Exact Minimum Density of Codes Identifying Vertices in the Square Grid

Ben-Haim, Yael; Litsyn, Simon

doi:10.1137/s0895480104444089

Cited by 58 publications

(92 citation statements)

References 19 publications

(31 reference statements)

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“…As before, we can improve the straightforward lower bound of 1 6 using the frame of Figure 10d and discharging rules:…”

Section: Casementioning

confidence: 99%

“…Then the frame on the left and on the top have necessarily four vertices. This is not enough to improve the lower bound of 1 6 , but we can use further analysis. Let C be a ( √ 5, √ 8 − √ 5)-identifying code.…”

Section: Casementioning

confidence: 99%

“…For special values of r, this graph has been considered in many papers, for example, [1,3,6,8,9]. For related results, see [2,12].…”

Section: Introductionmentioning

confidence: 99%

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Tolerant identification with Euclidean balls

2012

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The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. The identifying codes can be applied, for example, to sensor networks. In this paper, we consider as sensors the set Z 2 where one sensor can check its neighbours within Euclidean distance r. We construct tolerant identifying codes in this network that are robust against some changes in the neighbourhood monitored by each sensor. We give bounds for the smallest density of a tolerant identifying code for general values of r. We also provide infinite families of values r with optimal such codes and study the case of small values of r.

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“…As before, we can improve the straightforward lower bound of 1 6 using the frame of Figure 10d and discharging rules:…”

Section: Casementioning

confidence: 99%

Section: Casementioning

confidence: 99%

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Tolerant identification with Euclidean balls

2012

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“…The works in [2,[15][16][17] derive upper/lower bounds on size of the minimum identifying codes, with some providing graph constructions based on relating identifying codes to superimposed codes. The work in [17] focuses on random graphs, providing probabilistic conditions for existence together with bounds.…”

Section: Related Workmentioning

confidence: 99%

Connecting identifying codes and fundamental bounds

Fazlollahi

Starobinski

Trachtenberg

2011

2011 Information Theory and Applications Workshop

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Abstract-We consider the problem of generating a connected robust identifying code of a graph, by which we mean a subgraph with two properties: (i) it is connected, (ii) it is robust identifying, in the sense that the (subgraph-) induced neighborhoods of any two vertices differ by at least 2r + 1 vertices, where r is the robustness parameter. This particular formulation builds upon a rich literature on the identifying code problem but adds a property that is important for some practical networking applications. We concretely show that this modified problem is NP-complete and provide an otherwise efficient algorithm for computing it for an arbitrary graph. We demonstrate a connection between the the sizes of certain connected identifying codes and errorcorrecting code of a given distance. One consequence of this is that robustness leads to connectivity of identifying codes.

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“…Particular interest was dedicated to grids as many processor networks have a grid topology. Many results have been obtained on square grids [4,1,9,2,11], triangular grids [12,10], and hexagonal grids [5,7,8]. In this paper, we study king grids, which are strong products of two paths.…”

Section: Introductionmentioning

confidence: 99%

Minimum density of identifying codes of king grids

Dantas

Havet

Sampaio

2018

Discrete Mathematics

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To cite this version:Rennan Dantas, Rudini Sampaio, Frédéric Havet. The minimum density of an identifying code in G is denoted by d * (G). In this paper, we study the density of king grids which are strong product of two paths. We show that for every king grid G, d * (G) ≥ 2/9. In addition, we show this bound is attained only for king grids which are strong products of two infinite paths. Given k ≥ 3, we denote by K k the (infinite) king strip with k rows. We prove that

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Exact Minimum Density of Codes Identifying Vertices in the Square Grid

Cited by 58 publications

References 19 publications

Tolerant identification with Euclidean balls

Tolerant identification with Euclidean balls

Connecting identifying codes and fundamental bounds

Minimum density of identifying codes of king grids

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