1
$\begingroup$

Given a point set with $n$ points to build a k-d tree on. We have $n$ processors available. What is the time-optimal building time for the k-d tree?

A straight forward parallelization would be as follows: In each step $i$, we have to sort $i$ sets of $n^2/2^i$ points in the dimension with largest spread, which takes $\log(n^2)-i$ time for each of the $\log(n^2)$ levels of the tree, resulting in an upper bound for the total building time of $\Omega(\log^2(n))$.

However, this only holds for a straight forward parallelization. As a k-d tree can be used for sorting by placing all points along one dimension, by e.g. this paper, it therefore has a minimum build time of $\mathcal{O}(\log(n))$.

Can this gap be closed in either direction? That is: Can the k-d tree be built in $\Theta(\log(n))$ or $\Theta(\log^2(n))$?

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.