Reduction in Java

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A reduction operation reduces a collection of data items to a single data item by repeatedly combining
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the data items pairwise with a binary operator, usually one that is associative and commutative Examples of reductions include finding the sum, product, or maximum of all elements in an array In general, we can represent such an operation as the calculation of Equation 61
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where o is a binary operator The most general way to implement a reduction is to perform the calculation serially from left to right, an approach that offers no potential for concurrent execution If o is associative, however, the calculation in Eq 61 contains exploitable concurrency, in that partial results over subsets of v0, , vm 1 can be calculated concurrently and then combined If o is also commutative, the calculation can be not only regrouped, but also reordered, opening up additional possibilities for concurrent execution, as described later Not all reduction operators have these useful properties, however, and one question to be considered is whether the operator can be treated as if it were associative and/or commutative without significantly changing the result of the calculation For example, floating point addition is not strictly associative (because the finite precision with which numbers are represented can cause round off errors, especially if the difference in magnitude between operands is large), but if all the data items to be added have roughly the same magnitude, it is usually close enough to associative to permit the parallelization strategies discussed next If the data items vary considerably in magnitude, this may not be the case Most parallel programming environments provide constructs that implement reduction
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MPI provides general purpose functions MPI_Reduce and MPI_Allreduce as well as support for several common reduction operations (MPI_MIN, MPI_MAX, and MPI_SUM) OpenMP includes a reduction clause that can be applied on a parallel section or a workshare construct It provides for addition, subtraction, multiplication, and a number of bitwise and logical operators
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As an example of a reduction, consider the MPI program in Fig 616 This is a continuation of our earlier example where we used a barrier to enforce consistent timing (Fig 63) The timing is measured for each UE independently As an indication of the load imbalance for the program, it is useful to report the time as a minimum time, a maximum time, and an average time We do this in Fig 616 with three calls to MPI_Reduce one with the MPI_MIN function, one with the MPI_MAX function, and one with the MPI_SUM function An example of reduction in OpenMP can be found in Fig 617 This is an OpenMP version of the program in Fig 616 The variables for the number of threads (num_threads) and the average time (ave_time) are declared prior to the parallel region so they can be visible both inside and following the parallel region We use a reduction clause on the parallel pragma to indicate that we will be doing a reduction with the + operator on the variable ave_time The compiler will create a local copy of the variable and initialize it to the value of the identity of the operator in question (zero for
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addition) Each thread sums its values into the local copy Following the close of the parallel region, all the local copies are combined with the global copy to produce the final value Because we need to compute the average time, we need to know how many threads were used in the parallel region The only way to safely determine the number of threads is by a call to the omp_num_threads () function inside the parallel region Multiple writes to the same variable can interfere with each other, so we place the call to omp_num_threads () inside a single construct, thereby ensuring that only one thread will assign a value to the shared variable num_threads A single in OpenMP implies a barrier, so we do not need to explicitly include one to make sure all the threads enter the timed code at the same time Finally, note that we compute only the average time in Fig 617 because min and max operators are not included in the C/C++ OpenMP version 20
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Figure 616 MPI program to time the execution of a function called runit() We use MPI_Reduce to find minimum, maximum, and average runtimes
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