On the Identification of Sets of Points in the Square Lattice

Abstract

Identifying codes in the square lattice are considered. The motivation for these codes is the following: if a multiprocessor system is modelled by the square lattice, then we can locate faulty processors in the system with the aid of identifying codes. Constructions, some of which are optimal, are given.

Adaptive identification in graphs

“…The first one consists in locating not only one, but at most a fixed number t 1 of faulty vertices, that is to say to study adaptive (r, t)-identifying codes. This problem has already been studied for the non-adaptive version of these codes [10][11][12][13]16]. The second perspective for further research is to consider that vertices have probabilities of defection, which would lead to minimizing the expected number of queries and not only the greatest possible number of queries (worst-case analysis).…”

On a new class of identifying codes in graphs

“…When r = 1 and l 3, there are no (r, l) + -identifying codes, because always I 1 ((0, 0), (0, 2), (2, 1)) = I 1 ((0, 0), (0, 2), (2, 1), (1, 1)). The code C = Z 2 is (1, 3)-identifying (see [10]), and therefore (1, 2) + -identifying (see Theorem 4). No proper subset of Z 2 is (1, 2) + -identifying: if, e.g., (1, 1) / ∈ C, then I 1 ((0, 0), (2, 2)) = I 1 ((0, 0), (1, 1), (2, 2)).…”

“…The smallest possible density of a (1, 2)-identifying code in the infinite square grid is 1/2; see [10]. For (r, 2)-identifying codes the density must be at least 1/8 and there is a sequence of (r, 2)-identifying codes with densities D r such that D r → 1/8, when r → ∞; see [13].…”

On identification in the triangular grid

“…Several other architectures have also been considered: in particular the square lattice (see [1,3,6,7,10,12,14,17]), the king lattice (see [2,9,13]) and the hexagonal mesh (see [3,8]). For identifying codes in binary hypercubes, see, e.g., [15,17] and their references.…”