arXiv subjects

Ville Junnila

Publications and source records attributed to Ville Junnila.

21 records · Page 2Linked to original sources

arXiv2012

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

An $r$-identifying code in a graph $G = (V,E)$ is a subset $C \subseteq V$ such that for each $u \in V$ 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 $5/29 \approx 0.172$ and that there are no 2-identifying codes with density smaller than $3/20 = 0.15$. Recently, the lower bound has been improved to $6/37 \approx 0.162$ by Martin and Stanton (2010). 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 $6/35 \approx 0.171$.

math.CO↗

arXiv2012

Tolerant identification with Euclidean balls

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 and Delta. We also provide infinite families of values (r,Delta) with optimal such codes and study the case of small values of r.

cs.IT↗

arXiv2012

Optimal lower bound for 2-identifying code in the hexagonal grid

An $r$-identifying code in a graph $G = (V,E)$ is a subset $C \subseteq V$ such that for each $u \in V$ the intersection of $C$ and the ball of radius $r$ centered at $u$ is non-empty 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 hexagonal grid with density 4/19 and that there are no 2-identifying codes with density smaller than 2/11. Recently, the lower bound has been improved to 1/5 by Martin and Stanton (2010). In this paper, we prove that the 2-identifying code with density 4/19 is optimal, i.e. that there does not exist a 2-identifying code in the hexagonal grid with smaller density.

math.CO↗