PARALLEL SYSTEMS AND PROGRAMMING in .NET

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Dependence in Scheduling As will be seen later in Section 35, the graph representation of programs used in task scheduling only addresses real data dependence Other dependence types are bound to a given implementation and are not inherent in the represented computation Thus, a graph represents either an application speci cation with the inherent real data dependence relations or a concrete program, where all false dependence relations have been eliminated Based on this convention, every data dependence in a graph is equivalent to a data transfer, called communication, between the respective tasks
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252 Data Dependence in Loops Data dependence in loops (ie, iterative (or cyclic) computations) is conceptually identical to the data dependence in linear programs as described previously For the parallelization of a program, loops are very attractive, since they typically consume the lion s share of the total execution time of the program They are different from linear programs in two aspects: (1) loops form a regular computation the same statements are executed multiple times; and (2) loops often contain array variables Both aspects together have an impact on the dependence analysis of loops The following discussion of data dependence in loops is restricted to ow dependence, yet the de nitions and conclusions are valid for all types of data dependence Single Loops Example 5 presents a loop over the index variable i with a loop body, or loop kernel, consisting of the two statements, or tasks, S and T Example 5 Single Loop for i = 2 to 100 do S: A(i) = B(i+2) + 7 T: B(i 2+1) = A(i-1) + C(i+1) end for During the execution of the loop, i takes 2, 3, , 100 as its values For each value, an instance of the loop body is executed, and this instance is called an iteration of the loop In each iteration, instances of the tasks S and T are executed and the instances corresponding to the iteration for i = j are denoted by S( j) and T( j) Unlike in the previous discussion where the variables used in the examples are scalars, the variables in the tasks S and T are elements of the disjoint arrays A, B, and C Thus, data dependences arise from the reference to the same array element of different task instances The two tasks read from and write to elements of the arrays A and B, potentially creating a dependence relation Elements of C are only read in task T, therefore not causing any dependence relation Within one iteration, that is, between the instances S( j) and T(k), with j = k, it can easily be veri ed that there is no dependence: for any value of i = j, S( j) and T( j) access different elements of A and B S( j) writes to A( j) and T( j) reads from
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A( j 1), but i never equals i 1 for i = 2, , 100 Likewise for array B, since i + 2 = i 2 + 1 for i = 2, , 100 If such a dependence exists, it is referred to as intraiteration dependence Dependence does arise between instances of different iterations Example 6 shows the rst four iterations, similar to the linear programs studied in the previous section
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Example 6 First Iterations of Example 5 S(2): A(2) = B(4) + 7 T(2): B(5) = A(1) + C(3) S(3): A(3) = B(5) + 7 T(3): B(7) = A(2) + C(4) S(4): A(4) = B(6) + 7 T(4): B(9) = A(3) + C(5) S(5): A(5) = B(7) + 7 T(5): B(11) = A(4) + C(6)
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Thus, it is easy to spot that instance T(3) is ow dependent on S(2): the output variable of task S(2), A(2), is the input of T(3) The same can be observed for the relation between T(4) and S(3), and between T(5) and S(4), caused by other elements of array A The dependence distance, that is, the number of iterations between the two task instances forming a dependence relation, is 1 In general, it can be stated that the instance T(i + 1) is dependent on the instance S(i) Such a dependence is called uniform since its distance is constant Analogous to the intraiteration dependence, this type is referred to as interiteration dependence For ow dependence relations, intraand interiteration dependences correspond to intra- and interiteration communications, respectively Note that a task can even depend on itself, under the condition that the dependence distance is greater than 0 Another logical step is to consider intraiteration dependence as a special case of interiteration dependence, namely, with distance 0 In fact, dependence analysis for loops, for example, as presented by Banerjee et al [17], does not distinguish between intra- and interiteration dependence However, it will prove useful for the treatment of graph representations in 3 In Example 6 one observes further dependence relations The instance S(3) is ow dependent on T(2), caused by the array element B(5), and S(5) is ow dependent on T(3), caused by B(7) These are two examples of the dependence pattern caused by the output variable B(i 2 + 1) of T and the input variable B(i + 2) of S Other dependence pairs of this pattern are: S(7) depends on T(4), S(9) depends on T(5), , S(99) depends on T(50) Since the distance of these dependence relations is not constant the minimum distance is 1 and the maximum is 49 it is said to be nonuniform
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