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Problem Suppose the problem involves an operation on a recursive data structure (such as a list, tree, or graph) that appears to require sequential processing How can operations on these data structures be performed in parallel Context Some problems with recursive data structures naturally use the divide and conquer strategy described in the Divide and Conquer pattern with its inherent potential for concurrency Other operations on these data structures, however, seem to have little if any potential for concurrency because it appears that the only way to solve the problem is to sequentially move through the data structure, computing a result at one element before moving on to the next Sometimes, however, it is possible to reshape the operations in a way that a program can operate concurrently on all elements of the data structure An example from [J92] illustrates the situation: Suppose we have a forest of rooted directed trees (defined by specifying, for each node, its immediate ancestor, with a root node's ancestor being itself) and want to compute, for each node in the forest, the root of the tree containing that node To do this in a sequential program, we would probably trace depth first through each tree from its root to its leaf nodes; as we visit each node, we have the needed information about the corresponding root Total running time of such a program for a forest of N nodes would be O(N) There is some potential for concurrency (operating on subtrees concurrently), but there is no obvious way to operate on all elements concurrently, because it appears that we cannot find the root for a particular node without knowing its parent's root However, a rethinking of the problem exposes additional concurrency: We first define for each node a "successor", which initially will be its parent and ultimately will be the root of the tree to which the node belongs We then calculate for each node its "successor's successor" or nodes one "hop" from F the root, this calculation does not change the value of its successor (because a root's parent is itself) For nodes at least two "hops" away from a root, this calculation makes the node's successor its parent's parent We repeat this calculation until it converges (that is, the values produced by one step are the same as those produced by the preceding step), at which point every node's successor is the desired value Fig 421 shows an example requiring three steps to converge At each step we can operate on all N nodes in the tree concurrently, and the algorithm converges in at most log N steps.

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At the root of every graphical definition model is the Canvas This root element has a name property and containment references to one or more Figure Galleries, Nodes, Connections, Compartments, and Labels Note the distinction between figures and diagram elements In a GMF graphical definition, a Figure is defined within a Figure Gallery, which is then referenced by Node, Connections, Compartment, and Label diagram elements, via Figure Descriptors and Accessors These diagram elements are siblings to Figure Gallery elements, although they can reference figure definitions from other graphical definition models The idea is to promote reuse of figure definitions without adding the complexity of yet another GMF model

Generics support the ability to define constraints on type parameters These constraints enforce the types to conform to various rules Take, for example, the BinaryTree<T> class shown in Listing 1119

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As mentioned, a Canvas can contain a number of Figure Gallery elements, which, in turn, contain figures, figure descriptors, and an optional implementation bundle property

public class BinaryTree<T> { public BinaryTree ( T item) { Item = item; } public T Item { get{ return _Item; } set{ _Item = value; } } private T _Item; public Pair<BinaryTree<T>> SubItems { get{ return _SubItems; } set{ _SubItems = value; } } private Pair<BinaryTree<T>> _SubItems; }

Figure 421 Finding roots in a forest Solid lines represent the original parent-child relationships among nodes; dashed lines point from nodes to their successors

(An interesting side note is that BinaryTree<T> uses Pair<T> internally, which is possible because Pair<T> is simply another type) Suppose you want the tree to sort the values within the Pair<T> value as it is assigned to the SubItems property In order to achieve the sorting, the SubItems set accessor uses the CompareTo() method of the supplied key, as shown in Listing 1120

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Be careful not to name your Canvas the same as any of the contained Figure Gallery elements because this will cause a name clashThis will be detected during validation of the graphical definition model, but it will result in exceptions when trying to load the Canvas mapping in the

public class BinaryTree<T> { public Pair<BinaryTree<T>> SubItems { get{ return _SubItems; } set { IComparable<T> first; // ERROR: Cannot implicitly convert type first = valueFirstItem; // Explicit cast required if (firstCompareTo(valueSecondItem) < 0) { // first is less than second } else { // first and second are the same or // second is less than first } _SubItems = value; } } private Pair<BinaryTree<T>> _SubItems; }

mapping definition model Validation is performed upon transformation from the mapping model to generator model, or when invoked manually from the context menu

At compile time, the type parameter T is generic When the code is written as shown, the compiler assumes that the only members available on T are those inherited from the base type object, since every type has object as an ancestor (Only methods such as ToString(), therefore, are available to the key instance of the type parameter T) As a result, the compiler displays a compilation error because the CompareTo() method is not defined on type object

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