Defining Asymptotic Parallel Time Complexity of Data-dependent Algorithms
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The scientific research community has reached a stage of maturity where its strong need for high-performance computing has diffused into also everyday life of engineering and industry algorithms. In efforts to satisfy this need, parallel computers provide an efficient and economical way to solve large-scale and/or time-constrained problems. As a consequence, the end-users of these systems have a vested interest in defining the asymptotic time complexity of parallel algorithms to predict their performance on a particular parallel computer.
The asymptotic parallel time complexity of data-dependent algorithms depends on the number of processors, data size, and other parameters. Discovering the main other parameters is a challenging problem and the clue in obtaining a good estimate of performance order. Great examples of these types of applications are sorting algorithms, searching algorithms and solvers of the traveling salesman problem (TSP).
This article encompasses all the knowledge discovery aspects to the problem of defining the asymptotic parallel time complexity of data-dependent algorithms. The knowledge discovery methodology begins by designing a considerable number of experiments and measuring their execution times. Then, an interactive and iterative process explores data in search of patterns and/or relationships detecting some parameters that affect performance. Knowing the key parameters which characterise time complexity, it becomes possible to hypothesise to restart the process and to produce a subsequent improved time complexity model. Finally, the methodology predicts the performance order for new data sets on a particular parallel computer by replacing a numerical identification.
As a case of study, a global pruning traveling salesman problem implementation (GP-TSP) has been chosen to analyze the influence of indeterminism in performance prediction of data-dependent parallel algorithms, and also to show the usefulness of the defined knowledge discovery methodology. The subsequent hypotheses generated to define the asymptotic parallel time complexity of the TSP were corroborated one by one. The experimental results confirm the expected capability of the proposed methodology; the predictions of performance time order were rather good comparing with real execution time (in the order of 85%).
KeywordsComplexity Data-Dependent Algorithms Parallel Computing Performance Order
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- 1.Alizadeh, F., Karp, R. M., Newberg, L. A. and Weisser, D. K., “Physical Mapping of Chromosomes: a Combinatorial Problem in Molecular Biology” in SODA ’93: Proc. of the 4th Annual ACM-SIAM Symposium on Discrete algorithms, isbn: 0-89871-313-7, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, pp. 371–381, 1993.Google Scholar
- 3.Applegate, D. L., Bixby, R. E., Chvatal, V. and Cook, W. J., The Traveling Salesman Problem: A Computational Study (Princeton Series in Applied Mathematics), Princeton University Press, 2007.Google Scholar
- 4.Arkin, E. M., Chiang, Y., Mitchell, J. S. B., Skiena, S. S. and Yang, T., “On the Maximum Scatter TSP,” in SODA ’97: Proc. of the 8th Annual ACM-SIAM Symposium on Discrete algorithms, isbn: 0-89871-390-0, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, pp. 211–220, 1997.Google Scholar
- 6.Barvinok, A., Fekete, S., Johnson, D., Tamir, A., Woeginger, G. and Woodroofe, R., “The Geometric Maximum Traveling Salesman Problem,” Journal of the ACM, New York, NY, USA, 50, 5, pp. 641–664, 2003.Google Scholar
- 7.Biggs, N., Lloyd, E. K. and Wilson, R. J., Graph Theory, 1736-1936, isbn: 0-198-53916-9, Clarendon Press, New York, NY, USA, 1986.Google Scholar
- 10.Chin, D. A., “Complexity Issues in General Purpose Parallel Computing” University of Oxford, 1991.Google Scholar
- 11.Christofides, N., “Vehicle Routing,” The Traveling Salesman Problem (Lawler, L., Lenstra, J. K., Kan R and Shmoys, D. B. eds.), John Wiley, pp. 431–448, 1985.Google Scholar
- 13.Garey, M. R., Graham, R. L. and Johnson, D. S., “Some NP-complete Geometric Problems,” in STOC ’76: Proc. of the 8th Annual ACM Symposium on Theory of computing, pp. 10–22, 1976.Google Scholar
- 14.Geraerts, R. and Overmars, M. H., “Reachability Analysis of Sampling Based Planners,” in IEEE International Conference on Robotics and Automation, John Wiley & Sons, Inc., pp. 406–412, 2005.Google Scholar
- 15.Gilmore, P. C. and Gomory, R. E., “Sequencing a One-State-Variable Machine: A Solvable Case of the Traveling Salesman Problem,” Operations Research, 12, 5, pp. 655–679, 1964.Google Scholar
- 16.Fritzsche, P., “¿‘Podemos Predecir en Algoritmos Paralelos No Deterministas?” Computer Architecture and Operating Systems Department, University Autonoma of Barcelona, Spain, http://caos.uab.es/, 2007.
- 18.Groth, R., Data Mining: a Hands-on Approach for Business Professionals, Prentice Hall PTR, 1998.Google Scholar
- 19.Huttenlocher, D., Klanderman, G. and Rucklidge, W., “Comparing Images Using the Hausdorff Distance,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 15, 9, pp. 850–863, 1993.Google Scholar
- 20.Jarrard, R. D., Scientific Methods, an online book, Dept. of Geology and Geophysics, University of Utah, 2001.Google Scholar
- 21.Johnson, D. S. and McGeoch, L. A., “The Traveling Salesman Problem: A Case Study in Local Optimization,” in Local Search in Combinatorial Optimization, pp. 215–310, 1997.Google Scholar
- 22.Johnson, D. S. and McGeoch, L. A., “Experimental Analysis of Heuristics for the STSP,” isbn: 978-1-4020-0664-7, in The Traveling Salesman Problem and Its Variations, 12, pp. 369–443, 2004.Google Scholar
- 24.Korostensky, C. and Gonnet, G. H., “Using Traveling Salesman Problem Algorithms for Evolutionary Tree Construction,” BIOINF: Bioinformatics, 16, 7, pp. 619–627, 2000.Google Scholar
- 26.MacQueen, J. B., “Some Methods for Classification and Analysis of MultiVariate Observations,” in Proc. of the fifth Berkeley Symposium on Mathematical Statistics and Probability (Le Cam L. M. and Neyman J. eds.), University of California Press, 1, pp. 281–297, 1967.Google Scholar
- 27.Martello, S. and Toth, P., Knapsack Problems: Algorithms and Computer Implementations, isbn: 0-471-92420-2, John Wiley & Sons, Inc., New York, NY, USA, 1990.Google Scholar
- 29.Miller, R. and Boxer, L., Algorithms Sequential and Parallel: A Unified Approach, isbn: 1-58450-412-9, Charles River Media, Computer Engineering Series, 2005.Google Scholar
- 30.Ratliff, H. D. and Rosenthal, A. S., “Order-Picking in a RectangularWarehouse: A Solvable Case for the Traveling Salesman Problem,” Operations Research, 31, 3, pp. 507–521, 1983.Google Scholar
- 31.Sankoff, D. and Blanchette, M., “The Median Problem for Breakpoints in Comparative Genomics,” in COCOON ’97: Proc. of the 3rd Annual International Conference on Computing and Combinatorics, isbn: 3-540-63357-X, Springer-Verlag, London, UK, pp. 251–264, 1997.Google Scholar
- 32.Schaefer, T. J., “The Complexity of Satisfiability Problems,” STOC, pp. 216–226, 1978.Google Scholar
- 33.Schrijver, A., “On the History of Combinatorial Optimization (till 1960),” CWI (Centre for Mathematics and Computer Science), Amsterdam, The Netherlands, 1990.Google Scholar
- 34.Skiena, S. S., The Algorithm Design Manual, isbn: 0-387-94860-0, Springer-Verlag New York, Inc., New York, NY, USA, 1998.Google Scholar
- 35.Takaoka, T., “A New Measure of Disorder in Sorting – Entropy,” CATS, pp. 441–476, 1998.Google Scholar
- 36.TSP Page, http://www.tsp.gatech.edu/history/, 2008.