171 research outputs found
Connectivity of Graphs Induced by Directional Antennas
This paper addresses the problem of finding an orientation and a minimum
radius for directional antennas of a fixed angle placed at the points of a
planar set S, that induce a strongly connected communication graph. We consider
problem instances in which antenna angles are fixed at 90 and 180 degrees, and
establish upper and lower bounds for the minimum radius necessary to guarantee
strong connectivity. In the case of 90-degree angles, we establish a lower
bound of 2 and an upper bound of 7. In the case of 180-degree angles, we
establish a lower bound of sqrt(3) and an upper bound of 1+sqrt(3). Underlying
our results is the assumption that the unit disk graph for S is connected.Comment: 8 pages, 10 figure
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
Unfolding Orthogrids with Constant Refinement
We define a new class of orthogonal polyhedra, called orthogrids, that can be
unfolded without overlap with constant refinement of the gridded surface.Comment: 19 pages, 12 figure
Epsilon-Unfolding Orthogonal Polyhedra
An unfolding of a polyhedron is produced by cutting the surface and
flattening to a single, connected, planar piece without overlap (except
possibly at boundary points). It is a long unsolved problem to determine
whether every polyhedron may be unfolded. Here we prove, via an algorithm, that
every orthogonal polyhedron (one whose faces meet at right angles) of genus
zero may be unfolded. Our cuts are not necessarily along edges of the
polyhedron, but they are always parallel to polyhedron edges. For a polyhedron
of n vertices, portions of the unfolding will be rectangular strips which, in
the worst case, may need to be as thin as epsilon = 1/2^{Omega(n)}.Comment: 23 pages, 20 figures, 7 references. Revised version improves language
and figures, updates references, and sharpens the conclusio
Unfolding Manhattan Towers
We provide an algorithm for unfolding the surface of any orthogonal
polyhedron that falls into a particular shape class we call Manhattan Towers,
to a nonoverlapping planar orthogonal polygon. The algorithm cuts along edges
of a 4x5x1 refinement of the vertex grid.Comment: Full version of abstract that appeared in: Proc. 17th Canad. Conf.
Comput. Geom., 2005, pp. 204--20
Grid Vertex-Unfolding Orthogonal Polyhedra
An edge-unfolding of a polyhedron is produced by cutting along edges and
flattening the faces to a *net*, a connected planar piece with no overlaps. A
*grid unfolding* allows additional cuts along grid edges induced by coordinate
planes passing through every vertex. A vertex-unfolding permits faces in the
net to be connected at single vertices, not necessarily along edges. We show
that any orthogonal polyhedron of genus zero has a grid vertex-unfolding.
(There are orthogonal polyhedra that cannot be vertex-unfolded, so some type of
"gridding" of the faces is necessary.) For any orthogonal polyhedron P with n
vertices, we describe an algorithm that vertex-unfolds P in O(n^2) time.
Enroute to explaining this algorithm, we present a simpler vertex-unfolding
algorithm that requires a 3 x 1 refinement of the vertex grid.Comment: Original: 12 pages, 8 figures, 11 references. Revised: 22 pages, 16
figures, 12 references. New version is a substantial revision superceding the
preliminary extended abstract that appeared in Lecture Notes in Computer
Science, Volume 3884, Springer, Berlin/Heidelberg, Feb. 2006, pp. 264-27
Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm
We show that every orthogonal polyhedron homeomorphic to a sphere can be
unfolded without overlap while using only polynomially many (orthogonal) cuts.
By contrast, the best previous such result used exponentially many cuts. More
precisely, given an orthogonal polyhedron with n vertices, the algorithm cuts
the polyhedron only where it is met by the grid of coordinate planes passing
through the vertices, together with Theta(n^2) additional coordinate planes
between every two such grid planes.Comment: 15 pages, 10 figure
Unfolding Genus-2 Orthogonal Polyhedra with Linear Refinement
We show that every orthogonal polyhedron of genus g ā¤ 2 can be unfolded without overlap while using only a linear number of orthogonal cuts (parallel to the polyhedron edges). This is the first result on unfolding general orthogonal polyhedra beyond genus- 0. Our unfolding algorithm relies on the existence of at most 2 special leaves in what we call the āunfolding treeā (which ties back to the genus), so unfolding polyhedra of genus 3 and beyond requires new techniques
- ā¦