A Geometric Approach for Partitioning N-Dimensional Non-rectangular Iteration Spaces

Abstract

Parallel loops account for the greatest percentage of program parallelism. The degree to which parallelism can be exploited and the amount of overhead involved during parallel execution of a nested loop directly depend on partitioning, i.e., the way the different iterations of a parallel loop are distributed across different processors. Thus, partitioning of parallel loops is of key importance for high performance and efficient use of multiprocessor systems. Although a significant amount of work has been done in partitioning and scheduling of rectangular iteration spaces, the problem of partitioning of non-rectangular iteration spaces-e.g. triangular, trapezoidal iteration spaces-has not been given enough attention so far. In this paper, we present a geometric approach for partitioning N-dimensional non-rectangular iteration spaces for optimizing performance on parallel processor systems. Speedup measurements for kernels (loop nests) of linear algebra packages are presented.

Trends in Digital Video Steganography: A Survey

“…It is an image coding technique in which the video frames are divided into the rectangles with the varying dimension for approximating the values in sub-triangle optimal quadratic approximation [27] is used which is obtained using a bivariate polynomial. Each frame of the stego video will be rectangular partitioned the partitioned codes can be an encrypted version of the original frame.…”

A novel approach for partitioning iteration spaces with variable densities

“…The transformation reduces the expressions of l k , u k and s k into canonical form which makes the iteration space amenable for uniform partitioning along i k . The significance of presence of an invariant iterator lies in the fact it obviates geometric [12] and variable density considerations during the partitioning phase and the fact that partitioning along the axis corresponding to an invariant iterator yields perfect load balance.…”

A general approach for partitioning N-dimensional parallel nested loops with conditionals

“…Efficient execution of these programs on parallel machines requires partitioning of the iteration spaces of these nested loops, so that different sets of the partition can be mapped on to the different processors. Though partitioning of loop nests with both rectangular and nonrectangular iteration spaces have been addressed in the past [1,2,3,4,5,6], however, in previous research the underlying loop model was assumed to be free of any control flow. Clearly, this is very restrictive as most of the parallel loops in high performance applications contain conditionals.…”

“…The loop thus obtained is then partitioned amongst the different processors for minimizing execution time, using our geometric iteration partitioning approach proposed in [1,2]. The rest of the paper is organized as follows -Section 2 presents the terminology used in the rest of the paper.…”

“…The transformation plays a critical role during the (static) partitioning process as it helps to capture the "exact" lower and upper bounds (can be either a constant or symbolic) of the loop indices. We generate a canonical form of the loop nest using the transformation and employ the geometric approach we proposed earlier (in [1,2]) for partitioning the iteration space along an axis corresponding to the outermost loop. For cases in which such a transformation does not exist, we propose a general approach for loop canonicalization.…”