Tolerant identification with Euclidean balls

Abstract

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.

Codes for locating objects in sensor networks

“…For the second part of the proposition, we know by Proposition 6 that it will be true for r big enough, but it is harder to find the codes for all the values of r (this is done in [7], one code for r = 4 is given below). We now consider the case when S r,Δ ((0, 0), (−1, 0)) has four elements: Theorem 8.…”

“…In the following x 1 (r, Δ) denotes the point after which the set S r,Δ (u, v), u − v = (1, 0), always contains a point of Z 2 on every vertical line with abscissa x ∈ (x 1 (r, Δ), r). The exact value of x 1 (r, Δ) is easy to calculate (see the full paper [7]). …”

“…For special values of r, this graph has been considered in many papers, for example, for r = 1 in BenHaim and Litsyn [1] and for r = √ 2 in Charon et al [3] and in [6] for general r. For related results, see [2], [5] and [9].…”

“…This paper is based on a full paper [7] (available also in Arxiv), where the reader can find all the detailed proofs.…”

Identification of points using disks

“…The complexity of identifying codes in unit interval graphs is surprisingly still open [9] (but has been proved to be NP-complete for interval graphs). Junnila and Laihonen [11] studied identifying codes in the grid Z 2 using Euclidean balls. The underlying graph has the set Z 2 as vertices and the closed neighbourhood are given by the Euclidean balls of a fixed radius r. This graph can also be seen as an (infinite) unit disk graph.…”