# Parallel Nearest Neighbors in Low Dimensions with Batch Updates

arXiv: https://arxiv.org/abs/2111.04182 · HTML: https://mlsanity.com/paper/2111.04182/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Nov 2021.

## Built on by (most cited first, 2 of 2)

- [Monitoring Robustness and Individual Fairness](https://mlsanity.com/paper/2506.00496/) (2025)
- [ANNLib: A Development Framework for Efficient Approximate Nearest Neighbor Search](https://mlsanity.com/paper/2607.17582/) (2026)

## Abstract (arXiv)

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 textitzd-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(nε) span for constant ε<1, and searching for the k-nearest neighbors of a point takes expected O(klog 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(kε + polylog(n)) span.