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Optimizing Sorting With Genetic Algorithms

Xiaoming Li, Mar í a Jes ú s Garzar á n, and David Padua University of Illinois at Urbana-Champaign. Optimizing Sorting With Genetic Algorithms. ESSL on Power3. ESSL on Power4. Outline. Our Solution Primitives & Selection mechanisms Genetic Algorithm Performance results

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Optimizing Sorting With Genetic Algorithms

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  1. Xiaoming Li, María Jesús Garzarán, and David Padua University of Illinois at Urbana-Champaign Optimizing Sorting With Genetic Algorithms

  2. ESSL on Power3

  3. ESSL on Power4

  4. Outline Our Solution Primitives & Selection mechanisms Genetic Algorithm Performance results Classifier System Conclusion

  5. Motivation • No universally best sorting algorithm • Can we automatically GENERATE and tune sorting algorithms for each platform (such as FFTW and Spiral)? • Performance of sorting on the platform and on the input characteristics. • The algorithm selection may not be enough.

  6. Algorithm Selection (CGO’04) • Select the best algorithm from Quicksort, Multiway Merge Sort and CC-radix. • Relevant input characteristics: number of keys, entropy vector.

  7. Algorithm Selection (CGO’0

  8. Proposed Solution • We need different algorithms for different partitions • The best sorting algorithm should be the result of the composition of the these different best algorithms. • Build Composite Sorting algorithms • Identify primitives from the sorting algorithms • Design a general method to select an appropriate sorting primitive at runtime • Design a mechanism to combine the primitives and the selection methods to generate the composite sorting algorithm

  9. Outline • Our Solution • Primitives & Selection mechanisms • Genetic Algorithm • Performance results • Classifier System • Conclusion

  10. Sorting Primitives • Divide-by-Value • A step in Quicksort • Select one or multiple pivots and sort the input array around these pivots • Parameter: number of pivots • Divide-by-Position (DP) • Divide input into same-size sub-partitions • Use heap to merge the multiple sorted sub-partitions • Parameters: size of sub-partitions, fan-out and size of the heap

  11. Sorting Primitives • Divide-by-Radix (DR) • Non-comparison based sorting algorithm • Parameter: radix (r bits) • Step 1: Scan the input to get distribution array, which records how many elements in each of the 2r sub-partitions. • Step 2: Compute the accumulative distribution array, which is used as the indexes when copying the input to the destination array. • Step 3: Copy the input to the 2r sub-partitions. src. counter accum. dest. 0 1 2 3 0 1 2 3 11 23 30 12 1 1 1 1 0 1 2 3 1 2 3 4 30 11 12 23

  12. Sorting Primitives • Divide-by-radix-assuming-uniform-distribution (DU) • Step 1 and Step 2 in DR are expensive. • If the input elements are distributed among 2r sub-partitions near evenly, the input can be copied into the destination array directly assuming every partition have the same number of elements. • Overhead: partition overflow • Parameter: radix (r bits) src. accum. dest. 0 1 2 3 11 23 30 12 0 1 2 3 1 2 3 4 30 11 12 23

  13. Selection Primitives • Branch-by-Size • Branch-by-Entropy • Parameter: number of branches, threshold vector of the branches

  14. Leaf Primitives • When the size of a partition is small, we stick to one algorithm to sort the partition fully. • Two methods are used in the cleanup operation • Quicksort • CC-Radix

  15. Composite Sorting Algorithms • The composite sorting algorithms are built from these primitives. • The algorithms have shapes of tree.

  16. Outline • Our Solution • Primitives & Selection mechanisms • Genetic Algorithm • Performance results • Classifier System • Conclusion

  17. Search Strategy • Search the best tree • Search the best parameter values of the primitives • Good solutions for small size problem should be retained to use in the solution for larger problem. • Genetic algorithms are a natural solution that satisfy the requirements: • Preserve good sub-trees • Give good sub-trees more chances to propagate

  18. Composite Sorting Algorithms • Search the best parameter values to adapt • To the architectural features • To the input characteristics

  19. Search Strategy • Search for the best tree • Search for the best parameter values of the primitives • Good solutions for small size problem should be retained to use in the solution for larger problem. • Genetic algorithms are a natural solution that satisfy the requirements: • Preserve good sub-trees • Give good sub-trees more chances to propagate

  20. Genetic Algorithm • Mutation • Mutate the structure of the algorithm. • Change the parameter values of primitives.

  21. Crossover • Propagate good sub-trees

  22. Fitness Function • A fitness function measures the relative performance of the genomes in a population. • The average performance of a genome on the training inputs is the base for the fitness of the genome. • A genome which performs well across inputs is preferred • fitness is penalized when performance varies across the test inputs

  23. Library Generation • Installation phase: Use genetic algorithm to search for the sorting genome. • Set of genomes in initial population • Test the genomes in a set of inputs with different characteristics

  24. Outline • Our Solution • Primitives & Selection mechanisms • Genetic Algorithm • Performance results • Classifier System • Conclusion

  25. Platforms • AMD Athlon MP • Sun UltraSparcIII • SGI R12000 • IBM Power3 • IBM Power4 • Intel Itanium2 • Intel Xeon

  26. AMD Athlon MP

  27. Power3

  28. Multiple-peak Performance

  29. Outline • Our Solution • Primitives & Selection mechanisms • Genetic Algorithm • Performance results • Classifier System • Conclusion

  30. The best genomes in different regions

  31. Problems of Genetic Adaptation • Fitness function is the average performance of the genome on the test inputs. • Fitness function in our genetic algorithm prefers genomes with stable performance • The genetic algorithm is not powerful enough to evolve into the complex genome which chooses the best genome in each small region

  32. Using Classifier System • Search the best genomes for different regions of the input characteristics. • Selects the regions • Selects the best algorithm for each region • Nice feature: The fitness of a genomes in a region will not be affected by its fitness in other regions

  33. Map sorting composition into a classifier system • The input characteristics (number of keys and entropy vector) are encoded into bit strings. • A rule in the classifier system has two parts • Condition: A string consisting of ‘0’, ‘1’, and ‘*’. Condition string will be used to match the encoded input characteristics. • Action: Sorting genomes without branch primitives

  34. Example for Classifier Sorting • Example: • For inputs of up-to 16M keys • Encode number of keys with 4 bits. • 0000: 0~1M, 0001: 1~2M… • Number of keys = 10.5M. Encoded into “1100” 1100 01** 1100 1010 1100 110* (dv 2 ( lr 6 16))

  35. Performance of Classifier Sorting • Power3

  36. Power4

  37. Conclusions • Replace the complexity of finding an efficient algorithm with the task of defining a set of generic primitives. • Design methods to search in the space of the composition of the primitives. • Genetic algorithms • Classifier system

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