17,119 research outputs found
Computing largest circles separating two sets of segments
A circle separates two planar sets if it encloses one of the sets and its
open interior disk does not meet the other set. A separating circle is a
largest one if it cannot be locally increased while still separating the two
given sets. An Theta(n log n) optimal algorithm is proposed to find all largest
circles separating two given sets of line segments when line segments are
allowed to meet only at their endpoints. In the general case, when line
segments may intersect times, our algorithm can be adapted to
work in O(n alpha(n) log n) time and O(n \alpha(n)) space, where alpha(n)
represents the extremely slowly growing inverse of the Ackermann function.Comment: 14 pages, 3 figures, abstract presented at 8th Canadian Conference on
Computational Geometry, 199
Computing Largest Circles Separating Two Sets of Segments
International audienceA circle C separates two planar sets if it encloses one of the sets and its open interior disk does not meet the other set. A separating circle is a largest one if it cannot be locally increased while still separating the two given sets. An Theta(n log n) optimal algorithm is proposed to find all largest circles separating two given sets of line segments when line segments are allowed to meet only at their endpoints. In the general case, when line segments may intersect Omega(n^2) times, our algorithm can be adapted to work in O(n alpha(n) log n) time and O(n alpha(n)) space, where alpha(n) represents the extremely slowly growing inverse of the Ackermann function
Covering Points by Disjoint Boxes with Outliers
For a set of n points in the plane, we consider the axis--aligned (p,k)-Box
Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together
contain n-k points. In this paper, we consider the boxes to be either squares
or rectangles, and we want to minimize the area of the largest box. For general
p we show that the problem is NP-hard for both squares and rectangles. For a
small, fixed number p, we give algorithms that find the solution in the
following running times:
For squares we have O(n+k log k) time for p=1, and O(n log n+k^p log^p k time
for p = 2,3. For rectangles we get O(n + k^3) for p = 1 and O(n log n+k^{2+p}
log^{p-1} k) time for p = 2,3.
In all cases, our algorithms use O(n) space.Comment: updated version: - changed problem from 'cover exactly n-k points' to
'cover at least n-k points' to avoid having non-feasible solutions. Results
are unchanged. - added Proof to Lemma 11, clarified some sections - corrected
typos and small errors - updated affiliations of two author
Extensions of the Maximum Bichromatic Separating Rectangle Problem
In this paper, we study two extensions of the maximum bichromatic separating
rectangle (MBSR) problem introduced in \cite{Armaselu-CCCG, Armaselu-arXiv}.
One of the extensions, introduced in \cite{Armaselu-FWCG}, is called
\textit{MBSR with outliers} or MBSR-O, and is a more general version of the
MBSR problem in which the optimal rectangle is allowed to contain up to
outliers, where is given as part of the input. For MBSR-O, we improve the
previous known running time bounds of to . The other extension is called \textit{MBSR among circles} or
MBSR-C and asks for the largest axis-aligned rectangle separating red points
from blue unit circles. For MBSR-C, we provide an algorithm that runs in time.Comment: 14 pages, 14 figures, full version of CCCG pape
Motion Planning of Legged Robots
We study the problem of computing the free space F of a simple legged robot
called the spider robot. The body of this robot is a single point and the legs
are attached to the body. The robot is subject to two constraints: each leg has
a maximal extension R (accessibility constraint) and the body of the robot must
lie above the convex hull of its feet (stability constraint). Moreover, the
robot can only put its feet on some regions, called the foothold regions. The
free space F is the set of positions of the body of the robot such that there
exists a set of accessible footholds for which the robot is stable. We present
an efficient algorithm that computes F in O(n2 log n) time using O(n2 alpha(n))
space for n discrete point footholds where alpha(n) is an extremely slowly
growing function (alpha(n) <= 3 for any practical value of n). We also present
an algorithm for computing F when the foothold regions are pairwise disjoint
polygons with n edges in total. This algorithm computes F in O(n2 alpha8(n) log
n) time using O(n2 alpha8(n)) space (alpha8(n) is also an extremely slowly
growing function). These results are close to optimal since Omega(n2) is a
lower bound for the size of F.Comment: 29 pages, 22 figures, prelininar results presented at WAFR94 and IEEE
Robotics & Automation 9
Configuration spaces of rings and wickets
The main result in this paper is that the space of all smooth links in
Euclidean 3-space isotopic to the trivial link of n components has the same
homotopy type as its finite-dimensional subspace consisting of configurations
of n unlinked Euclidean circles (the "rings" in the title). There is also an
analogous result for spaces of arcs in upper half-space, with circles replaced
by semicircles (the "wickets" in the title). A key part of the proofs is a
procedure for greatly reducing the complexity of tangled configurations of
rings and wickets. This leads to simple methods for computing presentations for
the fundamental groups of these spaces of rings and wickets as well as various
interesting subspaces. The wicket spaces are also shown to be K(G,1)'s.Comment: 28 pages. Some revisions in the expositio
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