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Parallel kd-Tree Construction on the GPU with an Adaptive Split and Sort Strategy

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Abstract

We introduce a parallel kd-tree construction method for 3-dimensional points on a GPU which employs a sorting algorithm that maintains high parallelism throughout construction. Typically, large arrays in the upper levels of a kd-tree do not yield high performance when computing each node in one thread. Conversely, small arrays in the lower levels of the tree do not benefit from typical parallel sorts. To address these issues, the proposed sorting approach uses a modified parallel sort on the upper levels before switching to basic parallelization on the lower levels. Our work focuses on 3D point registration and our results indicate that a speed gain by a factor of 100 can be achieved in comparison to a naive parallel algorithm for a typical scene.

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Notes

  1. https://nvlabs.github.io/cub.

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Authors and Affiliations

  1. Virtual Reality Applications Center, Iowa State University, Ames, IA, 50011, USA

    David Wehr & Rafael Radkowski

Authors
  1. David Wehr
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  2. Rafael Radkowski
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Correspondence to Rafael Radkowski.

Appendix

Appendix

1.1 Median Splitting Determination

  1. I

    To determine which chunk a point belongs to, we use the technique described in [7] to compute a median. In brief, we consider the width w of a chunk to be a real number \(w=\frac{N}{2^l}\), where l is the zero-indexed tree level. Therefore, given a particular index i, we can determine the chunk by \(c = \frac{i}{w}\).

  2. II

    Determining whether a point is a median splitting element is also necessary during the histogram calculation. That can be determined with the following criteria, as per [7].

    $$\begin{aligned} splitting = {\left\{ \begin{array}{ll} {\left\lceil \frac{i}{w} \right\rceil < \frac{i+1}{w} \wedge i \ne 0}; &{} true \\ \mathrm {else}; &{} false \end{array}\right. } \end{aligned}$$
  3. III

    Finally, we need the ability to calculate the starting index of a chunk, excluding the splitting element. Because the chunk width is constant, the starting index \(i_s\) of a chunk c can be computed by:

    $$\begin{aligned} i_s = {\left\{ \begin{array}{ll} c = 0; &{} 0 \\ i \ne 0; &{} \left\lfloor (w \cdot c) \right\rfloor + 1 \end{array}\right. } \end{aligned}$$

1.2 Experimental Data

See Tables 2 and 3.

Table 2 Quantitative results for all construction operations
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Table 3 Quantitative results for all nearest-neighbor queries
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Wehr, D., Radkowski, R. Parallel kd-Tree Construction on the GPU with an Adaptive Split and Sort Strategy. Int J Parallel Prog 46, 1139–1156 (2018). https://doi.org/10.1007/s10766-018-0571-0

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