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s t a t i c void P a r a l l e l F o r ( int 10, int h i , Act ion < i nt > bod y , int p ) { c o n s t int c h u n k = 1 6 ; II C h u n k s i z e ( co n s t a n t ) Countdown Event l a t c h = new CountdownEvent ( p ) ; II S c he d u l e t h e t h r e a d s to r u n in p a r a l l e l for ( i nt i = 0 ; i < p ; i++ ) { Thread Pool QueueUserWo rk Item ( delegate ( ob j e c t p roc Id ) { int s t a rt = 10 + ( i nt ) p r o c I d * c h u n k ; f o r ( i nt j = s t a rt ; j < h i ; j + = c h u n k * ( p - 1 { for ( i nt k = 0 ; k < c h u n k && j + k < h i ; k++ ) { body ( j + k ) ; } } latch Signal ( ) ; }, i ) ; } l at c h Wa it ( ) ; I I Wait for t hem to f i n i s h }

The only difference between this and the earlier chunking example is that we use two loops to enumerate the indices in a given chunk The outer loop (with induction variable j ) begins at a starting index of our 10 + proc l d * c h u n k and continues until we reach h i It increments j by c h u n k * ( p 1) on each iteration, having the effect of skipping over all other threads' chunks each time that thread finishes with one of its own, as explained earlier Then, beginning at that index, we enumerate the indices in the current chunk by using another inner loop (with induction variable k) We must make sure we also stay within the bounds of the loop by checking that j + k is less than hi each iteration All of the other details, such as how we initialize and sig

We nd that E(X + Y ) = 2 So Var(X + Y ) = 2 7 2

Form 2555 (2003)

nal the latch, call the function, and so forth, remain the same And many of the same limitations explained above in the context of contiguous partitions also hold here

The previous approaches relied on an up front partitioning of the iteration space As we noted, this can lead to imperfect utilization in cases where work blocks or is uneven Overdecomposition was a suggested method for dealing with this But there are other approaches too One good approach for dealing with the uneven work problem is to dynamically decompose the iteration space by handing out chunks of work "on demand " This looks a lot like the striped iteration case seen earlier, with one difference: we need to use synchronization to communicate the current index among workers It also handles loops that are not index based

For Known Size Iteration Spaces The first case we will look at is when the iteration space is of a known size, such as with a traditional fo r loop

5 7 + 3 24 2

(Continued)

stat i c void P a r a l l e l F o r ( int 1 0 , int h i , Action < i n t > bod y , int p ) { const int c h u n k 1 6 j II C h u n k s i z e ( c onstant ) Countdown Event l a t c h new Cou ntdown Event ( p ) j int c u r rent 10j

= = =

I I S c h e d u l e t h e t h read s to r u n i n p a r a l l e l f o r ( i nt i 0 j i < p j i++ )

7 7 + 4 24 2

16 17 18