networks with Compact Routing Tables, in: G. Rozenberg and A. Salomaa (Eds.) The Book of L, Springer-Verlag, Berlin (1986), pp. 298–307.Google Scholar
  17. [LT85]
    J. van Leeuwen and R. B. Tan, Interval Routing, Tech. Rep. RUU-CS-85-16, Dept. of Computer Science, Utrecht University (1985). Also in: Computer Journal 30 (1987), pp. 298–307.Google Scholar
  18. [LT94]
    J. van Leeuwen and R. B. Tan, Compact Routing Methods: A Survey, Proc. Colloquium on Structural Information and Communication Complexity (SICC'94), Carleton University Press (1994).Google Scholar
  19. [MT90]
    D. May and P. Thompson, Transputers and Routers: Components for Concurrent Machines, Inmos (1990).Google Scholar
  20. [PU89]
    D. Peleg, E. Upfal. A trade-off between space and efficiency for routing tables. Journal of the ACM, 36 (3) (1989), pp. 510–530.CrossRefGoogle Scholar
  21. [R88]
    P. Ružička, On Efficiency of Interval Routing Algorithms, in: M.P. Chytil, L. Janiga, V. Koubek (Eds.), Mathematical Foundations of Computer Science 1988, Springer-Verlag LNCS 324 (1988), pp. 492–500.Google Scholar
  22. [RC81]
    J. P. Robinson and M. Cohn. Counting Sequences, IEEE Transactions on Computers, C-30 (1) (1981), pp. 17–23.Google Scholar
  23. [SK82]
    N. Santoro and R. Khatib, Routing Without Routing Tables, Tech. Rep. SCS-TR-6, School of Computer Science, Carleton University (1982). Also as: Labelling and Implicit Routing in Networks, Computer Journal 28 (1) (1985), pp. 5–8.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Michele Flammini
    • 1
    • 2
  • Giorgio Gambosi
    • 3
  • Umberto Nanni
    • 1
  • Richard B. Tan
    • 4
    • 5
  1. 1.Dipartimento di Informatica e SistemisticaUniversity of Rome “La Sapienza”RomeItaly
  2. 2.Dipartimento di Matematica Pura ed ApplicataUniversity of L'AquilaL'AquilaItaly
  3. 3.Dipartimento di MatematicaUniversity of Rome “Tor Vergata”RomeItaly
  4. 4.Department of Computer ScienceUtrecht UniversityCH UtrechtThe Netherlands
  5. 5.Department of Computer ScienceUniversity of Sciences & Arts of OklahomaChickashaUSA

Personalised recommendations