Abstract
In the context of mesh-like, parallel processing computers for (i) approximating continuous space and (ii) analog simulation of the motion of objects and waves in continuous space, the present paper is concerned with which mesh-like interconnection of processors might be particularly suitable for the task and why.
Processor interconnection schemes based on nearest neighbor connections in geometric lattices are presented along with motivation. Then two major threads are exploded regarding which lattices would be good: the regular lattices, for their symmetry and other properties in common with continuous space, and the well-known root lattices, for being, in a sense, the lattices required for physically natural basic algorithms for motion.
The main theorem of the present paper implies that the well-known lattice An is the regular lattice having the maximum number of nearest neighbors among the n-dimensional regular lattices. It is noted that the only n-dimensional lattices that are both regular and root are An and Zn (Zn is the lattice of n-cubes. The remainder of the paper specifies other desirable properties of An including other ways it is superior to Zn for our purposes.
- AHUJA, N., AND SWAMY, S. 1984. Multiprocessor pyramid architecture for bottom-up image analysis. IEEE Trans. PAMI PAMI-6, 463-474.Google ScholarDigital Library
- AURENHAMMER, F. 1990. Voronoi diagrams-A survey of a fundamental geometric data structure. Tech. Rep. B-90-9. Institute for Computer Science, Dept. of Mathematics, Freie Universit~t Berlin, Berlin, Germany, November.Google Scholar
- BRACHMAN, R., AND LEVESQUE,H.J.,EDS. 1985. Readings in Knowledge Representation. Morgan- Kaufmann Publishers, Inc., San Francisco, Calif. Google ScholarDigital Library
- CASE, J., CHITOOR, S., RAJAN, D., AND SHENDE, A. 1995. Multi-particle motion in lattice computers. (Revision in preparation.)Google Scholar
- CASE, J., RAJAN, D., AND SHENDE, A. 1991. Simulating uniform motion in lattice computers I: Constant speed particle translation. Tech. Rep. 91-17. Univ. Delaware, Newark, Del., June.Google Scholar
- CASE, J., RAJAN, D., AND SHENDE, A. 1994. Representing the spatial/kinematic domain and lattice computers. J. Exper. Theoret. Artif. Int. 6, 17-40.Google ScholarCross Ref
- CASE, J., RAJAN, D., AND SHENDE, A. 2000. Spherical wave front generation in lattice computers. J. Comput. Inf. 1 (1) (Special Issue: Proceedings of the 6th International Conference on Computing and Information, Peterborough, Ontario, Canada (http://www.cs.tufts.edu/icci/94).Google Scholar
- CHANDRUPATLA, T., AND BELEGUNDU, A. 1991. Introduction to Finite Elements in Engineering. Prentice-Hall, Englewood Cliffs, New Jersey.Google Scholar
- CONWAY, J., AND SLOANE, N. 1993. Sphere Packings, Lattices and Groups. 2nd ed. Springer-Verlag, New York. Google ScholarDigital Library
- COXETER, H. S. M. 1973. Regular Polytopes. Dover Publications, New York.Google Scholar
- DUFF, M. J. B., AND LEVIALDI, S., EDS. 1981. Languages and Architectures for Image Processing. Academic Press, Orlando, Fla.Google Scholar
- DYER, C. R. A quadtree machine for parallel image processing. Tech. Rep. KSL 51. Univ. of Illinois at Chicago Circle, Chicago, Ill.Google Scholar
- FEYNMAN, R. P. 1982. Simulating physics with computers. Int. J. Theoret. Phys. 21, 6/7.Google ScholarCross Ref
- FREDKIN, E., AND TOFFOLI, T. 1982. Conservative logic. Int. J. Theoret. Phys. 21, 3/4.Google ScholarCross Ref
- FRISH, U., HASSLACHER, B., AND POMEAU, Y. 1986. Lattice-gas automata for the Navier Stokes equation. Phys. Rev. Lett. 56, 14 (Apr.) 1505-1508.Google Scholar
- FUNT, B. V. 1980. Problem-solving with diagrammatic representations. Artif. Int. 13, 201-230.Google ScholarCross Ref
- GRUBER, P. M., AND LEKKERKERKER, G. 1987. Geometry of Numbers. North-Holland Mathematical Library, Amsterdam, The Netherlands.Google Scholar
- HASSLACHER, B. 1987. Discrete fluids. Los Alamos Sci. 15, Special Issue, 175-217.Google Scholar
- HERSTEIN, I. N. 1964. Topics in Algebra. Blaisdell Publishing Co., Waltham, Mass.Google Scholar
- HILL, E. L. 1955. Relativistic theory of discrete momentum space and discrete space-time. Phys. Rev. 100, 6 (Dec.).Google ScholarCross Ref
- LEVIALDI, S., ED. 1985. Integrated Technology for Parallel Image Processing. Academic Press, Inc., Orlando, Fla. Google ScholarDigital Library
- MARGOLUS, N. 1984. Physics-like models of computation. Physica 10D, 81-95.Google Scholar
- MINSKY, M. 1982. Cellular vacuum. Int. J. Theoret. Phys. 21, 6/7.Google ScholarCross Ref
- PRESTON, K., AND DUFF, M. J. B. 1984. Modern Cellular Automata: Theory and Applications. Plenum Publishers, New York.Google Scholar
- PRESTON, K., AND UHR, L., EDS. 1982. Multicomputers and Image Processing: Algorithms and Programs. Academic Press, Orlando, Fla. Google ScholarDigital Library
- POULTON, J., FUCHS, H., ET AL. 1985. PIXEL-PLANES: Building a VLSI-based graphic system. In Chapel Hill Conference on VLSI.Google Scholar
- RAJAN, D. S., AND SHENDE, A. M. 1996. A characterization of root lattices. Disc. Math. 161, 309-314. Google ScholarDigital Library
- RAJAN, D. S., AND SHENDE, A. M. 1997. Root lattices are efficiently generated. Int. J. Alg. Comput. 7, 1, 33-50.Google ScholarCross Ref
- RUMELHART, D., AND NORMAN, D. 1974. Representation in memory. In Stevens' Handbook of Experimental Psychology, Vol. 2: Learning and Cognition. R. Atkinson, R. Hernstein, G. Lindzey, and D. Luce, Eds. Wiley, New York.Google Scholar
- SALEM, J. B., AND WOLFRAM, S. 1986. Thermodynamics and hydrodynamics with cellular automata. In Theory and Applications of Cellular Automata, S. Wolfram, Ed. World Scientific.Google Scholar
- SHENDE, A. 1991. Digital analog simulation of uniform motion in representations of physical n-space by lattice-work MIMD computer architectures. TR 91-14, Department of Computer Science. Ph.D. dissertation. SUNY at Buffalo, Buffalo, N.Y. http://www.roanoke.edu/shende/ Thales/Papers/thesis.ps. Google ScholarDigital Library
- SNYDER, H. S. 1947. Quantized space-time. Phys. Rev. 71,1,38-41.Google ScholarCross Ref
- SVOZIL, K. 1986. Are quantum fields cellular automata? Phys. Lett. A, 119, 4 (Dec.).Google ScholarCross Ref
- THINKING MACHINES. 1986. Introduction to data level parallelism. Tech. Rep. 86.14. Thinking Machines, April.Google Scholar
- TOFFOLI, T. 1977a. Cellular automata machines. Tech. Rep. 208, Comp. Comm. Sci. Dept., Univ. Michigan.Google Scholar
- TOFFOLI, T. 1977b. Computation and construction universality of reversible cellular automata. J. Comput. Syst. Sci. 15, 213-231.Google ScholarCross Ref
- TOFFOLI, T. 1984. CAM: A high-performance cellular-automaton machine. Physica 10D, 195-204.Google Scholar
- TOFFOLI, T., AND MARGOLUS, N. 1987. Cellular Automata Machines. MIT Press, Cambridge, Mass. Google ScholarDigital Library
- UHR, L. 1972. Layered 'recognition cone' networks that preprocess, classify, and describe. IEEE Trans. Computers, C-21, 758-768.Google ScholarDigital Library
- UNGER, S. H. 1958. A computer oriented towards spatial problems. Proc. IRE 46, 1744-1750.Google ScholarCross Ref
- VICHNIAC, G. 1984. Simulating physics with cellular automata. Physica 10D, 96-116.Google Scholar
- VITANYI, P. M. B. 1988. Locality, communication and interconnect length in multicomputers. SIAM J. Comput. 17, 4, 659-672. Google ScholarDigital Library
- WOLFRAM, S. 1983. Statistical mechanics of cellular automata. Rev. Mod. Phys. 55, 3 (July) 601-644.Google ScholarCross Ref
- ZUSE, K. 1969/1970. Rechnender Raum. Vieweg, Braunshweig, 1969. (Translated as Calculating Space, Tech. Transl. AZT-70-164-GEMIT, MIT Project MAC, 1970.)Google ScholarCross Ref
Index Terms
- Lattice computers for approximating Euclidean space
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