Skip to main content
SpringerLink
Log in

Hypercube embedding heuristics: An evaluation

  • Published:
International Journal of Parallel Programming Aims and scope Submit manuscript

Abstract

The hypercube embedding problem, a restricted version of the general mapping problem, is the problem of mapping a set of communicating processes to a hypercube multiprocessor. The goal is to find a mapping that minimizes the length of the paths between communicating processes. Unfortunately the hypercube embedding problem has been shown to be NP-hard. Thus many heuristics have been proposed for hypercube embedding. This paper evaluates several hypercube embedding heuristics, including simulated annealing, local search, greedy, and recursive mincut bipartitioning. In addition to known heuristics, we propose a new greedy heuristic, a new Kernighan-Lin style heuristic, and some new features to enhance local search. We then assess variations of these strategies (e.g., different neighborhood structures) and combinations of them (e.g., greedy as a front end of iterative improvement heuristics). The asymptotic running times of the heuristics are given, based on efficient implementations using a priority-queue data structure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. Berman and L. Snyder, On mapping parallel algorithms into parallel architectures,J. of Parallel and Distributed Computing,4:439–458 (1987).

    Article  Google Scholar 

  2. L. Bhuyan and D. P. Agrawal, Generalized hypercube and hyperbus structures for a computer network,IEEE Transactions on Computers,C-33:323–333 (1984).

    Google Scholar 

  3. W. Hillis,The Connection Machine, MIT Press (1985).

  4. Y. Saad and M. H. Schultz, Topological properties of hypercubes,IEEE Transactions on Computers, Vol. C-37 (1988).

  5. A. Wagner, Embedding arbitrary binary trees in a hypercube,J. of Parallel and Distributed Computing,7:503–520 (1989).

    Article  Google Scholar 

  6. INTEL Scientific Computers, Direct-Connecttm routing solves node communications challenge,iSCurrents, pp. 5–6 (1987).

  7. D. Lenoski, J. Laudon, K. Gharachorloo, A. Gupta, and J. Hennessy, The directory-based cache coherence protocol for the DASH multiprocessor, ACM Computer Architecture News,18:148–159 (1990).

    Article  Google Scholar 

  8. L. W. Tucker and G. G. Robertson, Architecture and applications of the Connection Machine,IEEE Computer,21:26–38 (1988).

    Google Scholar 

  9. J. Hong, K. Mehlhorn, and A. Rosenberg, Cost trade-offs in graph embeddings with applications,J. Assoc. Comput. Mach.,30:709–728 (1983).

    Google Scholar 

  10. G. Cybenko, D. Krumme, and K. Venkataraman, Fixed hypercube embedding,Information Processing Letters,25:35–39 (1987).

    Article  Google Scholar 

  11. A. Wagner and D. Corneil, Embedding trees in the hypercube isNP-complete,SIAM J. on Computing,19:570–590 (1990).

    Article  Google Scholar 

  12. F. Berman, Experience with an automatic solution to the mapping problem,The Characteristics of Parallel Algorithms, L. Jamieson, D. Gannon, and R. Douglas, (eds.), MIT Press (1987).

  13. S. Bokhari, On the mapping problem,IEEE Transactions on Computers,C-30:207–214 (1981).

    Google Scholar 

  14. K. Fukunaga, S. Yamada, and T. Kasai, Asignment of job modules onto array processors,IEEE Transactions on Computers,C-36:888–891 (1987).

    Google Scholar 

  15. A. Gabriellian and D. Tyler, Optimal object allocation in distributed computer systems,Proc. Int'l Conf. on Distributed Computer Systems, pp. 88–95 (1984).

  16. H. Mühlenbein, M. Gorges-Schleuter, and O. Krämer, New solutions to the mapping problem of parallel systems: The evolution approach,Parallel Computing,4:269–279 (1987).

    Article  Google Scholar 

  17. J. Ramanujam, F. Ercal, and P. Sadayappan, Task allocation by simulated annealing,Proc. Int'l Conf. on Supercomputing (1988).

  18. P. Sadayappan and F. Ercal, Nearest-neighbor mapping of finite element graphs onto processor meshes,IEEE Trans. on Computers,C-36:1408–1424 (1987).

    Google Scholar 

  19. K. Schwan and C. Gaimon, Automating resource allocation for theCm * multiprocessor,Proc. Int'l Conf. on Distributed Computer Systems, pp. 310–320 (1984).

  20. W.-K. Chen and E. Gehringer, A graph-oriented mapping strategy for a hypercube,Proc. Third Conf. on Hypercube Concurrent Computers and Applications, pp. 200–209 (1988).

  21. F. Ercal, J. Ramanujam, and P. Sadayappan, Task allocation onto a hypercube by recursive mincut bipartitioning,J. of Parallel and Distributed Computing, to appear (1990).

  22. F. Ercal and P. Sadayappan, One-to-one mapping process graphs onto a hypercube,Proc. Supercomputing '89, ACM, pp. 91–98 (1989).

  23. O. Krämer and H. Mühlenbein, Mapping strategies in message-based multiprocessor systems.Parallel Computing,9:213–225 (1989).

    Article  Google Scholar 

  24. S.-Y. Lee and J. Aggarwal, A mapping strategy for parallel processing,IEEE Transactions on Computers,C-36:433–442 (1987).

    Google Scholar 

  25. M. Garey and R. Graham, On cubical graphs,J. of Combinatorial Theory, Vol. 18 (1975).

  26. K. Bhat, On the complexity of testing a graph forN-cube,Information Processing Letters,11:16–19 (1980).

    Article  Google Scholar 

  27. S. Bettayeb, z. Miller, and I. Sudborough, Embedding grids into hypercubes,VLSI Algorithms and Architectures: 3rd Aegean Workshop on Computing, Lecture Notes inComputer Science, Springer Verlag,319:201–211 (1988).

  28. J. Brandenburg and D. Scott, Embedding of communication trees and grids into hypercubes, Technical Report 280182-001, INTEL Scientific Computers (1985).

  29. M. Chan, Dilation-2 embeddings of grids into hypercubes,Proc. Int'l Conf. on Parallel Processing, Vol.III, pp. 295–298 (1988).

    Google Scholar 

  30. A. Wu, Embedding of tree networks into hypercubes,J. of Parallel and Distributed Computing,2:238–249 (1985).

    Article  Google Scholar 

  31. S. Bhatt, F. Chung, F. Leighton, and A. Rosenberg,Efficient embeddings of trees in hypercubes. Typescript, Department of Computer Science, Yale University, New Haven, Connecticut 06520.

  32. B. Monien and I. Sudborough, Simulating binary trees on hypercubes, inVLSI Algorithms and Architectures: 3rd Aegean Workshop on Computing, Lecture Notes inComputer Science, Springer Verlag,319:170–180 (1988).

  33. F. Afrati, C. Papadimitriou, and G. Papageorgiou, The Complexity of cubical graphs,Information and Control,66:53–60 (1985).

    Google Scholar 

  34. B. Kernighan and S. Lin, An efficient heuristic procedure for partitioning graphs,Bell System Technical Journal, pp. 291–307 (1970).

  35. S. Bollinger and S. Midkiff, Processor and link assignment in multiprocessors using simulated annealing,Proc. Int'l Conf. on Parallel Processing, Vol.I, pp. 1–7 (1988).

    Google Scholar 

  36. F. André, J. Pazat, and T. Priol, Experiments with mapping algorithms on a hypercube,Proc. Fourth Conf. on Hypercubes, Concurrent Computers, and Applications, Vol. I, pp. 39–46 (1989).

    Google Scholar 

  37. J.-L. Pazat,Outils pour la Programmation d'un Multiprocesseur à Mémoires Distribuées, Ph.D. thesis, Université de Bordeaux I (February 1989).

  38. D. S. Johnson, C. R. Aragon, L. A. McGeogh, and C. Schevon,Optimization by simulated annealing: An experimental evaluation (Part I). Typescript.

  39. C. Fiduccia and R. Mattheyses, A linear-time heuristic for improving network partitions,Proc. 19th Design Automation Conf., pp. 175–181 (1982).

  40. W.-K. Chen, A graph-oriented mapping strategy for a hypercube, Master's thesis, North Carolina State University (1988).

  41. F. Heath, Origins of the binary code,Scientific American,227:76–83 (1972).

    Google Scholar 

  42. S. Kirkpatrick, C. Gelatt, Jr., and M. Vecchi,Optimization by simulated annealing, Science, pp. 671–680 (1983).

  43. W.-K. Chen and M. Stallmann, Local search variants for hypercube embedding,Proc. Fifth Distributed Memory Computing Conf., to appear (1990).

  44. A. Wagner,Embedding Trees in a Hypercube, Ph.D. thesis, University of Toronto (1987).

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, North Carolina State University, 27695-7911, Raleigh, North Carolina

    Woei-Kae Chen

  2. Department of Computer Science, North Carolina State University, 27695-8206, Raleigh, North Carolina

    Matthias F. M. Stallmann & Edward F. Gehringer

Authors
  1. Woei-Kae Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthias F. M. Stallmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Edward F. Gehringer
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research is partially supported by the Office of Naval Research under Contract N00014-88-K-0555, which is gratefully acknowledged.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, WK., Stallmann, M.F.M. & Gehringer, E.F. Hypercube embedding heuristics: An evaluation. Int J Parallel Prog 18, 505–549 (1989). https://doi.org/10.1007/BF01381720

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01381720

Key Words

Search

Navigation