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Leeuwen, Arbitrary versus periodic storage schemes and tessellations of the plane using one type of polyomino
 Inform. and Control
, 1984
"... Given N distinct memory modules, the elements of an (infinite) array in storage are distributed 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 ..."
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Given N distinct memory modules, the elements of an (infinite) array in storage are distributed 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 Shapiro (IEEE Trans. Comput. C27 (1978), 42l~428) proved that there exists a valid skewing scheme for a template T if and only if T tessellates the plane. A conjecture of Shapiro is settled and it is proved 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. © 1984 Academic Press, Inc. 1.
Tiling Rectangles And Half Strips With Congruent Polyominoes
"... this paper, we present three new polyominoes that tile rectangles, as well as a new family of polyominoes that tile rectangles. We also give three families of polyominoes, each of which tiles an infinite half strip. All previous examples of polyominoes that tile half strips were either already known ..."
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Cited by 5 (4 self)
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this paper, we present three new polyominoes that tile rectangles, as well as a new family of polyominoes that tile rectangles. We also give three families of polyominoes, each of which tiles an infinite half strip. All previous examples of polyominoes that tile half strips were either already known to tile a rectangle, or were later found to tile a rectangle. It is still unknown if every polyomino that tiles a half strip also tiles a rectangle.
A Characterization of Recognizable Picture Languages By Tilings By Finite Sets
, 1999
"... As extension of the Kleene star to pictures, we introduce the operation of tiling. We give a characterization of recognizable picture languages by intersection of tilings by finite sets of pictures. ..."
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Cited by 4 (0 self)
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As extension of the Kleene star to pictures, we introduce the operation of tiling. We give a characterization of recognizable picture languages by intersection of tilings by finite sets of pictures.
TILING SIMPLY CONNECTED REGIONS WITH RECTANGLES
"... Abstract. In 1995, Beauquier, Nivat, Rémila, and Robson showed that tiling of general regions with two rectangles is NPcomplete, except for a few trivial special cases. In a different direction, in 2005, Rémila showed that for simply connected regions by two rectangles, the tileability can be solve ..."
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Abstract. In 1995, Beauquier, Nivat, Rémila, and Robson showed that tiling of general regions with two rectangles is NPcomplete, except for a few trivial special cases. In a different direction, in 2005, Rémila showed that for simply connected regions by two rectangles, the tileability can be solved in quadratic time (in the area). We prove that there is a finite set of at most 10 6 rectangles for which the tileability problem of simply connected regions is NPcomplete, closing the gap between positive and negative results in the field. We also prove that counting such rectangular tilings is #Pcomplete, a first result of this kind. 1.
Isohedral Polyomino Tiling of the Plane
"... A polynomial time algorithm is given for deciding, for a given polyomino P, whether there exists an isohedral tiling of the Euclidean plane by isometric copies of P. The decidability question for general tilings by copies of a single polyomino, or even periodic tilings by copies of a single polyomin ..."
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A polynomial time algorithm is given for deciding, for a given polyomino P, whether there exists an isohedral tiling of the Euclidean plane by isometric copies of P. The decidability question for general tilings by copies of a single polyomino, or even periodic tilings by copies of a single polyomino, remains open.
Tiling the Plane with a Fixed Number of polyominoes
, 2008
"... Deciding whether a finite set of polyominoes tiles the plane is undecidable by reduction from the Domino problem. In this paper, we prove that the problem remains undecidable if the set of instances is restricted to sets of 5 polyominoes. In the case of tiling by translations only, we prove that the ..."
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Deciding whether a finite set of polyominoes tiles the plane is undecidable by reduction from the Domino problem. In this paper, we prove that the problem remains undecidable if the set of instances is restricted to sets of 5 polyominoes. In the case of tiling by translations only, we prove that the problem is undecidable for sets of 11 polyominoes.
SPOTLIGHT TILING
, 711
"... Abstract. This article introduces spotlight tiling, a type of covering for a region which is similar to tiling. The distinguishing aspects of spotlight tiling are that the “tiles ” have elastic size, and that the order of placement is significant. Spotlight tilings are decompositions, or coverings, ..."
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Abstract. This article introduces spotlight tiling, a type of covering for a region which is similar to tiling. The distinguishing aspects of spotlight tiling are that the “tiles ” have elastic size, and that the order of placement is significant. Spotlight tilings are decompositions, or coverings, and can be considered dynamic as compared to typical static tiling methods. A thorough examination of spotlight tilings of rectangles is presented, including the distribution of such tilings according to size, and how the directions of the spotlights themselves are distributed. The spotlight tilings of several other regions are studied, and suggest that further analysis of spotlight tilings will continue to yield elegant results and enumerations. 1.