67,305 research outputs found
A sharp threshold for a modified bootstrap percolation with recovery
Bootstrap percolation is a type of cellular automaton on graphs, introduced
as a simple model of the dynamics of ferromagnetism. Vertices in a graph can be
in one of two states: `healthy' or `infected' and from an initial configuration
of states, healthy vertices become infected by local rules. While the usual
bootstrap processes are monotone in the sets of infected vertices, in this
paper, a modification is examined in which infected vertices can return to a
healthy state. Vertices are initially infected independently at random and the
central question is whether all vertices eventually become infected. The model
examined here is such a process on a square grid for which healthy vertices
with at least two infected neighbours become infected and infected vertices
with no infected neighbours become healthy. Sharp thresholds are given for the
critical probability of initial infections for all vertices eventually to
become infected.Comment: 45 page
Simple I/O-efficient flow accumulation on grid terrains
The flow accumulation problem for grid terrains takes as input a matrix of
flow directions, that specifies for each cell of the grid to which of its eight
neighbours any incoming water would flow. The problem is to compute, for each
cell c, from how many cells of the terrain water would reach c. We show that
this problem can be solved in O(scan(N)) I/Os for a terrain of N cells. Taking
constant factors in the I/O-efficiency into account, our algorithm may be an
order of magnitude faster than the previously known algorithm that is based on
time-forward processing and needs O(sort(N)) I/Os.Comment: This paper is an exact copy of the paper that appeared in the
abstract collection of the Workshop on Massive Data Algorithms, Aarhus, 200
Three dimensional extension of Bresenham’s algorithm with Voronoi diagram
Bresenham’s algorithm for plotting a two-dimensional line segment is elegant and efficient in its deployment of mid-point comparison and integer arithmetic. It is natural to investigate its three-dimensional extensions. In so doing, this paper uncovers the reason for little prior work. The concept of the mid-point in a unit interval generalizes to that of nearest neighbours involving a Voronoi diagram. Algorithmically, there are challenges. While a unit interval in two-dimension becomes a unit square in three-dimension, “squaring” the number of choices in Bresenham’s algorithm is shown to have difficulties. In this paper, the three-dimensional extension is based on the main idea of Bresenham’s algorithm of minimum distance between the line and the grid points. The structure of the Voronoi diagram is presented for grid points to which the line may be approximated. The deployment of integer arithmetic and symmetry for the three-dimensional extension of the algorithm to raise the computation efficiency are also investigated
Odometer of long-range sandpiles in the torus: mean behaviour and scaling limits
In \cite{Cipriani2016}, the authors proved that with the appropriate
rescaling, the odometer of the (nearest neighbours) Divisible Sandpile in the
unit torus converges to the bi-Laplacian field. Here, we study
-long-range divisible sandpiles similar to those introduced in
\cite{Frometa2018}. We obtain that for , the limiting field
is a fractional Gaussian field on the torus. However, for , we recover the bi-Laplacian field. The central tool for our
results is a careful study of the spectrum of the fractional Laplacian in the
discrete torus. More specifically, we need the rate of divergence of such
eigenvalues as we let the side length of the discrete torus goes to infinity.
As a side result, we construct the fractional Laplacian built from a long-range
random walk. Furthermore, we determine the order of the expected value of the
odometer on the finite grid. \end{abstract}Comment: 35 pages, 4 figure
Symmetry-Based Search Space Reduction For Grid Maps
In this paper we explore a symmetry-based search space reduction technique
which can speed up optimal pathfinding on undirected uniform-cost grid maps by
up to 38 times. Our technique decomposes grid maps into a set of empty
rectangles, removing from each rectangle all interior nodes and possibly some
from along the perimeter. We then add a series of macro-edges between selected
pairs of remaining perimeter nodes to facilitate provably optimal traversal
through each rectangle. We also develop a novel online pruning technique to
further speed up search. Our algorithm is fast, memory efficient and retains
the same optimality and completeness guarantees as searching on an unmodified
grid map
Visual Spike-based Convolution Processing with a Cellular Automata Architecture
this paper presents a first approach for
implementations which fuse the Address-Event-Representation
(AER) processing with the Cellular Automata using FPGA and
AER-tools. This new strategy applies spike-based convolution
filters inspired by Cellular Automata for AER vision
processing. Spike-based systems are neuro-inspired circuits
implementations traditionally used for sensory systems or
sensor signal processing. AER is a neuromorphic
communication protocol for transferring asynchronous events
between VLSI spike-based chips. These neuro-inspired
implementations allow developing complex, multilayer,
multichip neuromorphic systems and have been used to design
sensor chips, such as retinas and cochlea, processing chips, e.g.
filters, and learning chips. Furthermore, Cellular Automata is a
bio-inspired processing model for problem solving. This
approach divides the processing synchronous cells which
change their states at the same time in order to get the solution.Ministerio de Educación y Ciencia TEC2006-11730-C03-02Ministerio de Ciencia e Innovación TEC2009-10639-C04-02Junta de Andalucía P06-TIC-0141
Pore evolution in interstellar ice analogues: simulating the effects of temperature increase
Context. The level of porosity of interstellar ices - largely comprised of
amorphous solid water (ASW) - contains clues on the trapping capacity of other
volatile species and determines the surface accessibility that is needed for
solid state reactions to take place. Aims. Our goal is to simulate the growth
of amorphous water ice at low temperature (10 K) and to characterize the
evolution of the porosity (and the specific surface area) as a function of
temperature (from 10 to 120 K). Methods. Kinetic Monte Carlo simulations are
used to mimic the formation and the thermal evolution of pores in amorphous
water ice. We follow the accretion of gas-phase water molecules as well as
their migration on surfaces with different grid sizes, both at the top growing
layer and within the bulk. Results. We show that the porosity characteristics
change substantially in water ice as the temperature increases. The total
surface of the pores decreases strongly while the total volume decreases only
slightly for higher temperatures. This will decrease the overall reaction
efficiency, but in parallel, small pores connect and merge, allowing trapped
molecules to meet and react within the pores network, providing a pathway to
increase the reaction efficiency. We introduce pore coalescence as a new solid
state process that may boost the solid state formation of new molecules in
space and has not been considered so far.Comment: 9 pages, 8 figures Accepted for publication in A&
Minimum Weight Resolving Sets of Grid Graphs
For a simple graph and for a pair of vertices , we say
that a vertex resolves and if the shortest path from to
is of a different length than the shortest path from to . A set of
vertices is a resolving set if for every pair of vertices
and in , there exists a vertex that resolves and . The
minimum weight resolving set problem is to find a resolving set for a
weighted graph such that is minimum, where is
the weight of vertex . In this paper, we explore the possible solutions of
this problem for grid graphs where . We give
a complete characterisation of solutions whose cardinalities are 2 or 3, and
show that the maximum cardinality of a solution is . We also provide a
characterisation of a class of minimals whose cardinalities range from to
.Comment: 21 pages, 10 figure
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