### Book chapter

## Wavelet Trees Meet Suffix Trees

We present an improved wavelet tree construction algorithm and discuss its applications to a number of rank/select problems for integer keys and strings. Given a string of length n over an alphabet of size ω ≤ n, our method builds the wavelet tree in O(n log ω √log n) time, improving upon the state-of-the-art algorithm by a factor of √log n. As a consequence, given an array of n integers we can construct in O(n√log n) time a data structure consisting of O(n) machine words and capable of answering rank/select queries for the subranges of the array in O(log n/log log n) time. This is a log log n-factor improvement in query time compared to Chan and Pətraşcu (SODA 2010) and a √log n-factor improvement in construction time compared to Brodal et al. (Theor. Comput. Sci. 2011). Next, we switch to stringological context and propose a novel notion of wavelet suffix trees. For a string w of length n, this data structure occupies O(n) words, takes O (n√log n) time to construct, and simultaneously captures the combinatorial structure of substrings of w while enabling efficient top-down traversal and binary search. In particular, with a wavelet suffix tree we are able to answer in O(log |x|) time the following two natural analogues of rank/select queries for suffixes of substrings: 1) For substrings x and y of w (given by their endpoints) count the number of suffixes of x that are lexicographically smaller than y; 2) For a substring x of w (given by its endpoints) and an integer fc, find the κ-th lexicographically smallest suffix of x. We further show that wavelet suffix trees allow to compute a run-length-encoded Burrows-Wheeler transform of a substring x of w (again, given by its endpoints) in O(slog|x|) time, where s denotes the length of the resulting run-length encoding. This answers a question by Cormode and Muthukrishnan (SODA 2005), who considered an analogous problem for Lempel-Ziv compression. All our algorithms, except for the construction of wavelet suffix trees, which additionally requires O(n) time in expectation, are deterministic and operate in the word RAM model. Copyright © 2015 by the Society for Industrial and Applied Mathmatics.