R-identifying Code Research Articles

Bounds on r-identifying codes in q-ary Lee space

Identifying codes are used to locate malfunctioning processors in multiprocessor systems. In this paper, we study identifying codes in a $q$-ary hypercube which is used in parallel processing. Computing upper and lower bounds of $M_{r,q}(n),$ the smallest cardinality among all $r$-identifying codes in $\mathbb{Z}_q^n$ with respect to the Lee metric, is an important research problem in this area. Using our constructions, we produce tables for upper and lower bounds for $M_{r,q}(n)$. The upper and the lower bounds of $M_{r,4}(n)$ known only when $r=1$ but using our results, we compute the bounds for $M_{r,4}(n)$ for all $r\geq 1$. Also we improve upon the currently known upper bounds of $M_{1,4}(n)$ due to J. L. Kim and S. J. Kim. Upper bounds of $M_{r,q}(n)$ for $q>4$ are known previously for some cases of $n$. We improve some of these bounds and we also compute bounds for all $n$ by using our results.

The compared costs of domination, location-domination and identification

Let G = (V, E) be a finite graph and r ≥ 1 be an integer. For v ∈ V , let B r (v) = {x ∈ V : d(v, x) ≤ r} be the ball of radius r centered at v. A set C ⊆ V is an r-dominating code if for all v ∈ V , we have B r (v) ∩ C = ∅; it is an r-locating-dominating code if for all v ∈ V , we have B r (v) ∩ C = ∅, and for any two distinct non-codewords x ∈ V C, y ∈ V C, we have B r (x) ∩ C = B r (y) ∩ C; it is an r-identifying code if for all v ∈ V , we have B r (v) ∩ C = ∅, and for any two distinct vertices x ∈ V , y ∈ V , we have B r (x) ∩ C = B r (y) ∩ C. We denote by γ r (G) (respectively, ld r (G) and id r (G)) the smallest possible cardinality of an r-dominating code (respectively, an r-locating-dominating code and an r-identifying code). We study how small and how large the three differences id r (G)−ld r (G), id r (G)−γ r (G) and ld r (G) − γ r (G) can be.

Constructions of r-identifying codes and (r, ≤ l)-identifying codes

In binary Hamming space, we construct r-identifying codes from a $$\sum\limits_{i = 0}^{r - 1} {\left({\matrix{n \cr i \cr}} \right) + \left({\matrix{{n - 1} \cr {r - 1} \cr}} \right) + 1} $$ -fold r-covering code. By using this construction, we modify the construction of r-identifying codes of Charon et al. which helps to find codes of greater length. From this modified construction, we improve the upper bound of M5 (23) which is the smallest possible cardinality of a 5-identifying code of length 23. We also construct (2, ≤ l) -identifying codes and we define the length function L(r, l) as the smallest positive integer n for which there exists an (r, ≤ l)-identifying code in $$\mathbb{F}_{2}^{n}$$ . By using the construction of (2, ≤ l)-identifying codes, we improve the upper bounds of L(2, l) for all l ≥ 6. We also improve the upper bounds of M 2 (≤ ) (n) for all l ≥ 6 and when n is one more than the improved upper bound of L(2, l). We give the construction for (r, ≤ l)-identifying codes. From this result, we prove that M (≤ ) (2r +1) ≤ 22r for certain values of r and l.

R-Identifying codes in binary Hamming space, q-ary Lee space and incomplete hypercube

We study about monotonicity of [Formula: see text]-identifying codes in binary Hamming space, q-ary Lee space and incomplete hypercube. Also, we give the lower bounds for [Formula: see text] where [Formula: see text] is the smallest cardinality among all [Formula: see text]-identifying codes in [Formula: see text] with respect to the Lee metric. We prove the existence of [Formula: see text]-identifying code in an incomplete hypercube. Also, we give the construction techniques for [Formula: see text]-identifying codes in the incomplete hypercubes in Secs. 4.1 and 4.2. Using these techniques, we give the tables (see Tables 1–6) of upper bounds for [Formula: see text] where [Formula: see text] is the smallest cardinality among all [Formula: see text]-identifying codes in an incomplete hypercube with [Formula: see text] processors. Also, we give the exact values of [Formula: see text] for small values of [Formula: see text] and [Formula: see text] (see Sec. 4.3).

Unique (optimal) solutions: Complexity results for identifying and locating–dominating codes

We investigate the complexity of four decision problems dealing with the uniqueness of a solution in a graph: “Uniqueness of an r-Locating–Dominating Code with bounded size” (U-LDCr), “Uniqueness of an Optimal r-Locating–Dominating Code” (U-OLDCr), “Uniqueness of an r-Identifying Code with bounded size” (U-IdCr), “Uniqueness of an Optimal r-Identifying Code” (U-OIdCr), for any fixed integer r≥1.In particular, we describe a polynomial reduction from “Unique Satisfiability of a Boolean formula” (U-SAT) to U-OLDCr, and from U-SAT to U-OIdCr; for U-LDCr and U-IdCr, we can do even better and prove that their complexity is the same as that of U-SAT, up to polynomials. Consequently, all these problems are NP-hard, and U-LDCr and U-IdCr belong to the class DP.

More results on the complexity of identifying problems in graphs

We investigate the complexity of several problems linked with identification in graphs; for instance, given an integer r≥1 and a graph G=(V,E), the existence of, or search for, optimal r-identifying codes in G, or optimal r-identifying codes in G containing a subset of vertices X⊂V. We locate these problems in the complexity classes of the polynomial hierarchy.

On the ensemble of optimal identifying codes in a twin-free graph

Let G = (V, E) be a graph. For v ? V and r ? 1, we denote by BG,r(v) the ball of radius r and centre v. A set C⊆V$C \subseteq V$ is said to be an r-identifying code if the sets BG,r(v)?C$B_{G,r}(v)\cap C$, v ? V, are all nonempty and distinct. A graph G which admits an r-identifying code is called r-twin-free, and in this case the smallest size of an r-identifying code is denoted by ?r(G). We study the ensemble of all the different optimalr-identifying codes C, i.e., such that |C| = ?r(G). We show that, given any collection A$\mathcal {A}$ of k-subsets of V1={1,2,?,n}, there is a positive integer m, a graph G = (V, E) with V=V1?V2$V=V_{1}\cup V_{2}$, where V2 = {n + 1 ,?, n + m}, and a set S⊆V2$S\subseteq V_{2}$ such that C⊆V$C\subseteq V$ is an optimal r-identifying code in G if, and only if, C=A?S$C=A\cup S$ for some A?A$A\in \mathcal {A}$. This result gives a direct connection with induced subgraphs of Johnson graphs, which are graphs with vertex set a collection of k-subsets of V1, with edges between any two vertices sharing k?1 elements.

On the number of optimal identifying codes in a twin-free graph

Let G be a simple, undirected graph with vertex set V. For v∈V and r≥1, we denote by BG,r(v) the ball of radius r and centre v. A set C⊆V is said to be an r-identifying code in G if the sets BG,r(v)∩C, v∈V, are all nonempty and distinct. A graph G which admits an r-identifying code is called r-twin-free or r-identifiable, and in this case the smallest size of an r-identifying code in G is denoted by γrID(G).We study the number of different optimal r-identifying codes C, i.e., such that |C|=γrID(G), that a graph G can admit, and try to construct graphs having “many” such codes.

Minimum sizes of identifying codes in graphs differing by one edge

Let G be a simple, undirected graph with vertex set V. For v ? V and r ? 1, we denote by B G, r (v) the ball of radius r and centre v. A set 𝒞 ⊆ V ${\mathcal C} \subseteq V$ is said to be an r-identifying code in G if the sets B G , r ( v ) ? 𝒞 $B_{G,r}(v)\cap {\mathcal C}$ , v ? V, are all nonempty and distinct. A graph G admitting an r-identifying code is called r-twin-free, and in this case the size of a smallest r-identifying code in G is denoted by ? r (G). We study the following structural problem: let G be an r-twin-free graph, and G ? be a graph obtained from G by adding or deleting an edge. If G ? is still r-twin-free, we compare the behaviours of ? r (G) and ? r (G ?), establishing results on their possible differences and ratios.

Optimal identifying codes in the infinite 3-dimensional king grid

A subset C⊆V is an r-identifying code in a graph G=(V,E) if the sets Ir(v)={c∈C∣d(c,v)≤r} are distinct and non-empty for all vertices v⊆V. Here, d(c,v) denotes the number of edges on any shortest path from c to v. We consider the infinite n-dimensional king grid, i.e., the graph with vertex set V=Zn and the edge set E={{x=(x1,…,xn),y=(y1,…,yn)}∣|xi−yi|≤1for i=1,…,n,x≠y}, and give some lower bounds on the density of an r-identifying code. In particular, we prove that for n=3 and for all r≥15, the optimal density of an r-identifying code is 18r2. The problem finding a minimum identifying code in the 3-dimensional king grid is equivalent with a minimum packing problem of cubes in the 3-dimensional lattice so that every point is covered by a distinct and non-empty subset of cubes.

New lower bound for 2-identifying code in the square grid

An r-identifying code in a graph G=(V,E) is a subset C⊆V such that for each u∈V the intersection of C and the ball of radius r centered at u is nonempty and unique. Previously, r-identifying codes have been studied in various grids. In particular, it has been shown that there exists a 2-identifying code in the square grid with density 5/29≈0.172 and that there are no 2-identifying codes with density smaller than 3/20=0.15. Recently, the lower bound has been improved to 6/37≈0.162 by Martin and Stanton (2010) [11]. In this paper, we further improve the lower bound by showing that there are no 2-identifying codes in the square grid with density smaller than 6/35≈0.171.

On binary linear r-identifying codes

A subspace C of the binary Hamming space F n of length n is called a linear r-identifying code if for all vectors of F n the intersections of C and closed r-radius neighbourhoods are nonempty and different. In this paper, we give lower bounds for such linear codes. For radius r = 2, we give some general constructions. We give many (optimal) constructions which were found by a computer search. New constructions improve some previously known upper bounds for r-identifying codes in the case where linearity is not assumed.

Upper bounds for binary identifying codes

A nonempty set of words in a binary Hamming space F n is called an r-identifying code if for every word the set of codewords within distance r from it is unique and nonempty. The smallest possible cardinality of an r-identifying code is denoted by M r ( n ) . In this paper, we consider questions closely related to the open problem whether M t + r ( n + m ) ⩽ M t ( m ) M r ( n ) is true. For example, we show results like M 1 + r ( n + m ) ⩽ 4 M 1 ( m ) M r ( n ) , which improve previously known bounds. We also consider codes which identify sets of words of size at most ℓ. The smallest cardinality of such a code is denoted by M r ( ⩽ ℓ ) ( n ) . We prove that M r + t ( ⩽ ℓ ) ( n + m ) ⩽ M r ( ⩽ ℓ ) ( n ) M t ( ⩽ ℓ ) ( m ) is true for all ℓ ⩾ r + 3 when r ⩾ 1 and t = 1 . We also obtain a result M 1 ( n + 1 ) ⩽ ( 2 + ε n ) M 1 ( n ) where ε n → 0 when n → ∞ . This bound is related to the conjecture M 1 ( n + 1 ) ⩽ 2 M 1 ( n ) . Moreover, we give constructions for the best known 1-identifying codes of certain lengths.

Optimal t-Edge-Robust r-Identifying Codes in the King Lattice

Locating faulty processors in a multiprocessor system gives a motivation for identifying codes. The concept of a t-edge-robust r-identifying code was introduced in [8]. We consider these codes in the king lattice and give several optimal densities.

Extremal cardinalities for identifying and locating-dominating codes in graphs

Consider a connected undirected graph G = ( V , E ) , a subset of vertices C ⊆ V , and an integer r ⩾ 1 ; for any vertex v ∈ V , let B r ( v ) denote the ball of radius r centred at v , i.e., the set of all vertices linked to v by a path of at most r edges. If for all vertices v ∈ V (respectively, v ∈ V ⧹ C ), the sets B r ( v ) ∩ C are all nonempty and different, then we call C an r-identifying code (respectively, an r-locating-dominating code). We study the extremal values of the cardinality of a minimum r-identifying or r-locating-dominating code in any connected undirected graph G having a given number, n, of vertices. It is known that a minimum r-identifying code contains at least ⌈ log 2 ( n + 1 ) ⌉ vertices; we establish in particular that such a code contains at most n - 1 vertices, and we prove that these two bounds are reached. The same type of results are given for locating-dominating codes.

A family of optimal identifying codes in [formula omitted

Assume that G = ( V , E ) is a simple undirected graph, and C is a nonempty subset of V. For every v ∈ V , we denote I r ( v ) = { u ∈ C | d G ( u , v ) ⩽ r } , where d G ( u , v ) denotes the number of edges on any shortest path between u and v. If the sets I r ( v ) for v ∈ V are pairwise different, and none of them is the empty set, we say that C is an r-identifying code in G. If C is r-identifying in every graph G ′ that can be obtained by adding and deleting edges in such a way that the number of additions and deletions together is at most t, the code C is called t-edge-robust. Let K be the graph with vertex set Z 2 in which two different vertices are adjacent if their Euclidean distance is at most 2 . We show that the smallest possible density of a 3-edge-robust code in K is r + 1 2 r + 1 for all r > 2 .

On the Structure of Identifiable Graphs

Abstract Consider a connected undirected graph G = ( V , E ) , a subset of vertices C ⊆ V , and an integer r ≥ 1 ; for any vertex v ∈ V , let B r ( v ) denote the ball of radius r centered at v, i.e., the set of all vertices linked to v by a path of at most r edges. If for all vertices v ∈ V , the sets B r ( v ) ∩ C are all nonempty and different, then we call C an r-identifying code. A graph is said to be r-identifiable if it admits at least one r-identifying code. We prove the following structural properties of r-identifiable graphs. For any r ≥ 1 , any r-identifiable graph must have at least 2 r + 1 vertices. For r = 1 and for any r-identifiable graph G with at least 2 r + 2 vertices, or for any r ≥ 1 and for any r-identifiable tree G with at least 2 r + 2 vertices, there always exists at least one vertex such that its removing from G leaves an r-identifiable graph. This property is not true for r ≥ 3 in general. The case r = 2 remains open for general graphs.

Identifying codes of cycles

In this paper we deal with identifying codes in cycles. We show that for all r≥1, any r-identifying code of the cycle Cn has cardinality at least gcd(2r+1,n)⌈n2gcd(2r+1,n)⌉. This lower bound is enough to solve the case n even (which was already solved in [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating-dominating codes on chains and cycles, European Journal of Combinatorics 25 (7) (2004) 969–987]), but the case n odd seems to be more complicated. An upper bound is given for the case n odd, and some special cases are solved. Furthermore, we give some conditions on n and r to attain the lower bound.

Identifying and locating-dominating codes on chains and cycles

Consider a connected undirected graph G=( V, E), a subset of vertices C⊆ V, and an integer r≥1; for any vertex v∈ V, let B r ( v) denote the ball of radius r centered at v, i.e., the set of all vertices within distance r from v. If for all vertices v∈ V (respectively, v∈ V ⧹ C), the sets B r ( v)∩ C are all nonempty and different, then we call C an r-identifying code (respectively, an r-locating-dominating code). We study the smallest cardinalities or densities of these codes in chains (finite or infinite) and cycles.

The minimum density of an identifying code in the king lattice

Consider a connected undirected graph G=( V, E) and a subset of vertices C. If for all vertices v∈ V, the sets B r ( v)∩ C are all nonempty and different, where B r ( v) denotes the set of all points within distance r from v, then we call C an r-identifying code. For all r, we give the exact value of the best possible density of an r-identifying code in the king lattice, i.e., the infinite two-dimensional square lattice with two diagonals.