masterarbeit/dokumentation/generating_random_grids.md

Generating random grids

The Task

Given an amount p = p_1 \cdot p_2\cdot \dots \cdot p_n of random points in an $n$-dimensional unit-cube ([0..1]^n), find a regular grid^[i.e. no intersections of cells, each point inside the grid is connected to 2n points] with grid-dimensions of p_1 \times \dots \times p_n.

The Algorithm

This Algorithm is a simple greedy-algorithm recursing over the dimensionality to construct one valid solution.

Choose two dimensions i and j. Sort the points by dimension i, divide them into chunks of p_j points and sort the points inside these chunks along x_j-x_i.

Now you connect every $k$-th (0 \leq k < p_j) point of each chunk to generate lines along the $i$-th dimension. By definition these paths cannot intersect each other in an projection onto the $i$/$j$-plane (see Proof).

Let the "starting point" (i.e. the smallest in dimension i on the line) be the representative for the line.

Recurse over the other dimensions without choosing dimension i again in a similar way. In the recursive call the "1:1 point merging" will become "1:1 line-merging" (and "1:1 grid-merging") where you connect the $k$-th component of each line/grid.

The resulting grid will be regular because no cell will overlap with another one due to careful construction^[TODO: Proof the same argument as above holds for arbitrary grids using only properties of the representing point] and each point inside gets 2 new neighbors in each of the n calls of the function resulting in 2n neighbors.

Application

For our thesis we only need the cases of n=2 and n=3, but having a higher-dimensional solution around may come in handy in the future.

Proof

It suffices to show that given 2 points of chunk i and 1 point of chunk i+1 there exists no point s.t. the line-segments created by these four points intersect.

Keep in mind that the intersection is in the $i$/$j$-projection and we are working in a unit-square.

Let a,b be the points in the $i$th-chunk and x,y the points in the $i+1$-th chunk. Further shall a,b and x be fixed.

We know (w.l.o.g., by construction), that

• \lbrace a_i, b_i \rbrace < \lbrace x_i, y_i \rbrace
• a_j - a_i < b_j - b_i

Assume y is placed in a way that the generated lines intersect. We have two cases to consider:

1. x_j - x_i < y_j - y_i

In this case the algorithm says that we have to connect x to a and y to b. As we can only choose y, this point has to have a greater distance to b than the $x/a$-line.

\lambda a + (1-\lambda) x for \lambda \in [0..1] are all points on the line.