We have proposed the extended Karnaugh map representation (EKMH) scheme for multidimensional array representation. We propose two data compression schemes, EKMR compressed row/column storage (ECRS/ECCS), for multidimensional sparse arrays based on the EKMR scheme. To evaluate the proposed schemes, we compare them to the CRS/CCS schemes. Both theoretical analysis and experimental tests were conducted. In the theoretical analysis, we analyze the CRS/CCS and the ECRS/ECCS schemes in terms of the time complexity, the space complexity, and the range of their usability for practical applications. In experimental tests, we compare the compressing time of sparse arrays and the execution time of matrix-matrix addition and matrix-matrix multiplication based on the CRS/CCS and the ECRS/ECCS schemes. The theoretical analysis and experimental results show that the ECRS/ECCS schemes are superior to the CRS/CCS schemes for all the evaluated criteria, except the space complexity in some case.

Array operations are useful in a large number of important scientific codes, such as molecular dynamics, finite element methods, climate modeling, atmosphere and ocean sciences, etc. In our previous work, we have proposed a scheme of extended Karnaugh map representation (EKMR) for multidimensional array representation. We have shown that sequential multidimensional array operation algorithms based on the EKMR scheme have better performance than those based on the traditional matrix representation (TMR) scheme. Since parallel multidimensional array operations have been an extensively investigated problem, we present efficient data parallel algorithms for multidimensional array operations based on the EKMR scheme for distributed memory multicomputers. In a data parallel programming paradigm, in general, we distribute array elements to processors based on various distribution schemes, do local computation in each processor, and collect computation results from each processor. Based on the row, column, and 2D mesh distribution schemes, we design data parallel algorithms for matrix-matrix addition and matrix-matrix multiplication array operations in both TMR and EKMR schemes for multidimensional arrays. We also design data parallel algorithms for six Fortran 90 array intrinsic functions: All, Maxval, Merge, Pack, Sum, and Cshift. We compare the time of the data distribution, the local computation, and the result collection phases of these array operations based on the TMR and the EKMR schemes. The experimental results show that algorithms based on the EKMR scheme outperform those based on the TMR scheme for all test cases.

For sparse array operations, in general, the sparse arrays are compressed by some data compression schemes in order to obtain better performance. The Compressed Row/Column Storage (CRS/CCS) schemes are the two common used data compression schemes for sparse arrays in the traditional matrix representation (TMR). When extended to higher dimensional sparse arrays, array operations using the CRS/CCS schemes usually do not perform well. We propose two data compression schemes, extended Karnaugh map representation Compressed Row/Column Storage (ECRS/ ECCS) for multi-dimensional sparse arrays based on the EKMR scheme. To evaluate the proposed schemes, both theoretical analysis and experimental tests are conducted. In theoretical analysis, we analyze CRS/CCS and ECRS/ECCS schemes in terms of the time complexity, the space complexity, and the range of their usability for practical applications. In experimental test, we compare the performance of matrix-matrix addition and matrix-matrix multiplication sparse array operations that use the CRS/CCS and ECRS/ECCS schemes. The experimental results show that sparse array operations based on the ECRS/ECCS schemes outperform those based on the CRS/CCS schemes for all test samples.

Array operations are used in a large number of important scientific codes. To implement these array operations efficiently, many methods have been proposed in the literature, most of which are focused on two-dimensional arrays. When extended to higher dimensional arrays, these methods usually do not perform well. Hence, designing efficient algorithms for multidimensional array operations becomes an important issue. We propose a new scheme, extended Karnaugh map representation (EKMR), for the multidimensional array representation. The main idea of the EKMR scheme is to represent a multidimensional array by a set of two-dimensional arrays. Hence, efficient algorithm design for multidimensional array operations becomes less complicated. To evaluate the proposed scheme, we design efficient algorithms for multidimensional array operations, matrix-matrix addition/subtraction and matrix-matrix multiplications, based on the EKMR and the traditional matrix representation (TMR) schemes. Theoretical and experimental tests for these array operations were conducted. In the experimental test, we compare the performance of intrinsic functions provided by the Fortran 90 compiler with those based on the EKMR scheme. The experimental results show that the algorithms based on the EKMR scheme outperform those based on the TMR scheme and those provided by the Fortran 90 compiler.

In this paper we use the tensor product notation as the framework of a programming methodology for designing various parallel prefix algorithms. In this methodology, we first express a computational problem in its matrix form. Next, we formulate a matrix equation for the matrix of the computational problem. Then, solve the matrix equation to obtain some simple matrices. Finally, we recursively factorize the subproblem to obtain a tensor product formula representing an algorithm for this problem. We will use the parallel prefix computation problem to illustrate our methodology and derive various parallel prefix algorithms including divide-and-conquer and recursive doubling algorithms.

Abstract To efficiently execute a finite element program on a 2D torus, we need to map nodes of the corresponding finite element graph to processors of a 2D torus such that each processor has approximately the same amount of computational load and the communication among processors is minimized. If nodes of a finite element graph do not increase during the execution of a program, the mapping only needs to be performed once. However, if a finite element graph is solution‐adaptive, that is, nodes of a finite element graph increase discretely due to the refinement of some finite elements during the execution of a program, a dynamic load‐balancing algorithm has to be performed many times in order to balance the computational load of processors while keeping the communication cost as low as possible. In the paper we propose a parallel dynamic load‐balancing algorithm (LB) to deal with the load‐imbalancing problem of a solution‐adaptive finite element program on a 2D torus. The algorithm uses an iterative approach to achieve load‐balancing. We have implemented the proposed algorithm along with two parallel mapping algorithms, parallel orthogonal recursive bisection (ORB) and parallel recursive mincut bipartitioning (MC), on a simulated 2D torus. Three criteria, the execution time of load‐balancing algorithms, the computation time of an application program under different load balancing algorithms, and the total execution time of an application program (under several refinement phases) are used for performance evaluation. Simulation results show that (1) the execution of LB is faster than those of MC and ORB; (2) the mappings of LB are better than those of ORB and MC; and (3) the speedups of LB are better than those of ORB and MC.