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/ log n) 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 log n) time in the worst case.
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