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Let's say for example we have 1000 points and 50 dimensions.

And we build a quadtree where each node represents a 50-dimensional box and is divided by splitting the box into smaller boxes that are half the size in every dimension? How would i calculate how many children each internal node has?

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  • $\begingroup$ If you are interested in high-dimensional quadtree-type indexes, have a look at my PH-Tree. It is a somewhat optimized quadtree that works well with 50 dimensions, it was comparable to R-Tree performance for up to 1000 dimensions. Especially for highly clustered data (I tested with 40 dimensions) it seems to have much faster k-nearest-neighbor queries than kd-trees. The original paper (see link above) also discusses the number of subnodes in a quadtree. $\endgroup$
    – TilmannZ
    Jul 4, 2021 at 17:32

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If you split along each dimension into two parts, then in two dimensions you have $2\cdot 2 = 4$ children (that's why it's called a quadtree), in three dimensions you have $2\cdot 2\cdot 2 = 8$ children (that's why three-dimensional quadtrees are called octtrees), and for $k$ dimensions you have $2^k$ children.

That's independent of the numbers of points in your tree - each internal node in a quadtree splits in the same way.

And for higher dimensions that's a lot of children, and therefore you don't want to use a quadtree in that case, but something else, for example a k-d tree.

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  • $\begingroup$ Quadtrees can actually work quite well with 50+ dimensions if child nodes are only created when they are actually required. At least with the datasets that I tested, such a tree (called PH-Tree <-- self advertisement) they worked either as good (normal data) or a lot better )clustered data) than kd-trees, see here. $\endgroup$
    – TilmannZ
    Jul 4, 2021 at 17:28

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