Optimality of Fundamental Parallel Algorithms on the Hierarchical Memory Machine, with GPU Implementation

Abstract

The Hierarchical Memory Machine (HMM) is a theoretical parallel computing model that captures the essence of CUDA-enabled GPU architecture. It has multiple streaming multiprocessors with a shared memory, and the global memory that can be accessed by all threads. The HMM has several parameters: the number d of streaming multiprocessors, the number p of threads per streaming multiprocessor, the number w of memory banks of each shared memory and the global memory, shared memory latency l, and global memory latency L. The main purpose of this paper is to discuss optimality of fundamental parallel algorithms running on the HMM. We first show that image convolution for an image with n × n pixels using a filter of size (2v+1) × (2v+1) can be done in O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /w+n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> L/dp+n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> v <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /dw+n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> v <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> l/dp) time units on the HMM. Further, we show that this parallel implementation is time optimal by proving the lower bound of the running time. We then go on to show that the product of two n × n matrices can be computed in O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> /mw+n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> L/mdp+n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> /dw+n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> l/dp) time units on the HMM if the capacity of the shared memory in each streaming multiprocessor is O(m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). This implementation is also proved to be time optimal. We further clarify the conditions for image convolution and matrix multiplication to hide the memory access latency overhead and to maximize the global memory throughput and the parallelism. Finally, we provide experimental results on GeForce GTX Titan to support our theoretical analysis.