Abstract
Given a set of points, a covering path is a directed polygonal path that visits all the points. We show that for any n points in the plane, there exists a (possibly self-crossing) covering path consisting of n/2 + O(n/logn) straight line segments. If no three points are collinear, any covering path (self-crossing or non-crossing) needs at least n/2 segments. If the path is required to be non-crossing, n − 1 straight line segments obviously suffice and we exhibit n-element point sets which require at least 5n/9 − O(1) segments in any such path. Further, we show that computing a non-crossing covering path for n points in the plane requires Ω(n logn) time in the worst case.
Dumitrescu acknowledges support from the NSF grant DMS-1001667. Tóth acknowledges support from the NSERC grant RGPIN 35586 and the NSF grant CCF-0830734. Part of the research was conducted at the Fields Institute, Toronto, ON.
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Dumitrescu, A., Tóth, C.D. (2013). Covering Paths for Planar Point Sets. In: Didimo, W., Patrignani, M. (eds) Graph Drawing. GD 2012. Lecture Notes in Computer Science, vol 7704. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36763-2_27
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