CPM 2006: Combinatorial Pattern Matching pp 117-128

# Tiling an Interval of the Discrete Line

• Olivier Bodini
• Eric Rivals
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4009)

## Abstract

We consider the problem of tiling a segment {0, ..., n} of the discrete line. More precisely, we ought to characterize the structure of the patterns that tile a segment and their number. A pattern is a subset of ℕ. A tiling pattern or tile for {0, ..., n} is a subset $$A \in {{\mathcal P}({\mathbb{N}})}$$ such that there exists $$B \in {{\mathcal P}({\mathbb{N}})}$$ and such that the direct sum of A and B equals {0, ..., n}. This problem is related to the difficult question of the decomposition in direct sums of the torus ℤ/nℤ (proposed by Minkowski). Using combinatorial and algebraic techniques, we give a new elementary proof of Krasner factorizations. We combinatorially prove that the tiles are direct sums of some arithmetic sequences of specific lengths. Besides, we show there are as many tiles whose smallest tilable segment is {0, ..., n} as tiles whose smallest tilable segment is {0, ..., d}, for all strict divisors d of n. This enables us to exhibit an optimal linear time algorithm to compute for a given pattern the smallest segment that it tiles if any, as well as a recurrence formula for counting the tiles of a segment.

## Keywords

Linear Time Algorithm Irreducible Factor Integer Sequence Algebraic Technique Discrete Line
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Beauquier, D., Nivat, M.: Tiling the plane with one tile. In: Proc. 6th Annual Symposium on Computational Geometry (SGC 1990), Berkeley, CA, pp. 128–138. ACM Press, New York (1990)
2. 2.
Beauquier, D., Nivat, M., Remila, E., Robson, J.M.: Tiling figures of the plane with two bars. Computational Geometry: Theory and Applications 5 (1996)Google Scholar
3. 3.
Berger, R.: The undecidability of the domino problem. Mem. Amer. Math. Soc. 66, 1–72 (1966)Google Scholar
4. 4.
Coven, E.M., Meyerowitz, A.D.: Tiling the integers with translates of one finite set. Journal of Algebra 212, 161–174 (1999)
5. 5.
Culik II, K., Kari, J.: On Aperiodic Sets of Wang Tiles. In: Freksa, C., Jantzen, M., Valk, R. (eds.) Foundations of Computer Science. LNCS, vol. 1337, pp. 153–162. Springer, Heidelberg (1997)
6. 6.
de Bruijn, N.G.: On bases for the set of intergers. Publ. Math. Debrecen 1, 232–242 (1950)
7. 7.
De Felice, C.: An application of Hajós factorizations to variable-length codes. Theoretical Computer Science 164(1–2), 223–252 (1996)
8. 8.
Fuchs, L.: Abelian Groups. Oxford Univ. Press, Oxford (1960)
9. 9.
Hajós, G.: Sur la factorisation des groupes abéliens. Cas. Mat. Fys. 74(3), 157–162 (1950)Google Scholar
10. 10.
Hajós, G.: Sur le problème de factorisation des groupes cycliques. Acta Math. Acad. Sci. Hung. 1, 189–195 (1950)
11. 11.
Krasner, M., Ranulak, B.: Sur une propriété des polynômes de la division du cercle. Comptes rendus de l’Académie des Sciences Paris 240, 397–399 (1937)Google Scholar
12. 12.
Lagarias, J.C., Wang, Y.: Tiling the line with translates of one tile. Inventiones Mathematicae 124(1-3), 341–365 (1996)
13. 13.
Lam, N.H.: Hajós factorizations and completion of codes. Theoretical Computer Science 182(1–2), 245–256 (1997)
14. 14.
15. 15.
Penrose, R.: Pentaplexy. Bulletin of the Institute of Mathematics and its Applications 10, 266–271 (1974)Google Scholar
16. 16.
Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences (2004), available at: http://www.research.att.com/~njas/sequences/
17. 17.
Stein, S.K., Szabo, S.: Algebra and Tiling: Homomorphisms in the Service of Geometry. Carus Mathematical Monograph, vol. 25, MAA (1994)Google Scholar
18. 18.
Szabo, S.: Topics in factorization of abelian groups. Birkhauser, Basel (2004)Google Scholar
19. 19.
Thiant, N.: An O(n logn)-algorithm for finding a domino tiling of a plane picture whose number of holes is bounded. Theorical Computer Sciences 303(2-3), 353–374 (2003)
20. 20.
Thurston, W.P.: Conway’s tiling groups. Am. Math. Monthly, 757–773 (October 1990)Google Scholar
21. 21.
Tijdeman, R.: Decomposition of the integers as a direct sum of two subsets. In: David, S. (ed.) Number Theory, pp. 261–276. Oxford Univ. Press, Oxford (1995)Google Scholar