107 research outputs found
Detecting all regular polygons in a point set
In this paper, we analyze the time complexity of finding regular polygons in
a set of n points. We combine two different approaches to find regular
polygons, depending on their number of edges. Our result depends on the
parameter alpha, which has been used to bound the maximum number of isosceles
triangles that can be formed by n points. This bound has been expressed as
O(n^{2+2alpha+epsilon}), and the current best value for alpha is ~0.068.
Our algorithm finds polygons with O(n^alpha) edges by sweeping a line through
the set of points, while larger polygons are found by random sampling. We can
find all regular polygons with high probability in O(n^{2+alpha+epsilon})
expected time for every positive epsilon. This compares well to the
O(n^{2+2alpha+epsilon}) deterministic algorithm of Brass.Comment: 11 pages, 4 figure
Recognizing Weakly Simple Polygons
We present an O(n log n)-time algorithm that determines whether a given planar n-gon is weakly simple. This improves upon an O(n^2 log n)-time algorithm by [Chang, Erickson, and Xu, SODA, 2015]. Weakly simple polygons are required as input for several geometric algorithms. As such, how to recognize simple or weakly simple polygons is a fundamental question
The Complexity of Order Type Isomorphism
The order type of a point set in maps each -tuple of points to
its orientation (e.g., clockwise or counterclockwise in ). Two point sets
and have the same order type if there exists a mapping from to
for which every -tuple of and the
corresponding tuple in have the same
orientation. In this paper we investigate the complexity of determining whether
two point sets have the same order type. We provide an algorithm for
this task, thereby improving upon the algorithm
of Goodman and Pollack (1983). The algorithm uses only order type queries and
also works for abstract order types (or acyclic oriented matroids). Our
algorithm is optimal, both in the abstract setting and for realizable points
sets if the algorithm only uses order type queries.Comment: Preliminary version of paper to appear at ACM-SIAM Symposium on
Discrete Algorithms (SODA14
Reconfiguration of 3D Crystalline Robots Using O(log n) Parallel Moves
We consider the theoretical model of Crystalline robots, which have been
introduced and prototyped by the robotics community. These robots consist of
independently manipulable unit-square atoms that can extend/contract arms on
each side and attach/detach from neighbors. These operations suffice to
reconfigure between any two given (connected) shapes. The worst-case number of
sequential moves required to transform one connected configuration to another
is known to be Theta(n). However, in principle, atoms can all move
simultaneously. We develop a parallel algorithm for reconfiguration that runs
in only O(log n) parallel steps, although the total number of operations
increases slightly to Theta(nlogn). The result is the first (theoretically)
almost-instantaneous universally reconfigurable robot built from simple units.Comment: 21 pages, 10 figure
Meshes Preserving Minimum Feature Size
The minimum feature size of a planar straight-line graph is the minimum distance between a vertex and a nonincident edge. When such a graph is partitioned into a mesh, the degradation is the ratio of original to final minimum feature size. For an n-vertex input, we give a triangulation (meshing) algorithm that limits degradation to only a constant factor, as long as Steiner points are allowed on the sides of triangles. If such Steiner points are not allowed, our algorithm realizes \ensuremathO(lgn) degradation. This addresses a 14-year-old open problem by Bern, Dobkin, and Eppstein
Minimum feature size preserving decompositions
The minimum feature size of a crossing-free straight line drawing is the
minimum distance between a vertex and a non-incident edge. This quantity
measures the resolution needed to display a figure or the tool size needed to
mill the figure. The spread is the ratio of the diameter to the minimum feature
size. While many algorithms (particularly in meshing) depend on the spread of
the input, none explicitly consider finding a mesh whose spread is similar to
the input. When a polygon is partitioned into smaller regions, such as
triangles or quadrangles, the degradation is the ratio of original to final
spread (the final spread is always greater).
Here we present an algorithm to quadrangulate a simple n-gon, while achieving
constant degradation. Note that although all faces have a quadrangular shape,
the number of edges bounding each face may be larger. This method uses Theta(n)
Steiner points and produces Theta(n) quadrangles. In fact to obtain constant
degradation, Omega(n) Steiner points are required by any algorithm.
We also show that, for some polygons, a constant factor cannot be achieved by
any triangulation, even with an unbounded number of Steiner points. The
specific lower bounds depend on whether Steiner vertices are used or not.Comment: 12 pages, 4 figure
Bichromatic compatible matchings
ABSTRACT For a set R of n red points and a set B of n blue points, a BR-matching is a non-crossing geometric perfect matching where each segment has one endpoint in B and one in R. Two BR-matchings are compatible if their union is also noncrossing. We prove that, for any two distinct BR-matchings M and M , there exists a sequence of BR-matchings M = M1, . . . , M k = M such that Mi−1 is compatible with Mi. This implies the connectivity of the compatible bichromatic matching graph containing one node for each BR-matching and an edge joining each pair of compatible BR-matchings, thereby answering the open problem posed by Aichholzer et al. in [5]
Bichromatic compatible matchings
Abstract For a set R of n red points and a set B of n blue points, a BR-matching is a non-crossing geometric perfect matching where each segment has one endpoint in B and one in R. Two BRmatchings are compatible if their union is also non-crossing. We prove that, for any two distinct BRmatchings M and M , there exists a sequence of BR-matchings M = M 1 , . . . , M k = M such that M i−1 is compatible with M i . This implies the connectivity of the compatible bichromatic matching graph containing one node for each BR-matching and an edge joining each pair of compatible BR-matchings, thereby answering the open problem posed by Aichholzer et al. in [6]
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