On Codes Identifying Vertices in the Two-Dimensional Square Lattice with Diagonals

ÐFault diagnosis of multiprocessor systems motivates the following graph-theoretic definition. A subset C of points in an undirected graph G ˆ …V ;E† is called an identifying code if the sets B…v† \ C consisting of all elements of C within distance one from the vertex v are different. We also require that the sets B…v† \ C are all nonempty. We take G to be the infinite square lattice with diagonals and show that the density of the smallest identifying code is at least 2/9 and at most 4/17. Index TermsÐGraph, square lattice, code, identifying code. æ 1 INTRODUCTION A multiprocessor system can be modeled as an undirected graph G ˆ …V ;E† where V is the set of processors and E is the set of links in the system. Fault diagnosis consists of testing the system and locating faulty processors. For this purpose, a set of processors will be selected and they will be assigned the task of testing their neighbors for malfunctions. Whenever a selected processor detects a fault of any kind among its neighbors or if it malfunctions, an error message is issued that specifies only its origin. What is the minimum number of selected processors needed to identify any faulty processor? Let us reformulate the problem in graph-theoretic terms: G ˆ …V ;E† is an undirected graph (finite or infinite). We denote B…v† ˆ fx 2 V : d…x; v† 1g; the ball of radius one centered at the vertex v 2 V , where d…x; v† equals the number of edges in a shortest path between v and x. If d…x; v† 1, we say that x covers v (and vice versa). A code C is a nonempty subset of V . Its elements are called codewords. The code C is an identifying code if the sets B…v† \ C, v 2 V , are all nonempty and different. In this paper, we consider the case when G is the infinite square grid with diagonals. In other words, the vertex set is V ˆ Z Z a n d t h e e d g e s e t i s E ˆ ffu; vg : uÿ v 2 f…0; 1†; … 1; 0†; …1; 1†; …ÿ1; 1†gg: We denote this graph by T . If we view the vertices as the squares on an infinite chessboard, then B…v† denotes the squares that are a king's move away from v together with the square v itself. The density D of C is defined as D ˆ jCj=jV j if V is finite and, for the infinite graph T , the density is defined as D ˆ lim sup n!1 jC \Qnj jQnj ; where Qn is the set of vertices …x; y† such that jxj n and jyj n. For a given G, we would like to find a code C with minimal density. In fact, instead of this infinite grid, we can consider the finite graphs Tn;m with vertex set V ˆ Z =nZ Z =mZ and edge set E ˆ ffu; vg : uÿ v 2 f…0; 1†; … 1; 0†; …1; 1†; …ÿ1; 1†gg, w h e r e additions are now modulo n in the first and modulo m in the second coordinate. Assume that we have shown that the density of any identifying code in Tn;m has density at least and that this is true for all Tn;m such that n k and m k, where k is a constant. This implies that the density of any identifying code C in T has density at least . Indeed, let C2n‡1;2n‡1 consist of all codewords …x; y† of C such that ÿn < x < n and ÿn < y < n together with all the 8n points …ÿn; a†, …n; a†, …a;ÿn†, …a; n†, where ÿn a n. It is easy to verify that C2n‡1;2n‡1 is an identifying code in T2n‡1;2n‡1 (coordinates now viewed modulo 2n‡ 1). Therefore, jC \Qnj ‡ 8n jC2n‡1;2n‡1j …2n‡ 1†; i.e., jC \Qnj=jQnj ÿ 8n=…2n‡ 1†, from which the claim follows. If we consider the infinite two-dimensional square lattice (without any diagonals), it is proven in [4], [3], and [5] that the smallest identifying code has density at least 15=43 and at most 7=20. If we change the graph a little and add the main diagonal to each square, then the smallest identifying code has density 1=4 [8]. In this paper, we consider the case in which both the diagonals are included as edges. In this case, the easy lower and upper bounds are 1=5 (from inequality (2)) and 1=4. We prove that the lower bound can be improved to 2=9 and the upper bound to 4=17. For the problem of identifying codes, we refer to Karpovsky et al. [8] and [9]. The papers [1] and [2] give further results in the Hamming spaces and [6] for the hexagonal mesh. For a closely related problem in which B…v† \ C are required to be different only for noncodewords v 2 V n C, see Haynes et al. [7]. Such a set C is called a locating-dominating set. Any identifying code is a locatingdominating set. However, for instance, for the infinite twodimensional square lattice (without diagonals), it was shown by Slater [10] that the smallest possible density equals 3=10.