New bounds for (r, ≤ 2)-identifying codes in the infinite king grid

2009 ◽  
Vol 2 (1) ◽  
pp. 41-47 ◽  
Author(s):  
Mikko Pelto
Keyword(s):  
Identifying Codes ◽  
King Grid

scholarly journals An improved lower bound for $(1,\leq 2)$-identifying codes in the king grid

2014 ◽  
Vol 8 (1) ◽  
pp. 35-52 ◽  
Author(s):  
Aline Parreau ◽  
Tero Laihonen ◽  
Florent Foucaud
Keyword(s):  
Lower Bound ◽  
Identifying Codes ◽  
King Grid

scholarly journals On Identifying Codes in the King Grid that are Robust Against Edge Deletions

10.37236/727 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Iiro Honkala ◽  
Tero Laihonen
Keyword(s):  
Shortest Path ◽  
Euclidean Distance ◽  
Undirected Graph ◽  
Identifying Codes ◽  
Vertex Set ◽  
King Grid

Assume that $G = (V, E)$ is an undirected graph, and $C \subseteq V$. For every $v \in V$, we denote $I_r(G;v) = \{ u \in C: d(u,v) \leq r\}$, where $d(u,v)$ denotes the number of edges on any shortest path from $u$ to $v$. If all the sets $I_r(G;v)$ for $v \in V$ are pairwise different, and none of them is the empty set, the code $C$ is called $r$-identifying. If $C$ is $r$-identifying in all graphs $G'$ that can be obtained from $G$ by deleting at most $t$ edges, we say that $C$ is robust against $t$ known edge deletions. Codes that are robust against $t$ unknown edge deletions form a related class. We study these two classes of codes in the king grid with the vertex set ${\Bbb Z}^2$ where two different vertices are adjacent if their Euclidean distance is at most $\sqrt{2}$.

scholarly journals Improved bounds on identifying codes in binary Hamming spaces

2010 ◽  
Vol 31 (3) ◽  
pp. 813-827 ◽  
Author(s):  
Geoffrey Exoo ◽  
Ville Junnila ◽  
Tero Laihonen ◽  
Sanna Ranto
Keyword(s):  
Identifying Codes

Watching Systems in the King Grid

2012 ◽  
Vol 29 (3) ◽  
pp. 333-347 ◽  
Author(s):  
David Auger ◽  
Iiro Honkala
Keyword(s):  
King Grid

scholarly journals Lower Bounds for Identifying Codes in some Infinite Grids

10.37236/394 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ryan Martin ◽  
Brendon Stanton
Keyword(s):  
Lower Bounds ◽  
Finite Graph ◽  
Infinite Graphs ◽  
Locally Finite ◽  
Identifying Codes ◽  
Identifying Code ◽  
Hexagonal Grids ◽  
Closed Neighborhood ◽  
Definition Of ◽  
Infinite Grids

An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and unique. On a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.

scholarly journals General Bounds for Identifying Codes in Some Infinite Regular Graphs

10.37236/1583 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Irène Charon ◽  
Iiro Honkala ◽  
Olivier Hudry ◽  
Antoine Lobstein
Keyword(s):  
Undirected Graph ◽  
Upper Bounds ◽  
Regular Graphs ◽  
Lower And Upper Bounds ◽  
Identifying Codes ◽  
Identifying Code

Consider a connected undirected graph $G=(V,E)$ and a subset of vertices $C$. If for all vertices $v \in V$, the sets $B_r(v) \cap C$ are all nonempty and pairwise distinct, where $B_r(v)$ denotes the set of all points within distance $r$ from $v$, then we call $C$ an $r$-identifying code. We give general lower and upper bounds on the best possible density of $r$-identifying codes in three infinite regular graphs.

Sequences of optimal identifying codes

2002 ◽  
Vol 48 (3) ◽  
pp. 774-776 ◽  
Author(s):  
T.K. Laihonen
Keyword(s):  
Identifying Codes

Robust Localization Using Identifying Codes

2009 ◽  
pp. 321-347
Author(s):  
Moshe Laifenfeld ◽  
Ari Trachtenberg ◽  
David Starobinski
Keyword(s):  
Experimental Data ◽  
Information Theory ◽  
Real Life ◽  
Signal Propagation ◽  
Identifying Codes ◽  
Sensor Failures ◽  
Robust Localization ◽  
Potential Benefits ◽  
Set Up

Various real-life environments are exceptionally harsh for signal propagation, rendering well-known trilateration techniques (e.g. GPS) unsuitable for localization. Alternative proximity-based techniques, based on placing sensors near every location of interest, can be fairly complicated to set up, and are often sensitive to sensor failures or corruptions. The authors propose a different paradigm for robust localization based on identifying codes, a concept borrowed from the information theory literature. This chapter describes theoretical and practical considerations in designing and implementing such a localization infrastructure, together with experimental data supporting the potential benefits of the proposed technique.

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