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Multi-dimensional Interval Routing Schemes

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Distributed Algorithms (WDAG 1995)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 972))

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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.

Work supported by the EEC ESPRIT II Basic Research Action Program under contract No.8141 “Algorithms and Complexity II”, by the EEC “Human Capital and Mobility” MAP project, and by the Italian MURST 40% project “Algoritmi, Modelli di Calcolo e Strutture Informative”.

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References

  1. 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. B. Awerbuch, A. Bar-Noy, N. Linial, and D. Peleg, Improved Routing Strategies with Succinct Tables. Journal of Algorithms, 11 (1990), pp. 307–341.

    Article  Google Scholar 

  3. E. M. Bakker, J. van Leeuwen and R. B. Tan, Some Characterization Results in Compact Routing Schemes, Manuscript (1994).

    Google Scholar 

  4. 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. 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. 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. 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. 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. 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. 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. G. N. Frederickson and R. Janardan, Efficient Message Routing in Planar Networks, SIAM Journal on Computing 18 (1989), pp. 843–857.

    Article  Google Scholar 

  12. G. N. Frederickson and R. Janardan, Space Efficient Message Routing in c-Decomposable Networks, SIAM Journal on Computing 19 (1990), pp. 164–181.

    Article  Google Scholar 

  13. 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. The T9000 Transputer Products Overview Manual, Inmos (1991).

    Google Scholar 

  15. 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. 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. 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. 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. D. May and P. Thompson, Transputers and Routers: Components for Concurrent Machines, Inmos (1990).

    Google Scholar 

  20. D. Peleg, E. Upfal. A trade-off between space and efficiency for routing tables. Journal of the ACM, 36 (3) (1989), pp. 510–530.

    Article  Google Scholar 

  21. 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. J. P. Robinson and M. Cohn. Counting Sequences, IEEE Transactions on Computers, C-30 (1) (1981), pp. 17–23.

    Google Scholar 

  23. 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 

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Authors and Affiliations

  1. Dipartimento di Informatica e Sistemistica, University of Rome “La Sapienza”, via Salaria 113, I-00198, Rome, Italy

    Michele Flammini & Umberto Nanni

  2. Dipartimento di Matematica Pura ed Applicata, University of L'Aquila, via Vetoio loc. Coppito, I-67010, L'Aquila, Italy

    Michele Flammini

  3. Dipartimento di Matematica, University of Rome “Tor Vergata”, via della Ricerca Scientifica, I-00133, Rome, Italy

    Giorgio Gambosi

  4. Department of Computer Science, Utrecht University, Padualaan 14, 3584, CH Utrecht, The Netherlands

    Richard B. Tan

  5. Department of Computer Science, University of Sciences & Arts of Oklahoma, 73119, Chickasha, OK, USA

    Richard B. Tan

Authors
  1. Michele Flammini
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  2. Giorgio Gambosi
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  3. Umberto Nanni
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  4. Richard B. Tan
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Jean-Michel Hélary Michel Raynal

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© 1995 Springer-Verlag Berlin Heidelberg

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Flammini, M., Gambosi, G., Nanni, U., Tan, R.B. (1995). Multi-dimensional Interval Routing Schemes. In: Hélary, JM., Raynal, M. (eds) Distributed Algorithms. WDAG 1995. Lecture Notes in Computer Science, vol 972. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0022143

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  • DOI: https://doi.org/10.1007/BFb0022143

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60274-3

  • Online ISBN: 978-3-540-44783-2

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