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- W2951864268 abstract "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 $bf{ }$ 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." @default.
- W2951864268 created "2019-06-27" @default.
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- W2951864268 date "2009-09-22" @default.
- W2951864268 modified "2023-09-27" @default.
- W2951864268 title "Long non-crossing configurations in the plane" @default.
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