Parallel Nearest Neighbors in Low Dimensions with Batch Updates

TL;DR

This paper introduces a parallel algorithm for exact k-nearest neighbors in low dimensions using a novel zd-tree data structure, achieving high efficiency, speed, and support for dynamic batch updates.

Contribution

It presents the zd-tree, a new data structure combining kd-tree and Morton ordering, with proven efficiency and practical speed, and supports dynamic batch updates for k-nearest neighbor searches.

Findings

Achieves $O(n)$ construction time for the zd-tree.

Expected $O(k \, \log k)$ query time for k-nearest neighbors.

Up to 75x speedup over existing methods on 144 hyperthreads.

Abstract

We present a set of parallel algorithms for computing exact k-nearest neighbors in low dimensions. Many k-nearest neighbor algorithms use either a kd-tree or the Morton ordering of the point set; our algorithms combine these approaches using a data structure we call the \textit{zd-tree}. We show that this combination is both theoretically efficient under common assumptions, and fast in practice. For point sets of size $n$ with bounded expansion constant and bounded ratio, the zd-tree can be built in $O(n)$ work with $O(n^{\epsilon})$ span for constant $\epsilon<1$, and searching for the $k$-nearest neighbors of a point takes expected $O(k\log k)$ time. We benchmark our k-nearest neighbor algorithms against existing parallel k-nearest neighbor algorithms, showing that our implementations are generally faster than the state of the art as well as achieving 75x speedup on 144 hyperthreads. Furthermore, the zd-tree supports parallel batch-dynamic insertions and deletions; to our knowledge, it is the first k-nearest neighbor data structure to support such updates. On point sets with bounded expansion constant and bounded ratio, a batch-dynamic update of size $k$ requires $O(k \log n/k)$ work with $O(k^{\epsilon} + \text{polylog}(n))$ span.