# Simple, Fast and Lightweight Parallel Wavelet Tree Construction

## TL;DR

This paper introduces new practical sequential and parallel algorithms for wavelet tree construction that are faster, more space-efficient, and scalable to multiple cores, with adaptable methods for wavelet matrices.

## Contribution

It presents a unified bottom-up construction approach for wavelet trees and matrices, improving speed and space efficiency over existing methods.

## Key findings

- Sequential algorithm up to 2x faster than previous best
- Uses half the space of existing algorithms
- Scales efficiently to 32 cores with minimal space overhead

## Abstract

The wavelet tree (Grossi et al. [SODA, 2003]) and wavelet matrix (Claude et al. [Inf. Syst., 47:15--32, 2015]) are compact indices for texts over an alphabet $[0,\sigma)$ that support rank, select and access queries in $O(\lg \sigma)$ time. We first present new practical sequential and parallel algorithms for wavelet tree construction. Their unifying characteristics is that they construct the wavelet tree bottomup}, i.e., they compute the last level first. We also show that this bottom-up construction can easily be adapted to wavelet matrices. In practice, our best sequential algorithm is up to twice as fast as the currently fastest sequential wavelet tree construction algorithm (Shun [DCC, 2015]), simultaneously saving a factor of 2 in space. This scales up to 32 cores, where we are about equally fast as the currently fastest parallel wavelet tree construction algorithm (Labeit et al. [DCC, 2016]), but still use only about 75 % of the space. An additional theoretical result shows how to adapt any wavelet tree construction algorithm to the wavelet matrix in the same (asymptotic) time, using only little extra space.