1,069 research outputs found
Collective Construction of 2D Block Structures with Holes
In this paper we present algorithms for collective construction systems in
which a large number of autonomous mobile robots trans- port modular building
elements to construct a desired structure. We focus on building block
structures subject to some physical constraints that restrict the order in
which the blocks may be attached to the structure. Specifically, we determine a
partial ordering on the blocks such that if they are attached in accordance
with this ordering, then (i) the structure is a single, connected piece at all
intermediate stages of construction, and (ii) no block is attached between two
other previously attached blocks, since such a space is too narrow for a robot
to maneuver a block into it. Previous work has consider this problem for
building 2D structures without holes. Here we extend this work to 2D structures
with holes. We accomplish this by modeling the problem as a graph orientation
problem and describe an O(n^2) algorithm for solving it. We also describe how
this partial ordering may be used in a distributed fashion by the robots to
coordinate their actions during the building process.Comment: 13 pages, 3 figure
Towards the Erd\H{o}s-Gallai Cycle Decomposition Conjecture
In the 1960's, Erd\H{o}s and Gallai conjectured that the edges of any
-vertex graph can be decomposed into cycles and edges. We improve
upon the previous best bound of cycles and edges due to
Conlon, Fox and Sudakov, by showing an -vertex graph can always be
decomposed into cycles and edges, where is the
iterated logarithm function.Comment: Final version, accepted for publicatio
Fermions and Loops on Graphs. I. Loop Calculus for Determinant
This paper is the first in the series devoted to evaluation of the partition
function in statistical models on graphs with loops in terms of the
Berezin/fermion integrals. The paper focuses on a representation of the
determinant of a square matrix in terms of a finite series, where each term
corresponds to a loop on the graph. The representation is based on a fermion
version of the Loop Calculus, previously introduced by the authors for
graphical models with finite alphabets. Our construction contains two levels.
First, we represent the determinant in terms of an integral over anti-commuting
Grassman variables, with some reparametrization/gauge freedom hidden in the
formulation. Second, we show that a special choice of the gauge, called BP
(Bethe-Peierls or Belief Propagation) gauge, yields the desired loop
representation. The set of gauge-fixing BP conditions is equivalent to the
Gaussian BP equations, discussed in the past as efficient (linear scaling)
heuristics for estimating the covariance of a sparse positive matrix.Comment: 11 pages, 1 figure; misprints correcte
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