Counting integer points in parametric polytopes using Barvinok's rational functions

Algorithmica (New York)

Volumn 48, Issue 1, 2007, Pages 37-66

  Verdoolaege, Sven ^a   Seghir, Rachid ^b   Beyls, Kristof ^c   Loechner, Vincent ^b   Bruynooghe, Maurice ^a  

^a UNIVERSITY OF LEUVEN   (Belgium)

^b UNIVERSITÉ DE STRASBOURG   (France)

^c GHENT UNIVERSITY   (Belgium)

Author keywords

Barvinok's decomposition; Computer analysis; Ehrhart quasi polynomial; Parametric polytope; Polyhedral model; Signed unimodular decomposition; Vector partition function

Indexed keywords

BARVINOKS DECOMPOSITION; COMPUTER ANALYSIS; PARAMETRIC POLYTOPES; POLYHEDRAL MODELS; SIGNED UNIMODULAR DECOMPOSITION; VECTOR PARTITION FUNCTIONS;

COMPUTATION THEORY; CONSTRAINT THEORY; FUNCTION EVALUATION; INTEGER PROGRAMMING; INTERPOLATION; LINEAR SYSTEMS; PROGRAM COMPILERS;

POLYNOMIAL APPROXIMATION;

EID: 34547245432     PISSN: 01784617     EISSN: 14320541     Source Type: Journal    
DOI: 10.1007/s00453-006-1231-0     Document Type: Article

References (52)

1. S. Anantharaman and S. Pande. Compiler optimizations for real time execution of loops on limited memory embedded systems. In Proceedings of the 19th IEEE Systems Symposium (RTSS98), pages 154-164, 1998.
2. F. Balasa, F. Catthoor, and H. De Man. Background memory area estimation for multidimensional signal processing systems. IEEE Trans. Very Large Scale Integration (VLSI) Systems, 3(2):157-172, June 1995.
3. A. I. Barvinok. A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. In Proceedings of the 34th Annual Symposium on Foundations of Computer Science, pages 566-572. IEEE, New York, Nov. 1993.
4. A. Barvinok and J. Pommersheim. An algorithmic theory of lattice points in polyhedra. New Perspect. Algebraic Combin., 38:91-147, 1999.
5. M. Beck. The partial-fractions method for counting solutions to integral linear systems. Discrete Comp. Geom., 32:437-446, 2004 (special issue in honor of Louis Billera).
6. M. Bednara, F. Hannig, and J. Teich. Generation of distributed loop control. In In Embedded Processor Design Challenges: Systems, Architectures, Modeling, and Simulation (SAMOS), pages 154-170. Volume 2268 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 2002.
7. K. Beyls and E. D'Hollander. Generating cache hints for improved program efficiency. J. Systems Architecture, 51(4):223-250, 2005.
8. A. J. C. Bik. Compiler Support for Sparse Matrix Computations. Ph.D. thesis. University of Leiden. 1996.
9. B. Boigelot and L. Latour. Counting the solutions of Presburger equations without enumerating them. Theoret. Comput. Sci., 313(1): 17-29. Feb. 2004.
10. P. Boulet and X. Redon. Communication pre-evaluation in HPF. In Proceedings of EUROPAR '98. pages 263-272. Volume 1470 of Lecture Notes in Computer Science. Springer-Verlag. Berlin, 1998.
11. V. Braberman, D. Garbervetsky. and S. Yovine. On synthesizing parametric specifications of dynamic memory utilization. Technical Report TR-2004-03, VERIMAG, Oct. 2003.
12. M. Brion. Points entiers dans les polyèdres convexes. Ann. Sci. École Norm. Sup. (4), 21 (4):653-663, 1988.
13. S. Chatterjee, E. Parker, P. J. Hanlon, and A. R. Lebeck. Exact analysis of the cache behavior of nested loops. In Proceedings of the ACM SIGPLAN 2001 Conference on Programming Language Design and Implementation, pages 286-297. ACM Press, New York, 2001.
14. P. Clauss and V. Loechner. Parametric analysis of polyhedral iteration spaces. J. VLSI Signal Process., 19(2): 179-194. July 1998.
15. P. D'Alberto, A. Veidembaum, A. Nicolau, and R. Gupta. Static analysis of parameterized loop nests for energy efficient use of data caches. In Proceedings of the Second Workshop on Compilers and Operating Systems for Low Power (COLP01), pages 11.1-11.7. Sept. 2001.
16. J. A. De Loera, D. Haws, R. Hemmecke, P. Huggins, B. Sturmfels, and R. Yoshida. Short rational functions for toric algebra and applications. J. Symbolic Comput 38(2):959-973, 2004.
17. J. A. De Loera, D. Haws, R. Hemmecke, P. Huggins, J. Tauzer, and R. Yoshida. A user's guide for LattE vl.1, Nov. 2003. Software package LattE is available at http://www.math.ucdavis.edu/~latte/.
18. J. A. De Loera, R. Hemmecke, J. Tauzer, and R. Yoshida. Effective lattice point counting in rational convex polytopes. J. Symbolic Comput., 38(4): 1273-1302, 2004.
19. S. Derrien, A. Turjan, C. Zissulescu, B. Kienhuis, and E. Deprettere. Deriving efficient control in Kahn process networks. In Proceedings of the International Workshop on Systems. Architectures. Modeling. and Simulation (SAMOS 2003), pages 160-167, 2003.
20. H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Verlag, New York, 1987.
21. E. Ehrhart. Polynò̂mes arithmétiques et méthode des polyedres en combinatoire. In Volume 35 of International Series of Numerical Mathematics. Birkhauser Verlag, Basel/Stuttgart, 1977.
22. P. Feautrier. Parametric integer programming. Oper. Res., 22(3):243-268, 1988.
23. J. Ferrante, V. Sarkar, and W. Thrash. On estimating and enhancing cache effectiveness. In U. Banerjee, D. Gelernter, A. Nicolau, and D. Padua, editors. Proceedings of the Fourth International Workshop on Languages and Compilers for Parallel Computing, pages 328-343. Volume 589 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1991.
24. D. Fimmel and R. Merker. Design of processor arrays for real-time applications. In Proceedings of the International Conference Euro-Par '98. pages 1018-1028. Volume 1470 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1998.
25. B. Franke and M. O'Boyle. Array recovery and high-level transformations for DSP applications. ACM Trans. Embedded Comput. Systems. 2(2):132-162, May 2003.
26. Free Software Foundation, Inc. GMP. Available from ftp://ftp.gnu.org/gnu/ gmp.
27. S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Trans. Program. Languages Systems, 21(4):703-746, 1999.
28. N. Halbwachs, D. Merchat, and C. Parent-Vigouroux. Cartesian factoring of polyhedra in linear relation analysis. In Proceedings of the Static Analysis Symposium. SAS'03. San Diego, June 2003. Volume 2694 of Lecture Notes in Computer Science. Springer-Verlag, Berlin. 2003.
29. F. Hannig and J. Teich. Design space exploration for massively parallel processor arrays. In Proceedings of the Sixth International Conference on Parallel Computing Technologies, pages 51-65. Volume 2127 of Lecture Notes in Computer Science. Springer-Verlag. Berlin, 2001.
30. F. Heine and A. Slowik. Volume driven data distribution for NUMA-machines. In Proceedings from the 6th International Euro-Par Conference on Parallel Processing, pages 415-424, 2000.
31. P. Henrici. Applied and Computational Complex Analysis. Wiley Classics Library. Wiley, New York, 1974.
32. W. Kelly, V. Maslov, W. Pugh, E. Rosser, T. Shpeisman, and D. Wonnacott. The Omega calculator and library. Technical report, University of Maryland, Nov. 1996.
33. H. S. Kim, N. Vijaykrishnan, M. Kandemir, E. Brockmeyer, F. Catthoor, and M. J. Irwin. Estimating influence of data layout optimizations on SDRAM energy consumption. In Proceedings of the 2003 International Symposium on Low Power Electronics and Design, pages 40-43. ACM Press, New York, 2003.
34. P. O. Kjeldsberg, F. Catthoor, and E. J. Aas. Data dependency size estimation for use in memory optimization. IEEE Trans. Comput.-Aided Design Integrated Circuits Systems, 22(7):908-921, July 2003.
35. B. Lisper. Fully automatic, parametric worst-case execution time analysis. In J. Gustafsson, editor, Proceedings of the Third International Workshop on Worst-Case Execution Time (WCET) Analysis, pages 77-80, Porto, July 2003.
36. V. Loechner. Contribution à l'étude des polyèdres paramétrés et applications en parallélisation automatique. Ph.D. thesis, University Louis Pasteur, Strasbourg, 1997.
37. V. Loechner. Polylib: A library for manipulating parameterized polyhedra. Technical report, ICPS, Université Louis Pasteur, Strasbourg, Mar. 1999.
38. V. Loechner, B. Meister, and P. Clauss. Precise data locality optimization of nested loops. J. Supercomput., 21(1):37-76, 2002.
39. V. Loechner and D. K. Wilde. Parameterized polyhedra and their vertices. Internat. J. Parallel Program., 25(6):525-549, Dec. 1997.
40. B. Nootaert, K. Beyls, and E. D'Hollander. On the calculation of Erhart polynomials in degenerate domains. Technical Report 105.001, Ghent University, 2005.
41. E. Parker and S. Chatterjee. An automata-theoretic algorithm for counting solutions to Presburger formulas. In Compiler Construction 2004, pages 104-119. Volume 2985 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Apr. 2004.
42. W. Pugh. Counting solutions to Presburger formulas: how and why. In SIGPLAN Conference on Programming Language Design and Implementation (PLDI '94), pages 121-134, 1994.
43. A. Schrijver. Theory of Linear and Integer Programming. Wiley, New York, 1986.
44. V. Shoup. NTL. Available from http://www.shoup.net/ntl/.
45. R. P. Stanley. Enumerative Combinatorics, volume 1. Cambridge University Press, Cambridge, 1986.
46. B. Sturmfels. On vector partition functions. J. Combin. Theory Ser. A, 72(2):302-309, 1995.
47. E. Su, A. Lain, S. Ramaswamy, D. J. Palermo, E. W. Hodges IV, and P. Banerjee. Advanced compilation techniques in the PARADIGM compiler for distributed-memory multicomputers. In Proceedings of the International Conference on Super-computing, pages 424-433, 1995.
48. A. Turjan, B. Kienhuis, and E. Deprettere. A compile time based approach for solving out-of-order communication in Kahn process networks. In Proceedings of the IEEE 13/A International Conference on Application-Specific Systems, Architectures and Processors (ASAP 2002), pages 17-28, July 2002.
49. S. Verdoolaege, K. Beyls, M. Bruynooghe, and F. Catthoor. Experiences with enumeration of integer projections of parametric polytopes. In R. Bodik, editor, Proceedings of the 14th International Conference on Compiler Construction, Edinburgh, Scotland, pages 91-105. Volume 3443 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 2005.
50. S. Verdoolaege, R. Seghir, K. Beyls, V. Loechner, and M. Bruynooghe. Analytical computation of Ehrhart polynomials: enabling more compiler analyses and optimizations. In Proceedings of the International Conference on Compilers, Architectures, and Synthesis for Embedded Systems, Washington D.C., pages 248-258, Sept. 2004.
51. R. Yoshida. Barvinok's rational functions: algorithms and applications to optimization, statistics, and algebra. Ph.D. thesis, University of California, Davis, 2004.
52. Y. Zhao and S. Malik. Exact memory size estimation for array computations. IEEE Trans. Very Large Scale Integration (VLSI) Systems, 8(5):517-521, October 2000.