Skip to main content
SpringerLink
Log in

Mapping tree-structured combinatorial optimization problems onto parallel computers

  • Chapter
  • First Online:
Solving Combinatorial Optimization Problems in Parallel

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1054))

Abstract

The tree structured optimization problems encountered in operations research are difficult to parallelize, because the two goals ‘minimization of processor idle times’ and ‘minimization of communication overheads’ cannot both be dealt with efficiently at the same time. We have presented a number of methods to solve the dynamic embedding problem necessary to map the dynamic tree arising during computation onto a distributed computing system.

A number of methods were investigated in more detail. We presented three search schemes with different characteristics:

  • a best-first branch & bound search for the Vertex Cover Problem and TSP

  • a depth-first branch & bound search with search-frontier splitting for VLSI floor-plan optimization

  • an iterative depth-first search with dynamic tree splitting for the N×N puzzle

  • an iterative depth-first search with search-frontier splitting for the N×N puzzle

Considerable speedup for all problems even on a large scale computing system connecting 1024 processor could be achieved using the methods presented. The efficiency of the methods was presented by solving small problems. Proving good scalability for these small problems, one can argue that the methods will provide even better scalability features for practical applications using much longer computation times for most cases.

This work was partly supported by the EC Esprit Basic Research Action Nr. 7141 (ALCOM II), the EC Human Capital and Mobility Project: “Efficient Use of Parallel Computers: Architecture, Mapping and Communication (MAP)” and by the EU Human Capital and Mobility Project “Solving combinatorial optimization problems in parallel (SCOOP)”

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Arvindam, V. Kumar and V. Rao. Efficient parallel algorithms for searching problems: Applications in VLSI CAD. 3rd Symp. Frontiers Mass. Par. Comp., Maryland (1990), 166–169.

    Google Scholar 

  2. R. D. Blumofe, C. E. Leiseron. Scheduling Multithreaded Computations by Work Stealing. Foundations of Computer Science, 1994

    Google Scholar 

  3. N. Christofides and C. Whitlock. An algorithm for two-dimensional cutting problems. Operations Research 25, 1 (1977), 30–44.

    Google Scholar 

  4. D. Culler, R. Karp, D. Patterson, A. Sahay, K.E. Schauser, E. Santos, R. Subramonian and T. van Eicken. LogP: Towards a realistic model of parallel computation. ACM SIGPLAN Symp. Principles and Practice of Parallel Programming, San Diego, (May 1993).

    Google Scholar 

  5. E. Dijkstra, W.H.J. Feijen and A.J.M. van Gasteren. Derivation of a termination detection algorithm for distributed computation. Inf. Proc. Lett. 16 (1983), 217–219.

    Article  Google Scholar 

  6. O.I. El-Dessouki and W.H. Huen. Distributed enumeration on network computers. Procs. 1979 Intern. Conf. Par. Proc., 137–146.

    Google Scholar 

  7. D. Ferguson, Y. Yemini, C. Nikolaou. Microeconomic Algorithms for Load Balancing in Distributed Computer Systems. Proc. IEEE 8 th Int. Conf. on Distributed Computing Systems 1988, pp. 539–546.

    Google Scholar 

  8. R. Finkel and U. Manber. DIB — A distributed implementation of backtracking. 5th Conf. Distr. Comp. Systems, Denver, 1985, 446–452.

    Google Scholar 

  9. A. Grama, V. Kumar and P. Pardalos. Parallel Processing of Discrete Optimization Problems. Encyclopedia of Microcomputers, Vol. 13 (1993), pp. 129–147, Marcel Dekker Inc., New York.

    Google Scholar 

  10. A. Gupta and V. Kumar. Performance properties of large scale parallel systems. J. Parallel and Distributed Comp., 19(1993), 234–244.

    Article  Google Scholar 

  11. M. Held and R.M. Karp. The traveling salesman problem and minimum spanning trees. Operations Research 18 (1970), 1138–1162.

    Google Scholar 

  12. M. Held and R.M. Karp, The traveling salesman problem and minimum spanning trees: part II, Mathematical Programming 1 (1971) 6–25

    Article  Google Scholar 

  13. S. H. Hosseini, B. Litow, M. Malkawi, J. Mepherson, K. Vairavan. Analysis of a graph coloring based distributed load balancing algorithm. Journal of Parallel and Distributed Computing, vol 10, 1990, pp. 160–166

    Article  Google Scholar 

  14. G.A.P. Kindervater and J.K. Lenstra. Parallel computing in Combinatorial Optimization. Annals of Operations Research 14, 1988, 245–289.

    Article  Google Scholar 

  15. D.E. Knuth and R.W. Moore. An analysis of alpha-beta pruning. Artif. Intell. 6,4(1975), 293–326.

    Article  Google Scholar 

  16. R.E. Korf. Depth-first iterative-deepening: An optimal admissible tree search. Art. Intell. 27 (1985), 97–109.

    Article  Google Scholar 

  17. V. Kumar and V. Rao. Scalable parallel formulations of depth-first search. Kumar, Gopalakrishnan, Kanal, eds., Par. Alg. for Mach. Intell. and Vision, Springer 1990, 1–41.

    Google Scholar 

  18. V. Kumar, A. Grama, A. Gupta and G. Karypis. Introduction to Parallel Computing. Design and Analysis of Algorithms. Benjamin/Cummings Publ., Redwood City, CA (1994).

    Google Scholar 

  19. V. Kumar, D.S. Nau and L. Kanal. A general branch-and-bound formulation for AND/OR graph and game-tree search. In L. Kanal, V. Kumar (eds.), Search in Artificial Intelligence. Springer-Verlag, Berlin (1988), 91–130.

    Google Scholar 

  20. E.L. Lawler and D.E. Wood. Branch and Bound methods: A survey. Operations Research 14 (1966), 600–719.

    Google Scholar 

  21. R. Lüling and B. Monien. Load balancing for distributed branch & bound algorithms. Intern. Par. Processing Symp., IPPS 1992.

    Google Scholar 

  22. R. Lüling, B. Monien and F. Ramme. Load Balancing in Large Networks: A Comparative Study. Proc. of 3rd IEEE Symp. on Parallel and Distributed Processing, 1991

    Google Scholar 

  23. R. Lüling, B. Monien and S. Tschöcke. Load balancing for distributed branch & bound algorithms: Experiments and theory. DIMACS Workshop “Parallel Processing of Discrete Optimization Problems”, (April 1994).

    Google Scholar 

  24. A. Mahanti, S. Ghosh, D.S. Nau, A.K. Pal and L. Kanal. Performance of IDA* on trees and graphs. 10th Nat. Conf. on Art. Int., AAAI-92, San Jose, (1992), 539–544.

    Google Scholar 

  25. R.N. Morabito, M.N. Arenales and V.F. Arcaro. An and-or-graph approach for two dimensional cutting problems. Europ. J. Oper. Res. 58 (1992), 263–271.

    Article  Google Scholar 

  26. N.J. Nilsson. Principles of Artificial Intelligence. Tioga Publ., Palo Alto, CA, 1980.

    Google Scholar 

  27. C.H. Papadimitriou and K. Steiglitz. Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, 1982.

    Google Scholar 

  28. P. M. Pardalos, A. Phillips and J.B. Rosen. Topics in Parallel Computing in Mathematical Programming. Science Press, (1992).

    Google Scholar 

  29. P.M. Pardalos M.G.C. Resende and K.G. Ramakrishnan (Editors). Parallel Processing of Discrete Optimization Problems. DIMACS Series, American Mathematical Society, (1995).

    Google Scholar 

  30. J. Pearl. Heuristics. Intelligent Search Strategies for Computer Problem Solving. Addison-Wesley, Reading, MA, (1984).

    Google Scholar 

  31. C. Powley and R.E. Korf. Single-agent parallel window search. IEEE Trans. Pattern Anal. Mach. Int., PAMI-13,5 (1991), 466–477.

    Article  Google Scholar 

  32. A. Ranade. Optimal speedup for backtracking search on a butterfly network. Procs. 3rd ACM Symp. Parallel Alg. and Architect. (1991), 40–48.

    Google Scholar 

  33. V.N. Rao, V. Kumar and K. Ramesh. A parallel implementation of iterative-deepening A*. AAAI-87, 878–882.

    Google Scholar 

  34. V.N. Rao and V. Kumar. On the efficiency of parallel backtracking. IEEE Trans. Par. Distr. Systems 4,4(1993), 427–437.

    Article  Google Scholar 

  35. D. Ratner and M. Warmuth. Finding a shortest solution for the N×N extension of the 15-puzzle is intractable. AAAI-86, 168–172.

    Google Scholar 

  36. A. Reinefeld and T.A. Marsland. Enhanced iterative-deepening search. IEEE Trans. Pattern Analysis Mach. Intell., IEEE-PAMI, July 1994.

    Google Scholar 

  37. A. Reinefeld and V. Schnecke. AIDA* — Asynchronous Parallel IDA*. Procs. 10th Canadian Conf. on Art. Intell. AI'94, (May 1994), Banff, Canada, Morgan Kaufman, 295–302.

    Google Scholar 

  38. A. Reinefeld and V. Schnecke. Work-load balancing in highly parallel depth-first search. Procs. Scalable High Perf. Comp. Conf. SHPCC'94, Knoxville, Te, 773–780.

    Google Scholar 

  39. G. Reinelt. TSPLIB — A Traveling Salesman Problem Library. ORSA Journal on Computing, 3 1991, pp. 376–284

    Google Scholar 

  40. V. Saletore and L.V. Kale. Consistent linear speedup to a first solution in parallel state-space search. Procs. 1990 Nat. Conf. Artif. Intell. (1990), 227–233.

    Google Scholar 

  41. J.A. Stankovic, I.S. Sidhu. An Adaptive Bidding Algorithm for Processes, Clusters and Distributed Groups. Proc. IEEE 4 th Int. Conf. on Distributed Computing Systems 1984, pp. 49–59

    Google Scholar 

  42. G.C. Stockman. A minimax algorithm faster than alpha-beta? Artificial Intelligence 12,2(1979), 179–196

    Article  Google Scholar 

  43. L. Stockmeyer. Optimal orientations of cells in silicon floorplan designs. Inform. and Control 57 (1983), 97–101.

    Article  Google Scholar 

  44. S. Tschöke, R. Lüling, B. Monien. Solving the Traveling Salesman Problem with a Distributed Branch and Bound Algorithm on a 1024 Processor Network. Proc. of Int. Parallel Processing Symposium (IPPS), 1995.

    Google Scholar 

  45. T. Volgenant, R. Jonker, A branch and bound algorithm for the symmetric traveling salesman problem based on the 1-tree relaxation, European J. Operational Res. 9 (1982) 83–89

    Article  Google Scholar 

  46. T. Volgenant, R. Jonker, The symmetric traveling salesman problem and edge exchange in minimal 1-trees, European J. Operational Res. 12 (1983) 394–403

    Article  Google Scholar 

  47. T. Volgenant, R. Jonker, Nonoptimal Edges for the Symmetric Traveling Salesman Problem, Operations Research Vol. 32 No. 4 (1984) 65–74

    Google Scholar 

  48. S. Wimer, I. Koren and I. Cederbaum. Optimal aspect ratios of building blocks in VLSI. 25th ACM/IEEE Design Automation Conference, (1988), 66–72.

    Google Scholar 

  49. C. Z. Xu, F.C.M. Lau. Analysis of the generalized dimension exchange method for dynamic load balancing. Journal of Parallel and Distributed Computing, vol 16, 1992, pp. 385–393

    Article  MathSciNet  Google Scholar 

  50. C. Z. Xu, B. Monien, R. Lüling, F.C.M. Lau. An analytical comparison of nearest neighbor algorithms for load balancing in parallel computers. Proc of Int. Parallel Processing Symposium (IPPS), 1995

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Paderborn Center for Parallel Computing, Warburger Str. 100, D-33095, Paderborn, Germany

    Reinhard Lüling, Burkhard Monien, Alexander Reinefeld & Stefan Tschöke

Authors
  1. Reinhard Lüling
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Burkhard Monien
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexander Reinefeld
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Stefan Tschöke
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Afonso Ferreira Panos Pardalos

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Lüling, R., Monien, B., Reinefeld, A., Tschöke, S. (1996). Mapping tree-structured combinatorial optimization problems onto parallel computers. In: Ferreira, A., Pardalos, P. (eds) Solving Combinatorial Optimization Problems in Parallel. Lecture Notes in Computer Science, vol 1054. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0027120

Download citation

  • DOI: https://doi.org/10.1007/BFb0027120

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61043-4

  • Online ISBN: 978-3-540-49875-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation