Abstract
This paper presents a new approach for computing the transitive closure of a union of relations describing all the dependences in both uniform and quasi-uniform perfectly-nested parameterized loops. This approach is based on calculating the basis of a dependence distance vectors set. The procedure has polynomial time complexity for most steps of calculations. This allows us to effectively extract both fine- and coarse-grained parallelism in loops using techniques based on applying the transitive closure of dependence relations. The effectiveness and time complexity of the approach are evaluated for loops provided by the NAS Parallel Benchmark Suite.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Kelly, W., Pugh, W., Rosser, E., Shpeisman, T.: Transitive closure of infinite graphs and its applications. In: Huang, C.-H., Sadayappan, P., Banerjee, U., Gelernter, D., Nicolau, A., Padua, D.A. (eds.) LCPC 1995. LNCS, vol. 1033, pp. 126–140. Springer, Heidelberg (1996)
Beletska, A., Bielecki, W., Cohen, A., Pałkowski, M., Siedlecki, K.: Coarse-grained loop parallelization: Iteration space slicing vs affine transformations. Parallel Computing 37, 479–497 (2011)
Bielecki, W., Pałkowski, M., Klimek, T.: Free scheduling for statement instances of parameterized arbitrarily nested affine loops. Parallel Computing 38, 518–532 (2012), http://dx.doi.org/10.1016/j.parco.2012.06.001
Verdoolaege, S., Cohen, A., Beletska, A.: Transitive Closures of Affine Integer Tuple Relations and Their Overapproximations. In: Yahav, E. (ed.) SAS 2011. LNCS, vol. 6887, pp. 216–232. Springer, Heidelberg (2011)
Beletska, A., Barthou, D., Bielecki, W., Cohen, A.: Computing the transitive closure of a union of affine integer tuple relations. In: Du, D.-Z., Hu, X., Pardalos, P.M. (eds.) COCOA 2009. LNCS, vol. 5573, pp. 98–109. Springer, Heidelberg (2009)
Bielecki, W., Klimek, T., Trifunovic, K.: Calculating exact transitive closure for a normalized affine integer tuple relation. Electronic Notes in Discrete Mathematics 33, 7–14 (2009)
Wlodzimierz, B., Tomasz, K., Marek, P., Beletska, A.: An iterative algorithm of computing the transitive closure of a union of parameterized affine integer tuple relations. In: Wu, W., Daescu, O. (eds.) COCOA 2010, Part I. LNCS, vol. 6508, pp. 104–113. Springer, Heidelberg (2010)
Lombardy, S., Regis-Ginas, Y., Sakarovitch, J.: Introducing VAUCANSON. Theoretical Computer Science 328(1-2), 77–96 (2004)
Bardin, S., Finkel, A., Leroux, J., Petrucci, L.: Fast: acceleration from theory to practice. STTT 10(5), 401–424 (2008)
Bozga, M., Gîrlea, C., Iosif, R.: Iterating octagons. In: Kowalewski, S., Philippou, A. (eds.) TACAS 2009. LNCS, vol. 5505, pp. 337–351. Springer, Heidelberg (2009)
Feautrier, P., Gonnord, L.: Accelerated invariant generation for c programs with aspic and c2fsm. Electronic Notes in Theoretical Computer Science 267, 3–13 (2010)
Ancourt, C., Coelho, F., Irigoin, F.: A modular static analysis approach to affine loop invariants detection. Electronic Notes in Theoretical Computer Science 267, 3–16 (2010)
NASA Advanced Supercomputing Division, http://www.nas.nasa.gov
Schrijver, A.: Theory of Linear and Integer Programming. Series in Discrete Mathematics (1999)
Shoup, V.: A Computational Introduction to Number Theory. Cambridge University Press (2005)
Rotman, J.: Advanced Modern Algebra, 2nd edn. Prentice Hall (2003)
Presburger, M.: Über de vollständigkeit eines gewissen systems der arithmetik ganzer zahlen, in welchen, die addition als einzige operation hervortritt. In: Comptes Rendus du Premier Congrès des Mathématicienes des Pays Slaves, Warsaw, vol. 395, pp. 92–101 (1927)
Griebl, M.: Automatic Parallelization of Loop Programs for Distributed Memory Achitectures. Habilitation. Fakultät für Mathematik und Informatik Universität Passau (2004)
Bondhugula, U.K.R.: Effective Automatic Parallelization and Locality Optimization Using the Polyhedral Model. Dissertation. The Ohio State University (2010)
Kelly, W., Maslov, V., Pugh, W., Rosser, E., Shpeisman, T., Wonnacott, D.: The Omega Library Interface Guide. Technical Report CS–TR–3445, Dept. of Computer Science, University of Maryland College Park (1995)
Skiena, S.: The Algorithm Design Manual, 2nd edn. Springer (2008)
Ramanujam, J.: Beyond Unimodular Transformations. Journal of Supercomputing 9, 365–389 (1995)
Cohen, E., Megiddo, N.: Recognizing Properties of Periodic Graphs. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, pp. 135–146. American Mathematical Society (1991)
Loechner, V.: PolyLib - A library for manipulating parameterized polyhedra. Technical report, ICPS, Université Louis Pasteur de Strasbourg (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bielecki, W., Kraska, K., Klimek, T. (2013). Transitive Closure of a Union of Dependence Relations for Parameterized Perfectly-Nested Loops. In: Malyshkin, V. (eds) Parallel Computing Technologies. PaCT 2013. Lecture Notes in Computer Science, vol 7979. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39958-9_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-39958-9_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39957-2
Online ISBN: 978-3-642-39958-9
eBook Packages: Computer ScienceComputer Science (R0)