An implementation of "The Wavelet Matrix" (Claude and Navarro) http:https://www.dcc.uchile.cl/~gnavarro/ps/spire12.4.pdf.

The wavelet matrix is a data structure to represent an immutable sequence of unsigned integers that supports some queries efficiently.

The WaveletMatrix type is defined as follows:

struct WaveletMatrix{w,T<:Unsigned,B<:AbstractBitVector} <: AbstractVector{T}

where

• w: the number of bits required to encode the unsigned integers (elements)
• T: the type of elements
• B: the type of bit vectors used to store elements.

To efficiently pack a sequence of unsigned integers, w should be as small as possible but enough to encode those integers. For example, if you want to store integers between 0x00 and 0x03 (i.e. four distinct integers), setting w = 2 (= ceil(log2(4))) is the best choice.

The basic operations available on the wavelet matrix are:

• getindex(wm::WaveletMatrix, i::Integer): Return i-th element of wm.
• rank(a::Unsigned, wm::WaveletMatrix, i::Integer): Count the number of a's occurrences in wm between 1:i.
• select(a::Unsigned, wm::WaveletMatrix, j::Integer): Return the position of the j-th occurrence of a in wm.

data = rand(0x00:0x03, 100_000)
wm = WaveletMatrix{2}(data)

wm[129]
rank(0x02, wm, 5555)
partialsort(0x01, wm, 9876)