194 research outputs found
Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles
We consider a coloring problem on dynamic, one-dimensional point sets: points
appearing and disappearing on a line at given times. We wish to color them with
k colors so that at any time, any sequence of p(k) consecutive points, for some
function p, contains at least one point of each color.
We prove that no such function p(k) exists in general. However, in the
restricted case in which points appear gradually, but never disappear, we give
a coloring algorithm guaranteeing the property at any time with p(k)=3k-2. This
can be interpreted as coloring point sets in R^2 with k colors such that any
bottomless rectangle containing at least 3k-2 points contains at least one
point of each color. Here a bottomless rectangle is an axis-aligned rectangle
whose bottom edge is below the lowest point of the set. For this problem, we
also prove a lower bound p(k)>ck, where c>1.67. Hence for every k there exists
a point set, every k-coloring of which is such that there exists a bottomless
rectangle containing ck points and missing at least one of the k colors.
Chen et al. (2009) proved that no such function exists in the case of
general axis-aligned rectangles. Our result also complements recent results
from Keszegh and Palvolgyi on cover-decomposability of octants (2011, 2012).Comment: A preliminary version was presented by a subset of the authors to the
European Workshop on Computational Geometry, held in Assisi (Italy) on March
19-21, 201
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
Manipulations of task switches during acquisition: A test of the elaboration hypothesis of contextual interference
The effects of practice composition on retention performance of motor skills have been investigated by many researchers (e.g., Shea, Kohl, & Indermill, 1990) and the order of task presentation during practice has been determined to be critical in skill retention. Specifically, presenting a subject with several versions of a task typically produces poorer performance during practice but superior performance during retention, relative to a situation in which the same version of the task is presented repeatedly. This concept is known as contextual interference. Typically, it is the switching between task versions that is considered the critical manipulation, while the consistency of switching is ignored. This experiment was designed to examine the consistency of task switching. Four groups were examined to determine if the consistency of task switches is also important in the retention of a force production task. Although no statistical differences were found between the groups in this experiment, explanations are provided and potential future studies are proposed
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
Polychromatic Coloring for Half-Planes
We prove that for every integer , every finite set of points in the plane
can be -colored so that every half-plane that contains at least
points, also contains at least one point from every color class. We also show
that the bound is best possible. This improves the best previously known
lower and upper bounds of and respectively. We also show
that every finite set of half-planes can be colored so that if a point
belongs to a subset of at least of the half-planes then
contains a half-plane from every color class. This improves the best previously
known upper bound of . Another corollary of our first result is a new
proof of the existence of small size \eps-nets for points in the plane with
respect to half-planes.Comment: 11 pages, 5 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
Colorful Strips
Given a planar point set and an integer , we wish to color the points with
colors so that any axis-aligned strip containing enough points contains all
colors. The goal is to bound the necessary size of such a strip, as a function
of . We show that if the strip size is at least , such a coloring
can always be found. We prove that the size of the strip is also bounded in any
fixed number of dimensions. In contrast to the planar case, we show that
deciding whether a 3D point set can be 2-colored so that any strip containing
at least three points contains both colors is NP-complete.
We also consider the problem of coloring a given set of axis-aligned strips,
so that any sufficiently covered point in the plane is covered by colors.
We show that in dimensions the required coverage is at most .
Lower bounds are given for the two problems. This complements recent
impossibility results on decomposition of strip coverings with arbitrary
orientations. Finally, we study a variant where strips are replaced by wedges
- …