PARALLEL SYSTEMS AND PROGRAMMING in .NET

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PARALLEL SYSTEMS AND PROGRAMMING
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Example 1 1: a = 2 2: u = a 3: v = a 4: x = u
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Program for x = a 7 + (a + 2) + 2 7 + v
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Every line of this program is a statement of the form variable = expression In the dependence analysis literature, statements are the entities between which dependence relations are de ned, especially since classical dependence analysis is strongly linked to cyclic computations (ie, loops), which will be covered in Section 252 In scheduling, the notion of a task is preferred, as it re ects a more exible concept in terms of the size of the represented computation A task can range from a small statement to basic blocks, loops, and sequences of these To emphasize the fact that the ideas of this section have general validity, a statement will be referred to as a task The concepts of dependence presented here are unaffected by the choice of notation The variable of each statement variable = expression holds the result of the task and thereby acts as its output, while the variables in the expression are the input of the task In Example 1, the output variable of task 1 (a) is an input variable of tasks 2 and 3, and the output variables of task 2 (u) and task 3 (v) are the input variables of task 4 In other words, tasks 2 and 3 depend on the outcome of task 1, and task 4 on the outcomes of tasks 2 and 3 It is said that tasks 2 and 3 are ow dependent on task 1, caused by variable a Correspondingly, task 4 is ow dependent on tasks 2 and 3, caused by the variables u and v Consequently, only tasks 2 and 3 are not dependent on each other and can therefore be executed in parallel Flow dependence is not the only type of dependence encountered in programs Consider the equation in Example 2 and its corresponding formulation as a program Program for x = a 7 + (a 5 + 2) + + 5 2 7 v
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Example 2 1: a = 2 2: v = a 3: u = v 4: v = a 5: x = u
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While task 3 is ow dependent on task 2 (caused by variable v), there is a new form of dependence between tasks 3 and 4 In ow dependence, a write occurs before a read, but task 3 reads variable v before task 4 writes it The supposed input of task 3 is the value computed by task 2 and not that of task 4, but both use variable v for their output In order to calculate the correct value of x, task 3 must be executed before task 4 Task 4 is said to be antidependent on task 3
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DEPENDENCE ANALYSIS
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Another type of dependence is output dependence Consider the program extract in Example 3 Example 3 Output Dependence 2: o = 10 4: o = 7 Suppose the variable o is only utilized in the two tasks 2 and 4 of the program After a sequential execution, the value of o is 7 In a parallel execution, however, the value of o is determined by the execution order of the two tasks: if task 4 is executed before task 2, the nal value of o is 10! Obviously, the sequential and the parallel versions should produce the same result Thus, task 4 is said to be output dependent on task 2 for a correct semantic of the program task 2 must be executed before task 4 The antidependence in Example 2 and the output dependence of Example 3 can be resolved by modifying the programs For instance, if Example 2 is rewritten as displayed in Example 4, its antidependence is eliminated task 4 is no longer antidependent on task 3 Example 4 1: a = 2 2: v = a 3: u = v 4: w = a 5: x = u Eliminated Antidependence from Program in Example 2 + + 5 2 7 w
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The dependence was eliminated by substituting variable v in tasks 4 and 5 with the new variable w Likewise, the output dependence of Example 3 is eliminated by the utilization of an additional variable Real Dependence In general, antidependence and output dependence are caused by variable (ie, memory) reuse or reassignment For this reason, they are sometimes called false dependence as ow dependence is called real dependence (Wolfe [204]) Flow dependence is inherent in the computation and cannot be eliminated by renaming variables Nevertheless, the concepts of output and antidependence are important in dependence analysis of concrete programs, for example, in a parallelizing compiler (Banerjee et al [13], Polychronopoulos et al [153]) There, the program is a concrete implementation of an application speci cation and might contain these types of false dependence The compiler must either eliminate the dependence relations or arrange the tasks in precedence order
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