Long Non-crossing Configurations in the Plane

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 2 π+1 ≈ 0.4829. The previous best ratio, due to Alon et al., was 1/π ≈ 0.3183. Moreover, the ratio of our algorithm is close to 2/π 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(n7/3 log n) time. (ii) For the longest non-crossing spanning tree prob...

• Type: Conference
• Year: 2010
• Where: STACS
• Authors: Noga Alon, Sridhar Rajagopalan, Subhash Suri
• Keywords: Algorithm, Alon Et, Approximation Algorithm, STACS 2010, Theoretical Computer Science