An Efficient Representation of Higher Dimensional Arrays and its Evaluation

Abstract

Scientilic and engineering computing requires storing and operating on flooded amount of data having very high number of dimensions. Traditional multidimensional array is widely popular for implementing higher dimensional data but its performance diminishes with increased number of dimensions. On the other side, traditional row-column view of two-dimensional data is facile for implementation, imagination and visualization. This thesis represents a scheme for higher dimensional array implementation and operation with row-column abstraction which can fit an n-dimensional array into a single 2-dimensional array. A mathematical function fits odd dimensions along row-direction and even dimensions along column direction which gives lower index computation cost, higher data locality and better sequential access of memory. Performance of the proposed matricization is measured with matrix-matrix addition/subtraction and multiplication operation which give 70% and 72% improvement respectively for dense data. But most real world data is sparse and degree of data sparsity increases with increased number of dimensions. A loop transformation technique which access odd dimensions fast and then even dimensions is proposed to store any dimensional sparse arrays. In traditional scheme, n numbers of one-dimensional auxiliary arrays are necessary to store n-dimensional array but our scheme requires two one-dimensional auxiliary arrays only which gives 16 times space improvement for 32-dimensional sparse data. Traditionally, the compression ratio is inversely proportional to the number of dimensions but it is independent of number of dimensions in our scheme. The operation on stored sparse data is measured with matrix-matrix addition/subtraction and multiplication which show up to 70% improvement.