Advertisement

On Symbolic Scheduling Independent Tasks with Restricted Execution Times

  • Daniel Sawitzki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3503)

Abstract

Ordered Binary Decision Diagrams (OBDDs) are a data structure for Boolean functions which supports many useful operations. It finds applications in CAD, model checking, and symbolic graph algorithms. We present an application of OBDDs to the problem of scheduling N independent tasks with k different execution times on m identical parallel machines while minimizing the over-all finishing time. In fact, we consider the decision problem if there is a schedule with makespan D. Leung’s dynamic programming algorithm solves this problem in time \({\mathcal O}({\rm log} m \cdot N^{2(k-1)})\). In this paper, a symbolic version of Leung’s algorithm is presented which uses OBDDs to represent the dynamic programming table T. This heuristical approach solves the scheduling problem by executing \({\mathcal O}(k {\rm log} m {\rm log}(mD))\) operations on OBDDs and is expected to use less time and space than Leung’s algorithm if T is large but well-structured. The only known upper bound of \({\mathcal O}((m \cdot D)^{3k+2})\) on its resource usage is trivial. Therefore, we report on experimental studies in which the symbolic method was applied to random scheduling problem instances.

Keywords

Execution Time Schedule Problem Model Check Schedule Algorithm Boolean Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Albers, S., Schröder, B.: An experimental study of online scheduling algorithms. Journal of Experimental Algorithms 7, 3 (2002)CrossRefGoogle Scholar
  2. 2.
    Bloem, R., Gabow, H.N., Somenzi, F.: An algorithm for strongly connected component analysis in nlogn symbolic steps. In: Johnson, S.D., Hunt Jr., W.A. (eds.) FMCAD 2000. LNCS, vol. 1954, pp. 37–54. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Bryant, R.E.: Symbolic manipulation of Boolean functions using a graphical representation. In: Design Automation Conference, pp. 688–694. ACM Press, New York (1985)Google Scholar
  4. 4.
    Bryant, R.E.: Graph-based algorithms for Boolean function manipulation. IEEE Transactions on Computers 35, 677–691 (1986)zbMATHCrossRefGoogle Scholar
  5. 5.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  6. 6.
    Gentilini, R., Piazza, C., Policriti, A.: Computing strongly connected components in a linear number of symbolic steps. In: Symposium on Discrete Algorithms, pp. 573–582. ACM Press, New York (2003)Google Scholar
  7. 7.
    Gentilini, R., Policriti, A.: Biconnectivity on symbolically represented graphs: A linear solution. In: Ibaraki, T., Katoh, N., Ono, H. (eds.) ISAAC 2003. LNCS, vol. 2906, pp. 554–564. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Hachtel, G.D., Somenzi, F.: Logic Synthesis and Verification Algorithms. Kluwer Academic Publishers, Boston (1996)zbMATHGoogle Scholar
  9. 9.
    Hachtel, G.D., Somenzi, F.: A symbolic algorithm for maximum flow in 0–1 networks. Formal Methods in System Design 10, 207–219 (1997)CrossRefGoogle Scholar
  10. 10.
    Hojati, R., Touati, H., Kurshan, R.P., Brayton, R.K.: Efficient ω-regular language containment. In: Probst, D.K., von Bochmann, G. (eds.) CAV 1992. LNCS, vol. 663, pp. 396–409. Springer, Heidelberg (1993)Google Scholar
  11. 11.
    Ishiura, N., Sawada, H., Yajima, S.: Minimization of binary decision diagrams based on exchanges of variables. In: International Conference on Computer Aided Design, pp. 472–475. IEEE Press, Los Alamitos (1991)Google Scholar
  12. 12.
    Jin, H., Kuehlmann, A., Somenzi, F.: Fine-grain conjunction scheduling for symbolic reachability analysis. In: Katoen, J.-P., Stevens, P. (eds.) TACAS 2002. LNCS, vol. 2280, pp. 312–326. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Leung, J.Y.-T.: On scheduling independent tasks with restricted execution times. Operations Research 30(1), 163–171 (1982)zbMATHCrossRefGoogle Scholar
  14. 14.
    McMillan, K.L.: Symbolic Model Checking. Kluwer Academic Publishers, Boston (1994)Google Scholar
  15. 15.
    Moon, I., Kukula, J.H., Ravi, K., Somenzi, F.: To split or to conjoin: The question in image computation. In: Design Automation Conference, pp. 23–28. ACM Press, New York (2000)Google Scholar
  16. 16.
    Ravi, K., Bloem, R., Somenzi, F.: A comparative study of symbolic algorithms for the computation of fair cycles. In: Johnson, S.D., Hunt Jr., W.A. (eds.) FMCAD 2000. LNCS, vol. 1954, pp. 143–160. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  17. 17.
    Sawitzki, D.: Experimental studies of symbolic shortest-path algorithms. In: Ribeiro, C.C., Martins, S.L. (eds.) WEA 2004. LNCS, vol. 3059, pp. 482–497. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Sawitzki, D.: Implicit flow maximization by iterative squaring. In: Van Emde Boas, P., Pokorný, J., Bieliková, M., Štuller, J. (eds.) SOFSEM 2004. LNCS, vol. 2932, pp. 301–313. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  19. 19.
    Sawitzki, D.: On graphs with characteristic bounded-width functions. Technical report, Universität Dortmund (2004)Google Scholar
  20. 20.
    Sawitzki, D.: A symbolic approach to the all-pairs shortest-paths problem. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 154–167. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  21. 21.
    Sawitzki, D.: Lower bounds on the OBDD size of graphs of some popular functions. In: Vojtáš, P., Bieliková, M., Charron-Bost, B., Sýkora, O. (eds.) SOFSEM 2005. LNCS, vol. 3381, pp. 298–309. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  22. 22.
    Wegener, I.: Branching Programs and Binary Decision Diagrams. SIAM, Philadelphia (2000)zbMATHCrossRefGoogle Scholar
  23. 23.
    Woelfel, P.: Symbolic topological sorting with OBDDs. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 671–680. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  24. 24.
    Xie, A., Beerel, P.A.: Implicit enumeration of strongly connected components. In: International Conference on Computer Aided Design, pp. 37–40. ACM Press, New York (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Daniel Sawitzki
    • 1
  1. 1.Computer Science 2University of DortmundDortmundGermany

Personalised recommendations