I have two sets $S,T$ of points in the 2-dimensional plane. I want to find the closest pair of points $s,t$ such that $s \in S$, $t \in T$, and the Euclidean distance between $s,t$ is as small as possible. How efficiently can this be done? Can it be done in $O(n \log n)$ time, where $n = |S|+|T|$?

I know that if I'm given a single set $S$, then it's possible to find the closest pair of points $s,s' \in S$ in $O(n \log n)$ time using a standard divide-and-conquer algorithm. However, that algorithm doesn't seem to generalize to the case of two sets, because there's no connection between the distance between the two closest points within $S$ or $T$ vs. the distance between the two closest points across those sets.

I thought of storing the set $T$ in a $k$-d tree, then for each $s \in S$, using a nearest-neighbor query to find the closest point in $T$ to $s$. However, the worst-case running time of this might be as bad as $O(n^2)$ time. There are results saying that if the points of $T$ are randomly distributed, then the expected running time for each query is $O(\log n)$, so we'd obtain an algorithm with expected running time $O(n \log n)$ if we were guaranteed that the points are randomly distributed -- but I'm looking for an algorithm that will work for any collection of points (not necessarily randomly distributed).

Motivation: An efficient algorithm would be useful for this other question.


Yes, this can be $O(n \log n)$ time. Build a Voronoi diagram for $T$. Then, for each point $s \in S$, find which cell of the Voronoi diagram it is contained in. The center of that cell is the point $t \in T$ that is closest to $s$.

You can build a Voronoi diagram in $O(n \log n)$ time, and each query (find the cell containing $s$) can be done in $O(\log n)$ time, so the total running time is $O(n \log n)$ time.

  • $\begingroup$ Nice, much simpler than what I could come up with :). $\endgroup$ – aelguindy Nov 5 '16 at 4:05
  • $\begingroup$ Nice approach! Links would help though, especially for the query side of things. I could find a Wikipedia page showing that the general point location problem can be solved in $O(\log n)$ time, but is there a nicer way for the special case of Voronoi cells? My searching only turned up this answer, which does it the $O(n)$ way. $\endgroup$ – j_random_hacker Nov 5 '16 at 14:49

I am expanding my comment into an answer, since I figured out a semi-satisfactory answer. This only solves the problem for $L^1$-distance. This answer is basically wrong.

This paper solves the problem of finding the closest pair of points in $d$ dimensions for the case when the sets are separated by a hyperplane in $O(n \log^{d-1} n)$.

For two dimensions, this solves the case in the answer you reference as your primary motivation for your question in $O(n \log n)$. It can also be used to solve the general case of the 2D problem in $O(n \log^2 n)$.

Given two sets $S, T$ of points in 2D, embed them in 3D space, displacing set $S$ by some $-\delta_z$ and set $T$ by $\delta_z$ in the $z$ direction. The choice of $\delta_z$ can be made to not affect the choice of the closest pair of points by taking $\delta_z$ to be smaller than the precision of your input points (and doubling the precision bits for each input coordinate). Use the 3D algorithm from the cited paper.

  • $\begingroup$ +1, but a couple of things about that paper (which I've only just started reading): (i) they only claim to solve the problem for the rectilinear (Manhattan) distance case; (ii) I don't understand why they think that region $P_2$ on p. 2 contains exactly 1 point. I had assumed that $p_m$ is the point in $P$ with median y co-ord in $P$, and $q_m$ is the point in $Q$ with median y co-ord in $Q$, but I don't see how the above could follow from this. $\endgroup$ – j_random_hacker Nov 5 '16 at 14:45
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    $\begingroup$ @j_random_hacker the paper only solves the problem for L1 distance and this answer is wrong :). And I think that's the letter $l$ :). $\endgroup$ – aelguindy Nov 5 '16 at 15:44
  • $\begingroup$ Link is broken :( $\endgroup$ – Keerthana Gopalakrishnan Dec 6 '18 at 7:35

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