Tiling an Interval of the Discrete Line

Abstract

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 ∈ P(N) such that there exists B ∈ P(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.

Tiling Periodicity

“…In particular we get a recursive formula for computing function L(n) that gives the maximal number of tiling periods for a unary word of length n. The value L(n) might be even more than the text length. Actually, Bodini and Rivals obtain this recursive formula a few months earlier: their paper [3] was submitted in January 2006, while our results were reported only in May 2006 [10]. Here we keep our proof since (comparing to [3]) it construct explicit one-to-one correspondence between tilers and length factorizations, and introduces levels in the set of all tilers.…”

“…Actually, Bodini and Rivals obtain this recursive formula a few months earlier: their paper [3] was submitted in January 2006, while our results were reported only in May 2006 [10]. Here we keep our proof since (comparing to [3]) it construct explicit one-to-one correspondence between tilers and length factorizations, and introduces levels in the set of all tilers.…”

“…(2) Does every word have a unique primitive tiling period? (3) How to find all tiling periods of minimal size? (4) How many periods can a word of length n have?…”

Remarks on Two Nonstandard Versions of Periodicity in Words

“…A list of the first 1000 elements of the sequence (F(n)) n≥1 can be found on the OnLine Encyclopedia of Integer Sequences (A067824) [8]. Bodini and Rivals [2] showed that F(n) is equal to the number of polynomials over x, p(x), with coefficients in the set {0, 1} such that ( ) ( ) also has coefficients in the set {0, 1}.…”

Two-dimensional Minkowski sum optimization of ganged stamping blank layouts for use on pre-cut sheet metal for convex and concave parts

“…More recent examples of a Minkowski representation include three-dimensional modeling of prosthetic teeth [19], onedimensional tiling of discrete lines [3], two-dimensional packing of polygons of dissimilar shapes [6], and laying out of shapes on a fabric or material with specific orientation constraints and tolerances [12,8]. The most relevant of these to the current work are those working in two dimensions, as this paper's problem is a twodimensional layout problem.…”