2,180 research outputs found
Detecting Weakly Simple Polygons
A closed curve in the plane is weakly simple if it is the limit (in the
Fr\'echet metric) of a sequence of simple closed curves. We describe an
algorithm to determine whether a closed walk of length n in a simple plane
graph is weakly simple in O(n log n) time, improving an earlier O(n^3)-time
algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary,
we obtain the first efficient algorithm to determine whether an arbitrary
n-vertex polygon is weakly simple; our algorithm runs in O(n^2 log n) time. We
also describe algorithms that detect weak simplicity in O(n log n) time for two
interesting classes of polygons. Finally, we discuss subtle errors in several
previously published definitions of weak simplicity.Comment: 25 pages and 13 figures, submitted to SODA 201
Diffuse Reflection Diameter in Simple Polygons
We prove a conjecture of Aanjaneya, Bishnu, and Pal that the minimum number
of diffuse reflections sufficient to illuminate the interior of any simple
polygon with walls from any interior point light source is . Light reflecting diffusely leaves a surface in all directions,
rather than at an identical angle as with specular reflections.Comment: To appear in Discrete Applied Mathematic
Memory-Constrained Algorithms for Simple Polygons
A constant-workspace algorithm has read-only access to an input array and may
use only O(1) additional words of bits, where is the size of
the input. We assume that a simple -gon is given by the ordered sequence of
its vertices. We show that we can find a triangulation of a plane straight-line
graph in time. We also consider preprocessing a simple polygon for
shortest path queries when the space constraint is relaxed to allow words
of working space. After a preprocessing of time, we are able to solve
shortest path queries between any two points inside the polygon in
time.Comment: Preprint appeared in EuroCG 201
Packing identical simple polygons is NP-hard
Given a small polygon S, a big simple polygon B and a positive integer k, it
is shown to be NP-hard to determine whether k copies of the small polygon
(allowing translation and rotation) can be placed in the big polygon without
overlap. Previous NP-hardness results were only known in the case where the big
polygon is allowed to be non-simple. A novel reduction from Planar-Circuit-SAT
is presented where a small polygon is constructed to encode the entire circuit
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
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