1,268 research outputs found
Chip-firing may be much faster than you think
A new bound (Theorem \ref{thm:main}) for the duration of the chip-firing game
with chips on a -vertex graph is obtained, by a careful analysis of the
pseudo-inverse of the discrete Laplacian matrix of the graph. This new bound is
expressed in terms of the entries of the pseudo-inverse.
It is shown (Section 5) to be always better than the classic bound due to
Bj{\"o}rner, Lov\'{a}sz and Shor. In some cases the improvement is dramatic.
For instance: for strongly regular graphs the classic and the new bounds
reduce to and , respectively. For dense regular graphs -
- the classic and the new bounds reduce to
and , respectively.
This is a snapshot of a work in progress, so further results in this vein are
in the works
The Lattice structure of Chip Firing Games and Related Models
In this paper, we study a famous discrete dynamical system, the Chip Firing
Game, used as a model in physics, economics and computer science. We use order
theory and show that the set of reachable states (i.e. the configuration space)
of such a system started in any configuration is a lattice, which implies
strong structural properties. The lattice structure of the configuration space
of a dynamical system is of great interest since it implies convergence (and
more) if the configuration space is finite. If it is infinite, this property
implies another kind of convergence: all the configurations reachable from two
given configurations are reachable from their infimum. In other words, there is
a unique first configuration which is reachable from two given configurations.
Moreover, the Chip Firing Game is a very general model, and we show how known
models can be encoded as Chip Firing Games, and how some results about them can
be deduced from this paper. Finally, we define a new model, which is a
generalization of the Chip Firing Game, and about which many interesting
questions arise.Comment: See http://www.liafa.jussieu.fr/~latap
On the Classification of Universal Rotor-Routers
The combinatorial theory of rotor-routers has connections with problems of
statistical mechanics, graph theory, chaos theory, and computer science. A
rotor-router network defines a deterministic walk on a digraph G in which a
particle walks from a source vertex until it reaches one of several target
vertices. Motivated by recent results due to Giacaglia et al., we study
rotor-router networks in which all non-target vertices have the same type. A
rotor type r is universal if every hitting sequence can be achieved by a
homogeneous rotor-router network consisting entirely of rotors of type r. We
give a conjecture that completely classifies universal rotor types. Then, this
problem is simplified by a theorem we call the Reduction Theorem that allows us
to consider only two-state rotors. A rotor-router network called the
compressor, because it tends to shorten rotor periods, is introduced along with
an associated algorithm that determines the universality of almost all rotors.
New rotor classes, including boppy rotors, balanced rotors, and BURD rotors,
are defined to study this algorithm rigorously. Using the compressor the
universality of new rotor classes is proved, and empirical computer results are
presented to support our conclusions. Prior to these results, less than 100 of
the roughly 260,000 possible two-state rotor types of length up to 17 were
known to be universal, while the compressor algorithm proves the universality
of all but 272 of these rotor types
The approach to criticality in sandpiles
A popular theory of self-organized criticality relates the critical behavior
of driven dissipative systems to that of systems with conservation. In
particular, this theory predicts that the stationary density of the abelian
sandpile model should be equal to the threshold density of the corresponding
fixed-energy sandpile. This "density conjecture" has been proved for the
underlying graph Z. We show (by simulation or by proof) that the density
conjecture is false when the underlying graph is any of Z^2, the complete graph
K_n, the Cayley tree, the ladder graph, the bracelet graph, or the flower
graph. Driven dissipative sandpiles continue to evolve even after a constant
fraction of the sand has been lost at the sink. These results cast doubt on the
validity of using fixed-energy sandpiles to explore the critical behavior of
the abelian sandpile model at stationarity.Comment: 30 pages, 8 figures, long version of arXiv:0912.320
Computational universality of fungal sandpile automata
Hyphae within the mycelia of the ascomycetous fungi are compartmentalised by
septa. Each septum has a pore that allows for inter-compartmental and
inter-hyphal streaming of cytosol and even organelles. The compartments,
however, have special organelles, Woronin bodies, that can plug the pores. When
the pores are blocked, no flow of cytoplasm takes place. Inspired by the
controllable compartmentalisation within the mycelium of the ascomycetous fungi
we designed two-dimensional fungal automata. A fungal automaton is a cellular
automaton where communication between neighbouring cells can be blocked on
demand. We demonstrate computational universality of the fungal automata by
implementing sandpile cellular automata circuits there. We reduce the Monotone
Circuit Value Problem to the Fungal Automaton Prediction Problem. We construct
families of wires, cross-overs and gates to prove that the fungal automata are
P-complete
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