Computational Issues in Construction of 4-D Projective Spaces with Perfect Access Patterns for Higher Primes
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Matrix operations are some of the important computations in scientific and engineering domains. Parallelization approaches for such operations have been a common topic of research. One of the novel approach proposed during the 90s is architectures based on finite projective spaces. A key benefit of this approach is the communication efficiency that can be achieved by exploiting perfect access patterns in the architecture. Such spaces of dimension 4 appear suitable for matrix-matrix multiplication and are amenable for distributions with good performance potential. The construction of such 4-dimensional spaces with perfect access patterns, however, has been reported only for the smallest space – the one corresponding to prime 2. In this paper, we explore the construction for primes greater than 2. We compare two alternative methods for computational construction of such spaces, based on their efficiency. We present the successful construction of such a space for prime 3 and indicate directions for future work.
KeywordsProjective spaces Parallel computations
The authors would like to thank Dr. B. S. Adiga for the many discussions on the concepts of projective geometry, and Prof Milind Sohani for the concepts of algebraic structure of the projective spaces.
- 1.Amrutur, B.S., Joshi, R., Karmarkar, N.K.: A projective geometry architecture for scientific computation. In: Proceedings of the International Conference on Application Specific Array Processors, pp. 64–80, August 1992. https://doi.org/10.1109/ASAP.1992.218581
- 2.D’Azevedo, E.F., Dongarra, J.: The design and implementation of the parallel out-of-core ScaLAPACK LU, QR and cholesky factorization routines. University of Tennessee, Knoxville. Technical report (1997)Google Scholar
- 3.Grama, A., Gupta, A., Karypis, G., Kumar, V.: Introduction to Parallel Computing. Addison–Wesley (2003)Google Scholar
- 4.Karmarkar, N.: A new parallel architecture for sparse matrix computation based on finite projective geometries. In: Proceedings of Supercomputing, pp. 358–369 (1991)Google Scholar
- 5.Karmarkar, N.: Massively parallel systems and global optimization (2008). http://math.mit.edu/crib/08/Extended-abstract.pdf
- 7.Sapre, S., Sharma, H., Patil, A., Adiga, B.S., Patkar, S.: Finite projective geometry based fast, conflict-free parallel matrix computations. https://arxiv.org/abs/1107.1127