Improved Parallel Construction of Wavelet Trees and Rank/Select Structures

TL;DR

This paper introduces improved parallel algorithms for constructing wavelet trees and rank/select structures, reducing work complexity and enabling efficient parallel processing for large datasets.

Contribution

It presents novel parallel algorithms with lower work complexity for wavelet tree construction and rank/select structures, including variants for different tree shapes and multiary trees.

Findings

Reduced work complexity compared to prior algorithms

Polylogarithmic and sub-linear depth algorithms

Efficient parallel construction of rank/select structures

Abstract

Existing parallel algorithms for wavelet tree construction have a work complexity of $O(n\log\sigma)$. This paper presents parallel algorithms for the problem with improved work complexity. Our first algorithm is based on parallel integer sorting and has either $O(n\log\log n\lceil\log\sigma/\sqrt{\log n\log\log n}\rceil)$ work and polylogarithmic depth, or $O(n\lceil\log\sigma/\sqrt{\log n}\rceil)$ work and sub-linear depth. We also describe another algorithm that has $O(n\lceil\log\sigma/\sqrt{\log n} \rceil)$ work and $O(\sigma+\log n)$ depth. We then show how to use similar ideas to construct variants of wavelet trees (arbitrary-shaped binary trees and multiary trees) as well as wavelet matrices in parallel with lower work complexity than prior algorithms. Finally, we show that the rank and select structures on binary sequences and multiary sequences, which are stored on wavelet tree nodes, can be constructed in parallel with improved work bounds, matching those of the best existing sequential algorithms for constructing rank and select structures.