Locating and identifying codes in dihedral graphs

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.

Download Full-text

General Bounds for Identifying Codes in Some Infinite Regular Graphs

The Electronic Journal of Combinatorics ◽

10.37236/1583 ◽

2001 ◽

Vol 8 (1) ◽

Cited By ~ 18

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.

Download Full-text

Sequences of optimal identifying codes

IEEE Transactions on Information Theory ◽

10.1109/18.986043 ◽

2002 ◽

Vol 48 (3) ◽

pp. 774-776 ◽

Cited By ~ 21

Author(s):

T.K. Laihonen

Keyword(s):

Identifying Codes

Download Full-text

Robust Localization Using Identifying Codes

Localization Algorithms and Strategies for Wireless Sensor Networks ◽

10.4018/978-1-60566-396-8.ch013 ◽

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.

Download Full-text