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Hypermatrix oriented supernode amalgamation

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Abstract

In this paper, we introduce a supernode amalgamation algorithm which takes into account the characteristics of a hypermatrix data structure. The resulting frontal tree is then used to create a variable-sized partitioning of the hypermatrix. The sparse hypermatrix Cholesky factorization obtained runs slightly faster than the one which uses a fixed-sized partitioning. The algorithm also reduces data dependencies which limit exploitation of parallelism.

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References

  1. Golub GH, Van Loan CF (1989) Matrix computations, 2nd edn. Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  2. Duff IS (1982) Full matrix techniques in sparse Gaussian elimination. In: Numerical analysis, Dundee, 1981. Lecture notes in math, vol 912. Springer, Berlin, pp 71–84

    Chapter  Google Scholar 

  3. Ng EG, Peyton BW (1993) Block sparse Cholesky algorithms on advanced uniprocessor computers. SIAM J Sci Comput 14(5):1034–1056

    Article  MATH  MathSciNet  Google Scholar 

  4. Rothberg E, Gupta A (1994) An efficient block-oriented approach to parallel sparse Cholesky factorization. SIAM J Sci Comput 15(6):1413–1439

    Article  MATH  MathSciNet  Google Scholar 

  5. Rothberg E (1996) Performance of panel and block approaches to sparse Cholesky factorization on the iPSC/860 and Paragon multicomputers. SIAM J Sci Comput 17(3):699–713

    Article  MATH  MathSciNet  Google Scholar 

  6. Gupta A, Joshi M, Kumar V (2001) WSMP: A high-performance shared- and distributed-memory parallel sparse linear equation solver. Technical Report, IBM Research Division, TJ Watson Research Center, April 2001

  7. Irony D, Shklarski G, Toledo S (2002) Parallel and fully recursive multifrontal sparse Cholesky. In: ICCS’02. Lecture notes in computer science, vol 2330. Springer, Berlin, pp 335–344

    Google Scholar 

  8. Gould NIM, Scott JA, Hu Y (2007) A numerical evaluation of sparse direct solvers for the solution of large sparse symmetric linear systems of equations. ACM Trans Math Softw 33(2), Article 10, 32 pages

  9. Liu JW, Ng EG, Peyton BW (1993) On finding supernodes for sparse matrix computations. SIAM J Matrix Anal Appl 14(1):242–252

    Article  MATH  MathSciNet  Google Scholar 

  10. Duff IS, Reid JK (1983) The multifrontal solution of indefinite sparse symmetric linear systems. ACM Trans Math Softw 9(3):302–325

    Article  MATH  MathSciNet  Google Scholar 

  11. Ashcraft C, Grimes RG (1989) The influence of relaxed supernode partitions on the multifrontal method. ACM Trans Math Softw 15:291–309

    Article  MATH  Google Scholar 

  12. Von Fuchs G, Roy JR, Schrem E (1972) Hypermatrix solution of large sets of symmetric positive-definite linear equations. Comput Meth Appl Mech Eng 1:197–216

    Article  MATH  Google Scholar 

  13. Noor A, Voigt S (1975) Hypermatrix scheme for the STAR–100 computer. Comput Struct 5:287–296

    Article  Google Scholar 

  14. Ast M, Fischer R, Manz H, Schulz U (1997) PERMAS: User’s reference manual. INTES publication no 450, rev.d

  15. Herrero JR, Navarro JJ (2007) Analysis of a sparse hypermatrix Cholesky with fixed-sized blocking. Appl Algebra Eng Commun Comput 18(3):279–295. doi:10.1007/s00200-007-0039-8

    Article  MATH  MathSciNet  Google Scholar 

  16. Liu JHW (1990) The role of elimination trees in sparse factorization. SIAM J Matrix Anal Appl 11(1):134–172

    Article  MATH  MathSciNet  Google Scholar 

  17. Ast M, Barrado C, Cela JM, Fischer R, Laborda O, Manz H, Schulz U (2000) Sparse matrix structure for dynamic parallelisation efficiency. In: Euro-Par 2000. Lecture notes in computer science, vol 1900, pp 519–526

  18. Herrero JR, Navarro JJ (2003) Automatic benchmarking and optimization of codes: an experience with numerical kernels. In: Int conf on software engineering research and practice. CSREA Press, pp 701–706

  19. Herrero JR, Navarro JJ (2003) Improving performance of hypermatrix Cholesky factorization. In: Euro-Par’03. Lecture notes in computer science, vol 2790. Springer, Berlin, pp 461–469

    Google Scholar 

  20. Herrero JR, Navarro JJ (2004) Reducing overhead in sparse hypermatrix Cholesky factorization. In: IFIP TC5 workshop on high performance computational science and engineering (HPCSE), world computer congress. Springer, Berlin, pp 143–154

    Google Scholar 

  21. Herrero JR, Navarro JJ (2005) Intra-block amalgamation in sparse hypermatrix Cholesky factorization. In: Int conf on computational science and engineering, pp 15–22

  22. NetLib. Linear programming problems

  23. Carolan WJ, Hill JE, Kennington JL, Niemi S, Wichmann SJ (1990) An empirical evaluation of the KORBX algorithms for military airlift applications. Oper Res 38:240–248

    Article  Google Scholar 

  24. Frangioni A Multicommodity min cost flow problems. Operations research group, Department of Computer Science, University of Pisa

  25. Badics T (1991) RMFGEN generator

  26. Lee Y, Orlin J (1991) GRIDGEN generator

  27. Goldberg AV, Oldham JD, Plotkin S, Stein C (1998) An implementation of a combinatorial approximation algorithm for minimum-cost multicommodity flow. In: Proceedings of the 6th international conference on integer programming and combinatorial optimization, IPCO’98, Houston, Texas, June 22–24, 1998. Lecture notes in computer science, vol 1412. Springer, Berlin, pp 338–352

    Google Scholar 

  28. Karypis G, Kumar V (1999) A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J Sci Comput 20(1):359–392

    Article  MATH  MathSciNet  Google Scholar 

  29. Smith JE (1988) Characterizing computer performance with a single number. Commun ACM, CACM 31(10):1202–1207

    Article  Google Scholar 

  30. Woo SC, Ohara M, Torrie E, Singh JP, Gupta A (1995) The SPLASH-2 programs: characterization and methodological considerations. In: Proceedings of the 22nd annual international symposium on computer architecture. ACM, New York, pp 24–36

    Chapter  Google Scholar 

  31. Liu JW-H (1992) The multifrontal method for sparse matrix solution: Theory and practice. SIAM Rev 34:82–109

    Article  MATH  MathSciNet  Google Scholar 

  32. Herrero JR, Navarro JJ (2006) Using non-canonical array layouts in dense matrix operations. In: PARA’06. Lecture notes in computer science, vol 4699. Springer, Berlin, pp 580–588

    Google Scholar 

  33. Gupta A (1997) Fast and effective algorithms for graph partitioning and sparse-matrix ordering. IBM J Res Dev 41(1–2):171–183

    Article  Google Scholar 

  34. Gupta A (1996) Graph partitioning based sparse matrix orderings for interior point algorithms. Technical Report RC 20467(90480), IBM Research Division

  35. Herrero JR, Navarro JJ (2007) Sparse hypermatrix Cholesky: Customization for high performance. IAENG Int J Appl Math 36(1):6–12

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Computer Architecture Department, Universitat Politècnica de Catalunya, Jordi Girona 1-3, Mòdul D6, 08034, Barcelona, Spain

    José R. Herrero & Juan J. Navarro

Authors
  1. José R. Herrero
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  2. Juan J. Navarro
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Correspondence to José R. Herrero.

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Herrero, J.R., Navarro, J.J. Hypermatrix oriented supernode amalgamation. J Supercomput 46, 84–104 (2008). https://doi.org/10.1007/s11227-008-0188-y

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  • DOI: https://doi.org/10.1007/s11227-008-0188-y

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