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# On-line adaptive parallel prefix computation PowerPoint PPT Presentation

On-line adaptive parallel prefix computation. Jean-Louis Roch, Daouda Traoré and Julien Bernard Presented by Andreas Söderström, ITN. The prefix problem. Given X = x 1 ,x 2 ,…,x n compute the n products π k =x 0 о x 1 о … ο x k for 1 ≤ k ≤ n where ο is some associative operation

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## On-line adaptive parallel prefix computation

Jean-Louis Roch, Daouda Traoré and Julien Bernard

Presented by Andreas Söderström, ITN

### The prefix problem

• Given X = x1,x2,…,xn compute the n productsπk=x0 о x1 о … ο xk for 1 ≤ k ≤ nwhere ο is some associative operation

• Example:o = + (i.e. addition)X = 1,3,5,7π1 = 1π2 = 1+3 = 4 π3 = 1+3+5 = 9 π4 = 1+3+5+7 = 16

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### Parallel prefix sum (second pass)

• For every even position use the value of the parent node

• For evey odd position pn compute pn-1+ pn

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### Parallel prefix computation

• Parallel time: 2n/p + O(log n)for p < n/(log n)

• Lower bound for parallel time:2n/(p+1) for n > p(p+1)/2

• Assumes identical processors!

### Parallel prefix computation

• Potential practical problems:

• Processor setup may be heterogenous

• Processor load may vary due to other users computing on the same machine

• Off-line optimal scheduling potentially not optimal anymore!

• Solution:

• Use on-line scheduling!

### The basic idea

• Combine a sequentially optimal algorithm with fine-grained parallellism using work stealing

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Steal work

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### The algorithm

Sequential process Ps:

• The sequential process Ps starts working on [π1, πk], i.e. value indices [1,k] where indices [k+1,m] has been stolen

• When Ps reaches the index k it communicates πk to the parallel process Pv that has stolen [k+1,m] and recoveres the last index n computed by Pv together with the local prefix result rn

• Ps uses associativity to calculate πn+1 = πko rnand continues with the computation from index n+1

### The algorithm

Parallel process Pv

• Pv scans for active processes (can be Ps or another Pv) and steals part of the work from that process.

• Pv computes the local prefix operation on the stolen interval

• The computation of Pv depends on a previous value and need to be finalized when that value is known

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### Performance

• If a processor is or becomes slow part of its work can be stolen by an idle processor

• Asymptotic optimality (proof provided in the paper)

### Performance

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### Performance

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