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word level parallelism, such as those found in the supercomputing industry, however, it often makes sense to use such techniques Another data parallel technique related to reductions is called a scan A scan is very much like a reduction except that the output of the operation is another list of values instead of a scalar Each element i in the result is the partial reduction of the list, obtained by applying the particular binary oper ator to all elements O i-I in the original list In the case of a sum scan (also called the partial sums of a list), for instance, the tenth element contains the sum of elements 0 through 9, the eleventh contains the sum of elements 0 through 1 0, and so on This seems like an inherently sequential problem, but again we can take advantage of associativity and commutativity in the same way we did to achieve parallelism (see Further Reading, Hillis, Steele)
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There are countless ways to sort a list This is true of sequential software, and holds true for parallel software too Parallel quick-sort, parallel merge-sort,
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Batcher's bitonic sort, and radix sort are just a few of the algorithms you can find written up in books and academic papers Instead of spending a great deal of time comparing and contrasting the different approaches, let's look briefly at one particular technique: parallel merge-sort A parallel merge-sort works a lot like an ordinary merge-sort The main difference is that we must partition the input among threads, have each of the threads locally sort their own copy, and each of the intermediary results must be merged The individual sorts are perfectly parallel, but the merge step contains a fair bit of communication This tends to be the limiting scal ing factor for this particular algorithm and prevents it from achieving lin ear speedup But it is the simplest to understand and implement, provided that you're somewhat familiar with the merge-sort algorithm already Before diving into the code, the two high level phases of the algorithm are as follows
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We first split the input into p chunks We use our P a r a l l e l F o r con struct to fork p workers, each of which uses the A r r a y S o rt algo rithm available in NET to sort the arrays locally (using a quick-sort) Depending on the partitioning used, this may or may not lead to the
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desired results Chunking, for example, will prevent some tasks from running in parallel We may be better off explicitly creating p tasks to ensure they run on separate processors
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At this point, we have p sorted chunks The next step is to merge them This takes log p steps Roughly speaking, adjacent tasks are paired up to merge: two tasks merge two arrays into one at a time The logic for this is somewhat complicated: we ensure that both threads merge up to the midpoint in the array Due to the way com parisons happen, we can be assured that this leads to an examina tion of all of the locally sorted inputs At the end, we copy this intermediate result so the next phase in merging has access to the output
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I I Step 1 : Sort t h e p c h u n k s of t h e i n p u t int c h u n k i n p u t Length I p j P a r a l l e l F o r ( e , p , d e legat e ( int i d x )
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{ I I Comp ute t h e bound s int s t a rt idx * c h u n k j int sizej if ( id x P 1) s i ze i n p u t Length - s t a rt j
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