1,804 research outputs found
Automated Discharging Arguments for Density Problems in Grids
Discharging arguments demonstrate a connection between local structure and
global averages. This makes it an effective tool for proving lower bounds on
the density of special sets in infinite grids. However, the minimum density of
an identifying code in the hexagonal grid remains open, with an upper bound of
and a lower bound of . We present a new, experimental framework for producing discharging
arguments using an algorithm. This algorithm replaces the lengthy case analysis
of human-written discharging arguments with a linear program that produces the
best possible lower bound using the specified set of discharging rules. We use
this framework to present a lower bound of on
the density of an identifying code in the hexagonal grid, and also find several
sharp lower bounds for variations on identifying codes in the hexagonal,
square, and triangular grids.Comment: This is an extended abstract, with 10 pages, 2 appendices, 5 tables,
and 2 figure
Improved Bounds for -Identifying Codes of the Hex Grid
For any positive integer , an -identifying code on a graph is a set
such that for every vertex in , the intersection of the
radius- closed neighborhood with is nonempty and pairwise distinct. For
a finite graph, the density of a code is , which naturally extends
to a definition of density in certain infinite graphs which are locally finite.
We find a code of density less than , which is sparser than the prior
best construction which has density approximately .Comment: 12p
On Vertex Identifying Codes For Infinite Lattices
PhD Thesis--A compilation of the papers: "Lower Bounds for Identifying Codes
in Some Infinite Grids", "Improved Bounds for r-identifying Codes of the Hex
Grid", and "Vertex Identifying Codes for the n-dimensional Lattics" along with
some other resultsComment: 91p
Finding codes on infinite grids automatically
We apply automata theory and Karp's minimum mean weight cycle algorithm to
minimum density problems in coding theory. Using this method, we find the new
upper bound for the minimum density of an identifying
code on the infinite hexagonal grid, down from the previous record of .Comment: 18 pages, 5 figure
An improved lower bound for (1,<=2)-identifying codes in the king grid
We call a subset of vertices of a graph a -identifying
code if for all subsets of vertices with size at most , the sets
are distinct. The concept of
identifying codes was introduced in 1998 by Karpovsky, Chakrabarty and Levitin.
Identifying codes have been studied in various grids. In particular, it has
been shown that there exists a -identifying code in the king grid
with density 3/7 and that there are no such identifying codes with density
smaller than 5/12. Using a suitable frame and a discharging procedure, we
improve the lower bound by showing that any -identifying code of
the king grid has density at least 47/111
Factoring Banded Permutations and Bounds on the Density of Vertex Identifying Codes on the Infinite Snub Hexagonal Grid
A permutation may be characterized as b-banded when it moves no element more than b places. Every permutation may be factored into 1-banded permutations. We prove that an upper bound on the number of tridiagonal factors necessary is 2b-1, verifying a conjecture of Gilbert Strang. A vertex identifying code of a graph is a subset D of the graph\u27s vertices with the property that for every pair of vertices v1 and v2, N[v1]∩D and N[v2]∩D are distinct and nonempty, where N[v] is the set of all vertices adjacent to v, including v. We compute an upper bound of 1/3 and a strict lower bound of 3/10 for the minimum density of a vertex identifying code on the infinite snub hexagonal grid
Optimal local identifying and local locating-dominating codes
We introduce two new classes of covering codes in graphs for every positive
integer . These new codes are called local -identifying and local
-locating-dominating codes and they are derived from -identifying and
-locating-dominating codes, respectively. We study the sizes of optimal
local 1-identifying codes in binary hypercubes. We obtain lower and upper
bounds that are asymptotically tight. Together the bounds show that the cost of
changing covering codes into local 1-identifying codes is negligible. For some
small optimal constructions are obtained. Moreover, the upper bound is
obtained by a linear code construction. Also, we study the densities of optimal
local 1-identifying codes and local 1-locating-dominating codes in the infinite
square grid, the hexagonal grid, the triangular grid, and the king grid. We
prove that seven out of eight of our constructions have optimal densities
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