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Asked: 2026-06-09T08:51:44+00:00

I am trying to take sets of points and split them up into smaller

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I am trying to take sets of points and split them up into smaller sets. The constraint is that each set has some minimum and some maximum for each of their dimensions. I want to generate all possible combinations of these sets (let’s call this a set of sets.) When I am done, each point appears in exactly one set in each set o’ sets.

As an example, let’s say I just have data points that have two independent variables, i and j. They are:

(1,1) (1,2) (2,2) (3,1),(2,1),(2,3)

Any of these splits are fine:

(1,1)(1,2) and (2,2)(3,2)(2,1)(2,3)
First set has i < 2, second set has i >= 2.

(1,1)(3,1)(2,1) and (1,2)(2,2)(2,3)
First set has j < 2, second set has j >= 2.

(1,1)(1,2) and (2,2)(3,1)(2,1) and empty and (2,3)
First set has (i < 2, j < 3), second set has (i >= 2, j < 3)
Third set has (i < 2, j >= 3), fourth set has (i >= 2, j >= 3)

How can I generate the entire set of splits without manually iterating through every point (distinct numbers)! times?

This isn’t homework, just a program I am trying to write as part of a data-fitter.

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1 Answer

  1. Editorial Team

    Assuming the intent is to have at most one dividing point in each dimension (thus partitioning the points into at most two sets with respect to that dimension), then:

    For each dimension, let V be the set of coordinates in that dimension. E.g., given the points (4, 10), (4, 20), (6, 10), (6, 18), (7, 3), then, for the first dimension, V is {4, 6, 7}. Iterate v through each value in the set.

    Nest those iterations, one for each dimension, so that, in the body of the innermost loop, you have a v for each dimension. We will number them, v0, v1, v2,…

    Each v forms a criterion: Either x < v or v <= x. For n dimensions, there are 2n combinations of these criteria. Each combination specifies a subset of the original points. E.g., one such subset is { p | x0 < v0 and v1 <= x1 and v2 <= x2 and… }, where the point p has coordinates (x0, x1, x2,…). So, iterate through the 2n potential subsets (some identical since they are empty), and collect them into a set. That is a partition of the original set of points.

    When you are done iterating the v’s through their values, you will have constructed each partition matching your criteria.

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