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When loops are nested, there is an interesting decision to make Considering a two-loop case, should we parallelize the outer loop, the inner loop, or both
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As with most things, there isn' t a simple one size fits all answer In many cases, parallelizing the outer loop will yield the most benefit This assumes that, in the above example, N is sufficiently large to expose enough paral lelism to achieve a speedup If N is less than the number of processors, for instance, then it is worth considering an alternative such as parallelizing the inner loop instead Again, this assumes M is sufficiently large If it isn't, then it may be worth at considering parallelizing both (When it comes to the parallelization process, we can use the techniques we have already reviewed ) A word of caution: a naive implementation of nested invoca tions of the above parallel loop examples will lead to terrible performance because the growth for units of work will be quadratic (Le, O(NM , and recursion and blocking will become a problem for many implementations (such as the thread pool, where such a scheme could easily lead to dead lock) There are alternative approaches One can "fuse" the inner with the outer loop, and then parallelize the single remaining loop This exposes more information to the parallel loops implementation, so that it can more accurately partition the entire space of the iteration at once, rather than dynamically
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This is typically the best approach for such blatant nesting It also leads to roughly the same cache access patterns as if the inner loop remained sequential It is also worth considering whether to rearrange the loop's structure If the data access pattern of the body is such that parallelizing on the inner loop but executing the outer loop inside each thread will lead to better
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cache efficiency, i t may b e desirable t o first restructure the above loop into the following code before parallelizing the outer loop (or even applying the fusion technique)
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As an example of why you might care, imagine we were indexing into a matrix in the body of our loop If the original inner loop (with j and M) controlled the row accessed and the original outer loop (with i and N) controlled the column, then partitioning on the row indices instead of the column would lead to better spatial and temporal cache locality for most dense matrix representations (eg, CLR rectangular arrays, such as i n t [ , ] ) due to the way individual elements in each row are stored adjacent to one another in memory Sometimes it may be useful to "tile" an array, for example, to assign AxB sections of the array to partitions at a time as the chunk unit size, such as 1 6x 1 6 This usually yields performance improvements due to locality and less frequent synchronization In other circumstances, this kind of chunk ing might be a correctness condition of the algorithm JPEG encoding, as an example, is a problem that can be parallelized (see Further Reading, Kodaka, Kimura, Kasahara), but requires that the input image be decoded into 8x8 chunks because of dependencies within individual chunks A plethora of additional loop restructurings is possible, often referred to by the general term loop blocking The idea is to optimize loops, partitioning, and chunk sizes, based on the data access patterns of the code itself Many exotic techniques have been explored over the years (see Further Reading, Lamport 1 973; 1 974), and much research has gone into the static optimization of such operations to achieve the best theoretical speedups (see Further Reading, Blelloch, Gibbons, Matias)
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