2,440 research outputs found
Quasi-Parallel Segments and Characterization of Unique Bichromatic Matchings
Given n red and n blue points in general position in the plane, it is
well-known that there is a perfect matching formed by non-crossing line
segments. We characterize the bichromatic point sets which admit exactly one
non-crossing matching. We give several geometric descriptions of such sets, and
find an O(nlogn) algorithm that checks whether a given bichromatic set has this
property.Comment: 31 pages, 24 figure
Euclidean TSP with few inner points in linear space
Given a set of points in the Euclidean plane, such that just points
are strictly inside the convex hull of the whole set, we want to find the
shortest tour visiting every point. The fastest known algorithm for the version
when is significantly smaller than , i.e., when there are just few inner
points, works in time [Knauer and Spillner,
WG 2006], but also requires space of order . The best
linear space algorithm takes time [Deineko, Hoffmann, Okamoto,
Woeginer, Oper. Res. Lett. 34(1), 106-110]. We construct a linear space
time algorithm. The new insight is extending the
known divide-and-conquer method based on planar separators with a
matching-based argument to shrink the instance in every recursive call. This
argument also shows that the problem admits a quadratic bikernel.Comment: under submissio
FC-Planner: A Skeleton-guided Planning Framework for Fast Aerial Coverage of Complex 3D Scenes
3D coverage path planning for UAVs is a crucial problem in diverse practical
applications. However, existing methods have shown unsatisfactory system
simplicity, computation efficiency, and path quality in large and complex
scenes. To address these challenges, we propose FC-Planner, a skeleton-guided
planning framework that can achieve fast aerial coverage of complex 3D scenes
without pre-processing. We decompose the scene into several simple subspaces by
a skeleton-based space decomposition (SSD). Additionally, the skeleton guides
us to effortlessly determine free space. We utilize the skeleton to efficiently
generate a minimal set of specialized and informative viewpoints for complete
coverage. Based on SSD, a hierarchical planner effectively divides the large
planning problem into independent sub-problems, enabling parallel planning for
each subspace. The carefully designed global and local planning strategies are
then incorporated to guarantee both high quality and efficiency in path
generation. We conduct extensive benchmark and real-world tests, where
FC-Planner computes over 10 times faster compared to state-of-the-art methods
with shorter path and more complete coverage. The source code will be open at
https://github.com/HKUST-Aerial-Robotics/FC-Planner.Comment: Submitted to ICRA2024. 6 Pages, 6 Figures, 3 Tables. Code:
https://github.com/HKUST-Aerial-Robotics/FC-Planner. Video:
https://www.bilibili.com/video/BV1h84y1D7u5/?spm_id_from=333.999.0.0&vd_source=0af61c122e5e37c944053b57e313025
The traveling salesman problem for lines, balls and planes
We revisit the traveling salesman problem with neighborhoods (TSPN) and
propose several new approximation algorithms. These constitute either first
approximations (for hyperplanes, lines, and balls in , for ) or improvements over previous approximations achievable in comparable times
(for unit disks in the plane).
\smallskip (I) Given a set of hyperplanes in , a TSP tour
whose length is at most times the optimal can be computed in
time, when is constant.
\smallskip (II) Given a set of lines in , a TSP tour whose
length is at most times the optimal can be computed in polynomial
time for all .
\smallskip (III) Given a set of unit balls in , a TSP tour
whose length is at most times the optimal can be computed in polynomial
time, when is constant.Comment: 30 pages, 9 figures; final version to appear in ACM Transactions on
Algorithm
A PTAS for Euclidean TSP with Hyperplane Neighborhoods
In the Traveling Salesperson Problem with Neighborhoods (TSPN), we are given
a collection of geometric regions in some space. The goal is to output a tour
of minimum length that visits at least one point in each region. Even in the
Euclidean plane, TSPN is known to be APX-hard, which gives rise to studying
more tractable special cases of the problem. In this paper, we focus on the
fundamental special case of regions that are hyperplanes in the -dimensional
Euclidean space. This case contrasts the much-better understood case of
so-called fat regions.
While for an exact algorithm with running time is known,
settling the exact approximability of the problem for has been repeatedly
posed as an open question. To date, only an approximation algorithm with
guarantee exponential in is known, and NP-hardness remains open.
For arbitrary fixed , we develop a Polynomial Time Approximation Scheme
(PTAS) that works for both the tour and path version of the problem. Our
algorithm is based on approximating the convex hull of the optimal tour by a
convex polytope of bounded complexity. Such polytopes are represented as
solutions of a sophisticated LP formulation, which we combine with the
enumeration of crucial properties of the tour. As the approximation guarantee
approaches , our scheme adjusts the complexity of the considered polytopes
accordingly.
In the analysis of our approximation scheme, we show that our search space
includes a sufficiently good approximation of the optimum. To do so, we develop
a novel and general sparsification technique to transform an arbitrary convex
polytope into one with a constant number of vertices and, in turn, into one of
bounded complexity in the above sense. Hereby, we maintain important properties
of the polytope
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