On a dispersion problem in grid labeling

Minghui Jiang (1) , Vincent Pilaud (2) , Pedro Tejada (1)

1 Departement of Computer Science
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

Given $k$ labelings of a finite $d$-dimensional cubical grid, define the combined distance between two labels to be the sum of the $l_1$-distance between the two labels in each labeling. We want to construct $k$ labelings which maximize the minimum combined distance between any two labels. When $d = 1$, this can be interpreted as placing $n$ nonattacking rooks in a $k$-dimensional chessboard of size $n$ in such a way to maximize the minimum $l_1$-distance between any two rooks. Rook placements are also known as Latin hypercube designs in the literature. In this paper, we revisit this problem with a more geometric approach. Instead of providing explicit but complicated formulas, we construct rook placements in a $k$-dimensional chessboard of size $n$ as certain lattice-like structures for certain well-chosen values of $n$. Then, we extend these constructions to any values of $n$ using geometric arguments. With this method, we present a clean and geometric description of the known optimal rook placements in the two-dimensional square grid. Furthermore, we provide asymptotically optimal constructions of $k$ labelings of $d$-dimensional cubical grids which maximize the minimum combined distance. Finally, we discuss the extension of this problem to labelings of an arbitrary graph. We prove that deciding whether a graph has two labelings with combined distance at least 3 is at least as hard as graph isomorphism.

Mots clés: Combined distance; Rook placement; Latin hypercube design; Graph labeling

Domaines: Combinatoire [math.CO]

Minghui Jiang, Vincent Pilaud, Pedro Tejada. On a dispersion problem in grid labeling. SIAM Journal on Discrete Mathematics, 2012, 26 (1), pp.39-51. ⟨10.1137/100815281⟩. ⟨hal-00778091⟩