Abstract
Given N parallel memory modules, we like to distribute the elements of an (infinite) array in storage such that any set of N elements arranged according to a given data template T can be accessed rapidly in parallel. Array embeddings that allow for this are called skewing schemes and have been studied in connection with vector-processing and SIMD machines. In 1975 H.D. Shapiro proved that there exists a valid skewing scheme for a template T if and only if T tessellates the plane. We settle a conjecture of Shapiro and prove that for polyominos P a valid skewing scheme exists if and only if there exists a valid periodic skewing scheme. (Periodicity implies a rapid technique to locate data elements.) The proof shows that when a polyomino P tessellates the plane without rotations or reflections, then it can tessellate the plane periodically, i.e., with the instances of P arranged in a lattice. It is also proved that there is a polynomial time algorithm to decide whether a polyomino tessellates the plane, assuming the polyominos in the tessellation should all have an equal orientation.
A full version of this paper is available as [10].
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
7. References
Budnik, P. and D.J. Kuck, The organisation and use of parallel memories, IEEE Trans. Comput. C-20 (1971) 1566–1569.
Gardner, M., Mathematical games, articles in the Scientific American: july 1975, pp. 112–117
Gardner, M., Mathematical games, articles in the Scientific American: august 1975, pp. 112–115
Gardner, M., Mathematical games, articles in the Scientific American: december 1975, pp. 116–119
Gardner, M., Mathematical games, articles in the Scientific American: january 1977, pp. 110–121
Göbel, F., Geometrical packing and covering problems, in: A. Schrijver (ed.), Packing and covering in combinatorics, Math. Centre Tracts 106, Mathematisch Centrum, Amsterdam, 1979, pp. 179–199.
Golomb, S.W., Tiling with polyominos, J. Combin. Theory 1 (1966) 280–296.
Golomb, S.W., Tiling with sets of polyominos, J. Combin. Theory 9 (1970) 60–71.
Levi, P., Sur une generalisation du theorème de Rolle, Comp. Rend. Acad. Sci. Paris 198 (1934) 424–425.
Shapiro, H.D., Theoretical limitations on the efficient use of parallel memories, IEEE Trans. Comput. C-27 (1978) 421–428.
Thurber, K.J., Large scale computer architecture: parallel and associative processors, Hayden Book Comp., Rochelle Park NJ, 1976.
Wang, H., Games, logic and computers, Scientific American 213 (1965) 98–106.
Wijshoff, H.A.G., and J. van Leeuwen, Periodic versus arbitrary tessellations of the plane using polyominos of a single type, Techn. Rep. RUU-CS-82-11, Dept. of Computer Science, University of Utrecht, Utrecht, 1982 (submitted for publication).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1982 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wijshoff, H.A.G., van Leeuwen, J. (1982). Periodic versus arbitrary tessellations of the plane using polyominos of a single type. In: Cremers, A.B., Kriegel, HP. (eds) Theoretical Computer Science. Lecture Notes in Computer Science, vol 145. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0036495
Download citation
DOI: https://doi.org/10.1007/BFb0036495
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-11973-9
Online ISBN: 978-3-540-39421-1
eBook Packages: Springer Book Archive