Covering Paths for Planar Point Sets

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/logn) straight line segments. If the path is required to be noncrossing, we prove that (1?ε)n straight line segments suffice for a small constant ε>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 Ω(nlogn) time in the worst case.