optimizing sorting with genetic algorithms xiaoming li, maría jesús garzarán, and david padua...
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Optimizing Sorting With Genetic Algorithms
Xiaoming Li, María Jesús Garzarán, and David Padua
University of Illinois at Urbana-Champaign
ESSL on Power3
ESSL on Power4
Outline
Our Solution Primitives & Selection mechanisms Genetic Algorithm Performance results Classifier System Conclusion
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.
Algorithm Selection (CGO’04)
Select the best algorithm from Quicksort, Multiway Merge Sort and CC-radix.
Relevant input characteristics: number of keys, entropy vector.
Algorithm Selection (CGO’0
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
Outline
Our Solution Primitives & Selection mechanisms Genetic Algorithm Performance results Classifier System Conclusion
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
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.1111
0123
counter
0123
0123
accum. dest.
11233012
src.
30111223
1234
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)
0123
0123
accum. dest.src.
1234
30111223
11233012
Selection Primitives
• Branch-by-Size• Branch-by-Entropy
– Parameter: number of branches, threshold vector of the branches
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
Composite Sorting Algorithms
• The composite sorting algorithms are built from these primitives.
• The algorithms have shapes of tree.
Outline
Our Solution Primitives & Selection mechanisms Genetic Algorithm Performance results Classifier System Conclusion
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
Composite Sorting Algorithms
• Search the best parameter values to adapt – To the architectural features– To the input characteristics
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
Genetic Algorithm
• Mutation– Mutate the structure of the algorithm.– Change the parameter values of primitives.
Crossover
• Propagate good sub-trees
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
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
Outline
Our Solution Primitives & Selection mechanisms Genetic Algorithm Performance results Classifier System Conclusion
Platforms
AMD Athlon MP Sun UltraSparcIII SGI R12000 IBM Power3 IBM Power4 Intel Itanium2 Intel Xeon
AMD Athlon MP
Power3
Multiple-peak Performance
Outline
Our Solution Primitives & Selection mechanisms Genetic Algorithm Performance results Classifier System Conclusion
The best genomes in different regions
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
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
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
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”
Condition Action Fitness
Accuracy
(dr 5 (lq 1 16)) … …
(dp 4 2 ( lr 5 16)) … …
… …
1100
1100
1100
01**
1010
110* (dv 2 ( lr 6 16))
Performance of Classifier Sorting• Power3
Power4
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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