10.4230/LIPICS.STACS.2010.2465
Dumitrescu, Adrian
Adrian
Dumitrescu
Tóth, Csaba D.
Csaba D.
Tóth
Long Non-crossing Configurations in the Plane
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
2010
Longest non-crossing Hamiltonian path
longest non-crossing Hamiltonian cycle
longest non-crossing spanning tree
approximation algorithm.
Marion, Jean-Yves
Jean-Yves
Marion
Schwentick, Thomas
Thomas
Schwentick
2010
2010-03-09
2010-03-09
2010-03-09
en
urn:nbn:de:0030-drops-24655
10.4230/LIPIcs.STACS.2010
978-3-939897-16-3
1868-8969
10.4230/LIPIcs.STACS.2010
LIPIcs, Volume 5, STACS 2010
27th International Symposium on Theoretical Aspects of Computer Science
2013
5
28
311
322
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Marion, Jean-Yves
Jean-Yves
Marion
Schwentick, Thomas
Thomas
Schwentick
1868-8969
Leibniz International Proceedings in Informatics (LIPIcs)
2010
5
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
12 pages
324178 bytes
application/pdf
Creative Commons Attribution-NoDerivs 3.0 Unported license
info:eu-repo/semantics/openAccess
We revisit several maximization problems for geometric networks design
under the non-crossing constraint, first studied by Alon, Rajagopalan
and Suri (ACM Symposium on Computational Geometry, 1993).
Given a set of $n$ points in the plane in general position (no three points collinear), compute a longest non-crossing configuration composed of straight line segments that is: (a) a matching (b) a Hamiltonian path (c) a spanning tree. Here we obtain new results for (b) and (c), as well as for the Hamiltonian cycle problem:
(i) For the longest non-crossing Hamiltonian path problem,
we give an approximation algorithm with ratio $\frac{2}{\pi+1} \approx 0.4829$. The previous best ratio, due to Alon et al., was $1/\pi \approx 0.3183$. Moreover, the ratio of our algorithm is close to $2/\pi$ on a relatively broad class of instances: for point sets whose perimeter (or diameter) is much shorter than the maximum length matching. The algorithm runs in $O(n^{7/3}\log{n})$ time.
(ii) For the longest non-crossing spanning tree problem, we give an
approximation algorithm with ratio $0.502$ which runs in $O(n \log{n})$ time. The previous ratio, $1/2$, due to Alon et al., was achieved by a quadratic time algorithm. Along the way, we first re-derive the result of Alon et al. with a faster $O(n \log{n})$-time algorithm and a very simple analysis.
(iii) For the longest non-crossing Hamiltonian cycle problem,
we give an approximation algorithm whose ratio is close to $2/\pi$ on a relatively broad class of instances: for point sets with the product
$\bf{\langle}$~diameter~$\times$ ~convex hull size $\bf{\rangle}$ much smaller than the maximum length matching. The algorithm runs in
$O(n^{7/3}\log{n})$ time. No previous approximation results
were known for this problem.
LIPIcs, Vol. 5, 27th International Symposium on Theoretical Aspects of Computer Science, pages 311-322