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Editorial Team
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Asked: 2026-06-17T03:26:07+00:00

I’m learning for the test and I can’t manage with this problem: We are

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I’m learning for the test and I can’t manage with this problem:

We are given a set of n < 1000 points and an integer d. The task is to create two disjoint sets of points A_1 and A_2 (which union is given set of n points) in such way that the distance (euclidean) between each pair of points from A_i (for i = 1 or 2) is less or equal to d. If it is not possible, print -1.

For example, input:

d = 3, and points:

(5,3), (1,1), (4,2), (1,3), (5,2), (2,3), (5,1)

we can create:

A_1 = { (2,3), (1,3), (1,1) }

A_2 = { (5,3), (4,2), (5,2), (5,1) }

since each pair of points from A_i (for i = 1 or 2) are close enough.

I really want to know how to solve it, but no idea. Since n is small, algorithm can be even O(n^2 log n), but I don’t know how to start. I was thinking that maybe start with counting the distance between each pair of points, then take two points with maximum distance and place them in to different sets (if their distance is greater than d). Then repeat this step for the rest of pairs, but the problem is how to decide where I can legally put next points. Can anyone help with this algorithm?

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1 Answer

  1. Editorial Team

    Let’s consider a simple graph with n nodes (corresponding to the n points). Two nodes are connected if the distance between the two corresponding points is greater that d.

    If is is possible to create the two disjoints sets, we must have a bi-partite graphe (if some nodes are not connected to the others, they can be put in any set).

    Thus, we only need to test the bipartiteness of the graph which is simple :

    http://en.wikipedia.org/wiki/Bipartite_graph#Testing_bipartiteness

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