2,807 research outputs found
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures
We study algorithmic aspects of bending wires and sheet metal into a
specified structure. Problems of this type are closely related to the question
of deciding whether a simple non-self-intersecting wire structure (a
carpenter's ruler) can be straightened, a problem that was open for several
years and has only recently been solved in the affirmative.
If we impose some of the constraints that are imposed by the manufacturing
process, we obtain quite different results. In particular, we study the variant
of the carpenter's ruler problem in which there is a restriction that only one
joint can be modified at a time. For a linkage that does not self-intersect or
self-touch, the recent results of Connelly et al. and Streinu imply that it can
always be straightened, modifying one joint at a time. However, we show that
for a linkage with even a single vertex degeneracy, it becomes NP-hard to
decide if it can be straightened while altering only one joint at a time. If we
add the restriction that each joint can be altered at most once, we show that
the problem is NP-complete even without vertex degeneracies.
In the special case, arising in wire forming manufacturing, that each joint
can be altered at most once, and must be done sequentially from one or both
ends of the linkage, we give an efficient algorithm to determine if a linkage
can be straightened.Comment: 28 pages, 14 figures, Latex, to appear in Computational Geometry -
Theory and Application
Probabilistic Bounds on the Length of a Longest Edge in Delaunay Graphs of Random Points in d-Dimensions
Motivated by low energy consumption in geographic routing in wireless
networks, there has been recent interest in determining bounds on the length of
edges in the Delaunay graph of randomly distributed points. Asymptotic results
are known for random networks in planar domains. In this paper, we obtain upper
and lower bounds that hold with parametric probability in any dimension, for
points distributed uniformly at random in domains with and without boundary.
The results obtained are asymptotically tight for all relevant values of such
probability and constant number of dimensions, and show that the overhead
produced by boundary nodes in the plane holds also for higher dimensions. To
our knowledge, this is the first comprehensive study on the lengths of long
edges in Delaunay graphsComment: 10 pages. 2 figures. In Proceedings of the 23rd Canadian Conference
on Computational Geometry (CCCG 2011). Replacement of version 1106.4927,
reference [5] adde
Universal Guard Problems
We provide a spectrum of results for the Universal Guard Problem, in which one is to obtain a small set of points ("guards") that are "universal" in their ability to guard any of a set of possible polygonal domains in the plane. We give upper and lower bounds on the number of universal guards that are always sufficient to guard all polygons having a given set of n vertices, or to guard all polygons in a given set of k polygons on an n-point vertex set. Our upper bound proofs include algorithms to construct universal guard sets of the respective cardinalities
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