What would be smart and fast way to find a pair of closest points in two sets?

Question

For example, I have two lists of points:

List<Point2D> a;
List<Point2D> b;

What would be the best way to find such i and j, so that a.get(i).distance(b.get(j)) is minimal?

The obvious solution is brute-force - calculate distance from each point in a to each point in b, keep the pair with shortest distance. But this algorithm is O(n^2), which is not good. Is there some better approach?

With algorithms like this one, you should always think about what happens if you *sort* your input. — alestanis, Nov 04 '12 at 09:45
@alestanis By what would you sort a set of x/y coordinates to make that happen? — Joachim Isaksson, Nov 04 '12 at 09:47
@alestanis - For example, I sort them by .getX. How can I then get closest pair? — Rogach, Nov 04 '12 at 09:47
@JoachimIsaksson right! I missed the `Point2D`! Sorry about that — alestanis, Nov 04 '12 at 09:48
If one of the lists is stable, then there's a largish literature about indexing low-dimensional spaces. You can then use the index to get better average If both of the lists are different each time, then from memory the best you can do it brute force. (It's an old memory, though, as evidenced by my wrong answer below). — Burleigh Bear, Nov 04 '12 at 09:58

score 3 · Answer 1 · answered Nov 04 '12 at 09:51

3

You can put one of the lists in a quad tree or some other spatial index to make each lookup fast.

As an alternative you could put all your data in a database with spatial index capabilites.

answered Nov 04 '12 at 09:51

Albin Sunnanbo

46,430
8
69
108

score 3 · Accepted Answer · edited May 23 '17 at 12:21

3

For every point of list a you can find the nearest point from list b as described in this answer. Time complexity is O((M+N) log M). N = |A|, M = |B|.

Then you just search the point in a, having the nearest neighbor. Time complexity is O(N).

edited May 23 '17 at 12:21

Community

1
1

answered Nov 04 '12 at 09:54

Evgeny Kluev

24,287
7
55
98

What would be smart and fast way to find a pair of closest points in two sets?

2 Answers2