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as mergesort, an inherently recursive algorithm-or because it is simply a convenient way of representing and processing certain kinds of problems and data structures-such as traversing a tree and doing something with each of its nodes Whatever the case, individual recursive calls are often completely inde pendent of other recursive calls in a tree of computations For example, the whole point of divide and conquer is to continually divide a problem space into smaller and smaller disjointed pieces so that they can be solved inde pendently, combining results as the recursion unwinds This is conducive to parallel execution of the individual parts In other nonembarrassingly parallel cases, some or all of the recursive calls share state, such as fields of shared objects, at which point all of the state management issues we've out lined earlier must be taken into account Without attention and care, this often leads to recursive lock usage, which is a bad idea for all the reasons outlined in 1 1 , Concurrency Hazards For what it's worth, recursion usually straddles the line between data and task parallelism In some cases, the depth of recursion and division of work is driven solely by the characteristics of the data being operated on, in which case recursion truly is a data parallel mechanism In other cases, the recursion may be completely program structure dependent and have nothing to do with data, in which case it appears as a task parallel problem Categorization aside, we discuss recursion in the task parallelism section because it is most typically reified using task parallel constructs; in fact, we'll make use of some of the task capabilities we just reviewed in the preceding paragraphs As an example of a simple recursive algorithm, imagine we have a binary tree and would like to mirror it in place That is, for each node in the.

tree, we would like to swap its left and right child subtrees with one another This is easy to parallelize, since there are no dependencies at all in the individual recursive calls and can be done in a divide and conquer style It is important that we ensure no two threads try to mirror the same node's children at once, which is done by virtue of the fact that the unit of work is an independent node For a graph that might have cycles, this would be far more difficult to do, perhaps requiring fine-grained node locks The sequential version might look like this

No Date To: (Controller) Authority is requested for destruction of the following records:

c l a s s TreeNode { internal TreeNode left ; i n t e r n a l TreeNode right ; } void Mirro r ( T reeNode node ) { if ( node == n U l l ) ret u r n ; M i r ro r ( node left ) ; M i r ro r ( node right ) ; TreeNode tmp = node left ; node left = node right ; node right = tmp ; }

Suppose we have part of a probability density function that is part of the parabola f (x) = 1 2 x, 2 0 x 1

n U l l ) ret u r n ;

DoAl l ( ( ) = > P a r a l l e l M i r ro r ( node l eft ) , ( ) = > P a r a l l e lMirror ( node right ) ); TreeNode tmp = node left ; node left node right ; node right = tmp ;

Destruction Recommended by Destruction Approved by I hereby certify that I have this day destroyed the accounts, records, and memoranda listed above, and further that no accounts, records, or memoranda other than those named were destroyed herewith (Witness) (Location) (Signed) (Position)

If, instead of performing side effects, the recursive function needed to compute values, we might consider using the F u t u r e < T > abstraction we

and we wish to approximate P(0 x 1) The exact value of this probability is 1/6 The graph of this function is shown in Figure 89

created above instead Executing this algorithm generates a tree-like structure of dependent computations, as shown in Figure 1 32 This entire problem could be generalized to any kind of binary traversal (or even arbitrary traversals) by adding more delegate invocations

and then list each box s contents on a separate index document However, it is possible that the index number will be improperly transcribed onto the master index document, so anyone referring to it will not know where some documents are stored To avoid this problem, it is useful to list the general contents of each box on the outside of the box, using a marker pen This way, there is no risk of losing track of documents and someone without access to the master index to still find stored documents with minimal search time

void Traverse < T > ( T c u r r , Action < T > bod y , F u n c < T , T> left , F u n c < T , T> right ) { if ( c u r r == defa u l t ( T DoAl l ( ( ) = > Traverse< T > ( left ( c u r r , ( ) = > Trave r s e< T > ( right ( c u r r ); body ( c u r r ) ; ret u r n ;

void P a r a l lelMi rro r ( TreeNode node ) { Trave r s e < T reeNod e > ( nod e , n => { T reeNode tmp = node left ; node left = node right ; node right = tmp ; }, n = > n left , n = > n right , );