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The closest pair of points problem deals with the task to find a pair of points with the global minimum distance. There is a problem, when all points share the same x-coordinate, or at least a large number of points.

I just don't know why, I heard the running time becomes $n^2$

What about it? What is the problem, since the proof for the algorithm shows, that at max. 8 points can reside in the $2 \delta \times \delta$ area.

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    $\begingroup$ I think you may be confused. $n^2$ is brute force... $\endgroup$ – Lev Reyzin Nov 17 '11 at 22:12
  • $\begingroup$ Yes I know that. But if all points share the same x-coordinate, all the points land in the same set in the deivide step, don't they? $\endgroup$ – GugenMinded Nov 17 '11 at 22:35
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Points with the same x-coordinate do not cause any substantial problem. However, if you implement the divide-and-conquer algorithm carelessly, they may cause a problem. One way to deal with them is by using symbolic perturbation.

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The input then is boring, and the algorithm is bored it takes it much longer to solve the problem.

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    $\begingroup$ There is no such thing as a boring input. $\endgroup$ – Tsuyoshi Ito Nov 18 '11 at 0:19
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    $\begingroup$ Theorem: There is no such thing as an interesting natural number. Proof: Let n be the smallest interesting natural number. Who cares? QED. $\endgroup$ – Jeffε Nov 22 '11 at 9:51

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