198 research outputs found
Separating bichromatic point sets in the plane by restricted orientation convex hulls
The version of record is available online at: http://dx.doi.org/10.1007/s10898-022-01238-9We explore the separability of point sets in the plane by a restricted-orientation convex hull, which is an orientation-dependent, possibly disconnected, and non-convex enclosing shape that generalizes the convex hull. Let R and B be two disjoint sets of red and blue points in the plane, and O be a set of k=2 lines passing through the origin. We study the problem of computing the set of orientations of the lines of O for which the O-convex hull of R contains no points of B. For k=2 orthogonal lines we have the rectilinear convex hull. In optimal O(nlogn) time and O(n) space, n=|R|+|B|, we compute the set of rotation angles such that, after simultaneously rotating the lines of O around the origin in the same direction, the rectilinear convex hull of R contains no points of B. We generalize this result to the case where O is formed by k=2 lines with arbitrary orientations. In the counter-clockwise circular order of the lines of O, let ai be the angle required to clockwise rotate the ith line so it coincides with its successor. We solve the problem in this case in O(1/T·NlogN) time and O(1/T·N) space, where T=min{a1,âŠ,ak} and N=max{k,|R|+|B|}. We finally consider the case in which O is formed by k=2 lines, one of the lines is fixed, and the second line rotates by an angle that goes from 0 to p. We show that this last case can also be solved in optimal O(nlogn) time and O(n) space, where n=|R|+|B|.Carlos AlegrĂa: Research supported by MIUR Proj. âAHeADâ no 20174LF3T8. David Orden:
Research supported by Project PID2019-104129GB-I00 / AEI / 10.13039/501100011033 of the Spanish
Ministry of Science and Innovation. Carlos Seara: Research supported by Project PID2019-104129GB-I00 /
AEI / 10.13039/501100011033 of the Spanish Ministry of Science and Innovation. Jorge Urrutia: Research
supported in part by SEP-CONACYThis project has received funding from the European Unionâs Horizon 2020 research and innovation
programme under the Marie SkĆodowskaâCurie Grant Agreement No 734922.Peer ReviewedPostprint (published version
Diseño de un algoritmo de minerĂa metaheurĂstico para separar puntos de dos colores en un entorno bidimensional
The separation of color points is one of the important issues in computational geometry, which is used in various parts of science; it can be used in facility locating, image processing and clustering. Among these, one of the most widely used computational geometry in the real-world is the problem of covering and separating points with rectangles. In this paper, we intend to consider the problemof separating the two-color points sets, using three rectangles. In fact, our goal is to separate desired blue points from undesired red points by three rectangles, in such a way that these three rectangles contain the most desire points. For this purpose, we provide a metaheuristic algorithm based on the simulated annealing method that could separates blue points from input points, , in time order O (n) with the help of three rectangles. The algorithm is executed with C# and also it has been compared and evaluated with the optimum algorithm results. The results show that our recommended algorithm responses is so close to optimal responses, and also in some cases we obtains the exact optimal response
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
Metric remote sensing experiments in preparation for Spacelab flights
Aerial and ground photographs of Wallis mountains and of Dolomiti di Cortina d'Ampezzo in Italy were made using spectrozonal emulsions and optical multichannel filters. A metric camera was used in the perspective of the first Spacelab flight aboard the space shuttle. Elementary forms of alpine geomorphology and ice or snow phenomena are detectable on these metric scenes
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On the Parameterized Complexity of Red-Blue Points Separation
We study the following geometric separation problem: Given a set R of red points and a set B of blue points in the plane, find a minimum-size set of lines that separate R from B. We show that, in its full generality, parameterized by the number of lines k in the solution, the problem is unlikely to be solvable significantly faster than the bruteforce nO(k) -time algorithm, where n is the total number of points. Indeed, we show that an algorithm running in time f(k)ná”(k/log k) , for any computable function f, would disprove ETH. Our reduction crucially relies on selecting lines from a set with a large number of different slopes (i.e., this number is not a function of k). Conjecturing that the problem variant where the lines are required to be axis-parallel is FPT in the number of lines, we show the following preliminary result. Separating R from B with a minimum-size set of axis-parallel lines is FPT in the size of either set, and can be solved in time Oâ(9|B|) (assuming that B is the smaller set)
On k-enclosing objects in a coloured point set
We introduce the exact coloured
k
-enclosing object
problem: given a set
P
of
n
points in
R
2
, each of
which has an associated colour in
f
1
;:::;t
g
, and a vec-
tor
c
= (
c
1
;:::;c
t
), where
c
i
2
Z
+
for each 1
i
t
,
nd a region that contains exactly
c
i
points of
P
of
colour
i
for each
i
. We examine the problems of nd-
ing exact coloured
k
-enclosing axis-aligned rectangles,
squares, discs, and two-sided dominating regions in a
t
-coloured point setPostprint (published version
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