On Separating Points by Lines

Given a set $$P$$ P of n points in the plane, its separability is the minimum number of lines needed to separate all its pairs of points from each other, denoted by $$\mathrm {sep}\left( {P}\right) $$ sep P . We show that the minimum number of lines needed to separate n points, picked randomly (and uniformly) in the unit square, is $$\Bigl .{\widetilde{\Theta }}( n^{2/3})$$ Θ ~ ( n 2 / 3 ) , where $${\widetilde{\Theta }}$$ Θ ~ hides polylogarithmic factors. In addition, we provide a fast $$O(\log (\mathrm {sep}\left( {P}\right) ))$$ O ( log ( sep P ) ) -approximation algorithm for computing the separability of a given point set in the plane. Finally, we point out the connection between separability and partitions.

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