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A Short Cut to Optimal Sequences

Abstract

We propose a method of developing efficient programs for finding the optimal sequence, such as the maximum valued one among those that are acceptable. We introduce a method of deriving efficient algorithms from naive enumerate-and-choose-style ones. Our method is based on shortcut fusion, which is a program transformation for eliminating intermediate data structures passed between functions, and a set of auxiliary transformations. As an implementation of our method, we introduce a library for finding optimal sequences. The library consists of proposed transformations, together with functions useful to describe desirable sequences, so that naive enumerate-and-choose-style programs will be automatically improved.

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References

  1. 1.

    Acar, U. A., Blelloch, G. E. and Harper, R., “Selective memoization,” in Conference Record of POPL 2003: The 30th SIGPLAN-SIGACT Symposium on Principles of Programming Languages, New Orleans, Louisisana, January 15-17, 2003, ACM Press, pp. 14–25, 2003.

  2. 2.

    Arnborg, S., Lagergren, J. and Seese, D., “Easy problems for tree-decomposable graphs,” J. Algorithms, 12, 2, pp. 308–340, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Bellman, R., Dynamic Programming, Princeton University Press, 1957.

  4. 4.

    Bentley, J., Programming Pearls, ACM, 1986.

  5. 5.

    Bird, R. S., “Maximum marking problems,” J. Funct. Program., 11, 4, pp. 411–424, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Bird, R. S. and de Moor, O., Algebra of Programming, Prentice Hall, 1997.

  7. 7.

    Borie, R. B., Parker, R. G. and Tovey, C. A., “Automatic generation of lineartime algorithms from predicate calculus descriptions of problems on recursively constructed graph families,” Algorithmica, 7, 5&6, pp. 555–581, 1992.

  8. 8.

    Chin, W.-N., Khoo, S.-C. and Jones, N., “Redundant call elimination via tupling,” Fundam. Inf., 69, 1–2, pp. 1–37, 2006.

    MATH  MathSciNet  Google Scholar 

  9. 9.

    Chitil, O., “Type inference builds a short cut to deforestation,” in Proc. of the 4th ACM SIGPLAN International Conference on Functional Programming, ICFP’99, Paris, France, September 27-29, 1999, ACM Press, pp. 249–260, 1999.

  10. 10.

    Cohen, N. H., “Eliminating redundant recursive calls,” ACM Trans. Program. Lang. Syst., 5, 3, pp. 265–299, 1983.

    Article  MATH  Google Scholar 

  11. 11.

    Cormen, T. H., Stein, C., Rivest, R. L., and Leiserson, C. E., Introduction to algorithms, Second edition, MIT Press, 2001.

  12. 12.

    de Moor, O., “Categories, Relations and Dynamic Programming,” Ph.D. thesis, Technical Monograph PRG-98, Oxford University Computing Laboratory, 1992.

  13. 13.

    de Moor, O., “A generic program for sequential decision processes,” in Programming Languages: Implementations, Logics and Programs, 7th International Symposium, PLILP’95, Utrecht, The Netherlands, September 20-22, 1995, Proceedings, LNCS, 982, Springer-Verlag, pp. 1–23, 1995.

  14. 14.

    Giegerich, R., Meyer, C. and Steffen, P., “A discipline of dynamic programming over sequence data,” Sci. Comput. Program., 51, 3, pp. 215–263, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Giegerich, R. and Steffen, P., “Implementing algebraic dynamic programming in the functional and the imperative programming paradigm,” in Mathematics of Program Construction, 6th International Conference, MPC 2002, Dagstuhl Castle, Germany, July 8-10, 2002, Proceedings, LNCS, 2386, Springer-Verlag, pp. 1–20, 2002.

  16. 16.

    Gill, A., “Cheap Deforestation for Non-strict Functional Languages,” Ph.D. thesis, Department of Computing Science, Glasgow University, 1996.

  17. 17.

    Gill, A., Launchbury, J. and Peyton Jones, S., “A short cut to deforestation,” in FPCA ’93 Conference on Functional Programming Languages and Computer Architecture. Copenhagen, Denmark, 9-11 June 1993, ACM Press, pp. 223–232, 1993.

  18. 18.

    Hu, Z., Iwasaki, H., Takeichi, M. and Takano, A., “Tupling calculation eliminates multiple data traversals,” in Proc. of the 2nd ACM SIGPLAN International Conference on Functional Programming, ICFP’97, Amsterdam, The Netherlands, June 9-11, 1997, ACM Press, pp. 164–175, 1997.

  19. 19.

    Kabanov, J. and Vene, V., “Recursion schemes for dynamic programming,” in Mathematics of Program Construction, 8th International Conference, MPC 2006, Kuressaare, Estonia, July 3-5, 2006, Proceedings, LNCS, 4014, Springer- Verlag, pp. 235–252, 2006.

  20. 20.

    Launchbury, J. and Sheard, T., “Warm fusion: Deriving build-catas from recursive definitions,” in Conference Record of FPCA ’95 SIGPLAN-SIGARCHWG2.8 Conference on Functional Programming Languages and Computer Architecture. La Jolla, CA, USA, 25-28 June 1995, ACM Press, pp. 314–323, 1995.

  21. 21.

    Liu, Y. A. and Stoller, S. D., “Dynamic programming via static incrementalization,” Higher-Order Symb. Comput., 16, 1–2, pp. 37–62, 2003.

    Article  MATH  Google Scholar 

  22. 22.

    Liu, Y. A., Stoller, S. D., Li, N., and Rothamel, T., “Optimizing aggregate array computations in loops,” ACM Trans. Program. Lang. Syst., 27, 1, pp. 91–125, 2005.

    Article  Google Scholar 

  23. 23.

    Morihata, A., “A short cut to optimal sequences,” in Proc. of the Seventh Asian Symposium on Programming Languages and Systems, APLAS 2009, LNCS, 5904, Springer-Verlag, pp. 63–78, 2009.

  24. 24.

    Morihata, A., “Calculational Approach to Automatic Algorithm Construction,” Ph.D. thesis, Department of Mathematical Informatics, University of Tokyo, 2009.

  25. 25.

    Morihata, A., Matsuzaki, K. and Takeichi, M., “Write it recursively: A generic framework for optimal path queries,” in Proc. of the 2008 ACM SIGPLAN International Conference on Functional Programming, ICFP 2008, Sept. 22-24, 2008, Victoria, BC, Canada, ACM Press, pp. 169–178, 2008.

  26. 26.

    Mu, S.-C., “Maximum segment sum is back: Deriving algorithms for two segment problems with bounded lengths,” in Proc. of the 2008 ACM SIGPLAN Symposium on Partial Evaluation and Semantics-based Program Manipulation, PEPM 2008, San Francisco, California, USA, January 7-8, 2008, ACM Press, pp. 31–39, 2008.

  27. 27.

    Peyton Jones, S. ed., Haskell 98 Language and Libraries: The Revised Report, Cambridge University Press, 2003.

  28. 28.

    Peyton Jones, S., Tolmach, A. and Hoare, T., “Playing by the rules: Rewriting as a practical optimisation technique in GHC,” in Proc. of 2001 ACM SIGPLAN Haskell Workshop (HW’2001), Firenze, Italy, 2nd September 2001, Technical Report UU-CS-2001-23, Institute of Information and Computing Sciences Utrecht University, pp. 203–233, 2001.

  29. 29.

    Puchinger, J. and Stuckey, P. J., “Automating branch-and-bound for dynamic programs,” in Proc. of the 2008 ACM SIGPLAN Symposium on Partial Evaluation and Semantics-based Program Manipulation, PEPM 2008, San Francisco, California, USA, January 7-8, 2008, ACM Press, pp. 81–89, 2008.

  30. 30.

    Sasano, I., Hu, Z., Takeichi, M. and Ogawa, M., “Make it practical: A generic linear-time algorithm for solving maximum-weightsum problems,” in Proc. of the 5th ACM SIGPLAN International Conference on Functional Programming, ICFP’00, ACM Press, pp. 137–149, 2000.

  31. 31.

    Sasano, I., Ogawa, M. and Hu, Z., “Maximum marking problems with accumulative weight functions,” in Theoretical Aspects of Computing - ICTAC 2005, Second International Colloquium, Hanoi, Vietnam, October 17-21, 2005, Proceedings, LNCS, 3722, Springer-Verlag, pp. 562–578, 2005.

  32. 32.

    Wadler, P., “Theorems for free!,” in FPCA ’89 Conference on Functional Programming Languages and Computer Architecture. Imperial College, London, England, 11-13 September 1989, ACM Press, pp. 347–359, 1989.

  33. 33.

    Yokoyama, T., Hu, Z. and Takeichi, M., “Calculation rules for warming-up in fusion transformation,” in the 2005 Symposium on Trends in Functional Programming, TFP 2005, Tallinn, Estonia, 23-24 September 2005, pp. 399–412, 2005.

  34. 34.

    Zantema, H., “Longest segment problems,” Sci. Comput. Program., 18, 1, pp. 39–66, 1992.

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Akimasa Morihata.

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A preliminary report of this work was appeared in 23).

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Morihata, A. A Short Cut to Optimal Sequences. New Gener. Comput. 29, 31–59 (2011). https://doi.org/10.1007/s00354-010-0098-4

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Keywords

  • Functional Programming
  • Dynamic Programming
  • Program Transformation
  • Shortcut Fusion