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Covering paths for planar point sets. (English) Zbl 1294.05102

Summary: Given \(n\) points in the plane, a covering path is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least \(n/2\) segments, and \(n-1\) straight line segments obviously suffice even if the covering path is required to be noncrossing. We show that every set of \(n\) points in the plane admits a (possibly self-crossing) covering path consisting of \(n/2+O(n/\log n)\) straight line segments. If the path is required to be noncrossing, we prove that \((1-{\varepsilon})n\) straight line segments suffice for a small constant \(\varepsilon > 0\), and we exhibit \(n\)-element point sets that require at least \(5n/9-O(1)\) segments in every such path. Further, the analogous question for noncrossing covering trees is considered and similar bounds are obtained. Finally, it is shown that computing a noncrossing covering path for \(n\) points in the plane requires \(\varOmega(n \log n)\) time in the worst case.

MSC:

05C38 Paths and cycles
05C05 Trees
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