Parallel $k$d-tree with Batch Updates

TL;DR

This paper introduces Pkd-tree, a highly parallel and cache-efficient in-memory $k$d-tree supporting fast construction, batch updates, and various queries, with strong theoretical guarantees and superior practical performance.

Contribution

The paper presents the Pkd-tree, a novel parallel $k$d-tree with optimized algorithms for construction and updates, backed by theoretical bounds and practical implementation.

Findings

Pkd-tree outperforms existing implementations in construction and update speed.

The algorithms achieve strong theoretical bounds on work, span, and cache complexity.

Experimental results show competitive query performance with significantly faster updates.

Abstract

The $k$d-tree is one of the most widely used data structures to manage multi-dimensional data. Due to the ever-growing data volume, it is imperative to consider parallelism in $k$d-trees. However, we observed challenges in existing parallel kd-tree implementations, for both constructions and updates. The goal of this paper is to develop efficient in-memory $k$d-trees by supporting high parallelism and cache-efficiency. We propose the Pkd-tree (Parallel $k$d-tree), a parallel $k$d-tree that is efficient both in theory and in practice. The Pkd-tree supports parallel tree construction, batch update (insertion and deletion), and various queries including k-nearest neighbor search, range query, and range count. We proved that our algorithms have strong theoretical bounds in work (sequential time complexity), span (parallelism), and cache complexity. Our key techniques include 1) an efficient construction algorithm that optimizes work, span, and cache complexity simultaneously, and 2) reconstruction-based update algorithms that guarantee the tree to be weight-balanced. With the new algorithmic insights and careful engineering effort, we achieved a highly optimized implementation of the Pkd-tree. We tested Pkd-tree with various synthetic and real-world datasets, including both uniform and highly skewed data. We compare the Pkd-tree with state-of-the-art parallel $k$d-tree implementations. In all tests, with better or competitive query performance, Pkd-tree is much faster in construction and updates consistently than all baselines. We released our code.