Faster Wavelet Trees with Quad Vectors

Given a text, rank and select queries return the number of occurrences of a character up to a position (rank) or the position of a character with a given rank (select). These queries have applications in, e.g., compression, computational geometry, and pattern matching in the form of the backwards search -- the backbone of many compressed full-text indices. A wavelet tree is a compact data structure that for a text of length $n$ over an alphabet of size $\sigma$ requires only $n\lceil\log\sigma\rceil(1+o(1))$ bits of space and can answer rank and select queries in $\Theta(\log \sigma)$ time. Wavelet trees are used in the applications described above. In this paper, we show how to improve query performance of wavelet trees by using a 4-ary tree instead of a binary tree as basis of the wavelet tree. To this end, we present a space-efficient rank and select data structure for quad vectors. The 4-ary tree layout of a wavelet tree helps to halve the number of cache misses during queries and thus reduces the query latency. Our experimental evaluation shows that our 4-ary wavelet tree can improve the latency of rank and select queries by a factor of $\approx 2$ compared to the wavelet tree implementations contained in the widely used Succinct Data Structure Library (SDSL).