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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

36

Step 1

10

26

Step 2

3

7

11

15

Step 3

1

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6

7

8

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

36

10

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3

6

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15

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36

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

P0

P1

P2

…

Pn

Steal work

Steal work

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

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

P homogenous processeors

Performance

P heterogenous processors

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