Memory Optimized Dynamic Matrix Chain Multiplication Using Shared Memory in GPU

Abstract

Number of multiplications needed for Matrix Chain Multiplication of $n$ matrices depends not only on the dimensions but also on the order to multiply the chain. The problem is to find the optimal order of multiplication. Dynamic programming takes $O\left( {n^{3} } \right)$ time, along with $O\left( {n^{2} } \right)$ space in memory for solving this problem. Now-a-days, Graphics Processing Unit (GPU) is very useful to the developers for parallel programming using CUDA computing architecture. The main contribution of this paper is to recommend a new memory optimized technique to solve the Matrix Chain Multiplication problem in parallel using GPU, mapping diagonals of calculation tables $m[][]$ and $s[][]$ into a single combined calculation table of size $O\left( {n^{2} } \right)$ for better memory coalescing in the device. Besides optimizing the memory requirement, a versatile technique of utilizing Shared Memory in Blocks of threads is suggested to minimize time for accessing dimensions of matrices in GPU. Our experiment shows best ever Speedup as compared to sequential CPU implementation, run on large problem size.