Burrows–Wheeler transform

The Burrows–Wheeler transform is an algorithm used to prepare data for use with data compression techniques such as bzip2. It was invented by Michael Burrows and David Wheeler in 1994 while Burrows was working at DEC Systems Research Center in Palo Alto, California. It is based on a previously unpublished transformation discovered by Wheeler in 1983. The algorithm can be implemented efficiently using a suffix array thus reaching linear time complexity.[1]

When a character string is transformed by the BWT, the transformation permutes the order of the characters. If the original string had several substrings that occurred often, then the transformed string will have several places where a single character is repeated multiple times in a row.

For example:

The output is easier to compress because it has many repeated characters. In this example the transformed string contains six runs of identical characters: XX, SS, PP, .., II, and III, which together make 13 out of the 44 characters.

The transform is done by sorting all the circular shifts of a text in lexicographic order and by extracting the last column and the index of the original string in the set of sorted permutations of S.

Given an input string S = ^BANANA| (step 1 in the table below), rotate it N times (step 2), where N = 8 is the length of the S string considering also the symbol ^ representing the start of the string and the red | character representing the 'EOF' pointer; these rotations, or circular shifts, are then sorted lexicographically (step 3). The output of the encoding phase is the last column L = BNN^AA|A after step 3, and the index (0-based) I of the row containing the original string S, in this case I = 6.

^BANANA|

^BANANA|
|^BANANA
A|^BANAN
NA|^BANA
ANA|^BAN
NANA|^BA
ANANA|^B
BANANA|^

ANANA|^B
ANA|^BAN
A|^BANAN
BANANA|^
NANA|^BA
NA|^BANA
^BANANA|
|^BANANA

ANANA|^B
ANA|^BAN
A|^BANAN
BANANA|^
NANA|^BA
NA|^BANA
^BANANA|
|^BANANA

BNN^AA|A

The following pseudocode gives a simple (though inefficient) way to calculate the BWT and its inverse. It assumes that the input string s contains a special character 'EOF' which is the last character and occurs nowhere else in the text.

function BWT (string s)
    create a table, where the rows are all possible rotations of s
    sort rows alphabetically
    return (last column of the table)

function inverseBWT (string s)
    create empty table
    repeat length(s) times
        // first insert creates first column
        insert s as a column of table before first column of the table
        sort rows of the table alphabetically
    return (row that ends with the 'EOF' character)

To understand why this creates more-easily-compressible data, consider transforming a long English text frequently containing the word "the". Sorting the rotations of this text will group rotations starting with "he " together, and the last character of that rotation (which is also the character before the "he ") will usually be "t", so the result of the transform would contain a number of "t" characters along with the perhaps less-common exceptions (such as if it contains "ache ") mixed in. So it can be seen that the success of this transform depends upon one value having a high probability of occurring before a sequence, so that in general it needs fairly long samples (a few kilobytes at least) of appropriate data (such as text).

The remarkable thing about the BWT is not that it generates a more easily encoded output—an ordinary sort would do that—but that it does this reversibly, allowing the original document to be re-generated from the last column data.

The inverse can be understood this way. Take the final table in the BWT algorithm, and erase all but the last column. Given only this information, you can easily reconstruct the first column. The last column tells you all the characters in the text, so just sort these characters alphabetically to get the first column. Then, the last and first columns (of each row) together give you all pairs of successive characters in the document, where pairs are taken cyclically so that the last and first character form a pair. Sorting the list of pairs gives the first and second columns. Continuing in this manner, you can reconstruct the entire list. Then, the row with the "end of file" character at the end is the original text. Reversing the example above is done like this:

BNN^AA|A

B
N
N
^
A
A
|
A

A
A
A
B
N
N
^
|

BA
NA
NA
^B
AN
AN
|^
A|

AN
AN
A|
BA
NA
NA
^B
|^

BAN
NAN
NA|
^BA
ANA
ANA
|^B
A|^

ANA
ANA
A|^
BAN
NAN
NA|
^BA
|^B

BANA
NANA
NA|^
^BAN
ANAN
ANA|
|^BA
A|^B

ANAN
ANA|
A|^B
BANA
NANA
NA|^
^BAN
|^BA

BANAN
NANA|
NA|^B
^BANA
ANANA
ANA|^
|^BAN
A|^BA

ANANA
ANA|^
A|^BA
BANAN
NANA|
NA|^B
^BANA
|^BAN

BANANA
NANA|^
NA|^BA
^BANAN
ANANA|
ANA|^B
|^BANA
A|^BAN

ANANA|
ANA|^B
A|^BAN
BANANA
NANA|^
NA|^BA
^BANAN
|^BANA

BANANA|
NANA|^B
NA|^BAN
^BANANA
ANANA|^
ANA|^BA
|^BANAN
A|^BANA

ANANA|^
ANA|^BA
A|^BANA
BANANA|
NANA|^B
NA|^BAN
^BANANA
|^BANAN

BANANA|^
NANA|^BA
NA|^BANA
^BANANA|
ANANA|^B
ANA|^BAN
|^BANANA
A|^BANAN

ANANA|^B
ANA|^BAN
A|^BANAN
BANANA|^
NANA|^BA
NA|^BANA
^BANANA|
|^BANANA

^BANANA|

A number of optimizations can make these algorithms run more efficiently without changing the output. There is no need to represent the table in either the encoder or decoder. In the encoder, each row of the table can be represented by a single pointer into the strings, and the sort performed using the indices. Some care must be taken to ensure that the sort does not exhibit bad worst-case behavior: Standard library sort functions are unlikely to be appropriate. In the decoder, there is also no need to store the table, and in fact no sort is needed at all. In time proportional to the alphabet size and string length, the decoded string may be generated one character at a time from right to left. A "character" in the algorithm can be a byte, or a bit, or any other convenient size.

One may also make the observation that mathematically, the encoded string can be computed as a simple modification of the suffix array, and suffix arrays can be computed with linear time and memory. The BWT can be defined with regards to the suffix array SA of text T as (1-based indexing):

${\displaystyle BWT[i]={\begin{cases}T[SA[i]-1],&{\text{if }}SA[i]>0\\\$,&{\text{otherwise}}\end{cases}}}$[3]

There is no need to have an actual 'EOF' character. Instead, a pointer can be used that remembers where in a string the 'EOF' would be if it existed. In this approach, the output of the BWT must include both the transformed string, and the final value of the pointer. The inverse transform then shrinks it back down to the original size: it is given a string and a pointer, and returns just a string.

A complete description of the algorithms can be found in Burrows and Wheeler's paper, or in a number of online sources.[1] The algorithms vary somewhat by whether EOF is used, and in which direction the sorting was done. In fact, the original formulation did not use an EOF marker.[4]

Since any rotation of the input string will lead to the same transformed string, the BWT cannot be inverted without adding an EOF marker to the end of the input or doing something equivalent, making it possible to distinguish the input string from all its rotations. Increasing the size of the alphabet (by appending the EOF character) makes later compression steps awkward.

There is a bijective version of the transform, by which the transformed string uniquely identifies the original, and the two have the same length and contain exactly the same characters, just in a different order.[5][6]

The bijective transform is computed by factoring the input into a non-increasing sequence of Lyndon words; such a factorization exists and is unique by the Chen–Fox–Lyndon theorem,[7] and may be found in linear time.[8] The algorithm sorts the rotations of all the words; as in the Burrows–Wheeler transform, this produces a sorted sequence of n strings. The transformed string is then obtained by picking the final character of each string in this sorted list. The one important caveat here is that strings of different lengths are not ordered in the usual way; the two strings are repeated forever, and the infinite repeats are sorted. For example, "ORO" precedes "OR" because "OROORO..." precedes "OROROR...".

For example, the text "^BANANA|" is transformed into "ANNBAA^|" through these steps (the red | character indicates the EOF pointer) in the original string. The EOF character is unneeded in the bijective transform, so it is dropped during the transform and re-added to its proper place in the file.

The string is broken into Lyndon words so the words in the sequence are decreasing using the comparison method above. (Note that we're sorting '^' as succeeding other characters.) "^BANANA" becomes (^) (B) (AN) (AN) (A).

^BANANA|

^^^^^^^^... (^)
BBBBBBBB... (B)
ANANANAN... (AN)
NANANANA... (NA)
ANANANAN... (AN)
NANANANA... (NA)
AAAAAAAA... (A)

AAAAAAAA... (A)
ANANANAN... (AN)
ANANANAN... (AN)
BBBBBBBB... (B)
NANANANA... (NA)
NANANANA... (NA)
^^^^^^^^... (^)

AAAAAAAA... (A)
ANANANAN... (AN)
ANANANAN... (AN)
BBBBBBBB... (B)
NANANANA... (NA)
NANANANA... (NA)
^^^^^^^^... (^)

ANNBAA^|

ANNBAA^

A
N
N
B
A
A
^

A
A
A
B
N
N
^

AA
NA
NA
BB
AN
AN
^^

AA
AN
AN
BB
NA
NA
^^

AAA
NAN
NAN
BBB
ANA
ANA
^^^

AAA
ANA
ANA
BBB
NAN
NAN
^^^

AAAA
NANA
NANA
BBBB
ANAN
ANAN
^^^^

AAAA
ANAN
ANAN
BBBB
NANA
NANA
^^^^

^BANANA

Up until the last step, the process is identical to the inverse Burrows-Wheeler process, but here it will not necessarily give rotations of a single sequence; it instead gives rotations of Lyndon words (which will start to repeat as the process is continued). Here, we can see (repetitions of) four distinct Lyndon words: (A), (AN) (twice), (B), and (^). (NANA... doesn't represent a distinct word, as it is a cycle of ANAN....) At this point, these words are sorted into reverse order: (^), (B), (AN), (AN), (A). These are then concatenated to get

^BANANA

The Burrows-Wheeler transform can indeed be viewed as a special case of this bijective transform; instead of the traditional introduction of a new letter from outside our alphabet to denote the end of the string, we can introduce a new letter that compares as preceding all existing letters that is put at the beginning of the string. The whole string is now a Lyndon word, and running it through the bijective process will therefore result in a transformed result that, when inverted, gives back the Lyndon word, with no need for reassembling at the end.

Relatedly, the transformed text will only differ from the result of BWT by one character per Lyndon word; for example, if the input is decomposed into six Lyndon words, the output will only differ in six characters. For example, applying the bijective transform gives:

The bijective transform includes eight runs of identical characters. These runs are, in order: XX, II, XX, PP, .., EE, .., and IIII.

In total, 18 characters are used in these runs.

When a text is edited, its Burrows–Wheeler transform will change. Salson et al.[9] propose an algorithm that deduces the Burrows–Wheeler transform of an edited text from that of the original text, doing a limited number of local reorderings in the original Burrows–Wheeler transform, which can be faster than constructing the Burrows–Wheeler transform of the edited text directly.

This Python implementation sacrifices speed for simplicity: the program is short, but takes more than the linear time that would be desired in a practical implementation. It essentially does what the pseudocode section does.

Using the STX/ETX control codes to mark the start and end of the text, and using s[i:] + s[:i] to construct the ith rotation of s, the forward transform takes the last character of each of the sorted rows:

def bwt(s: str) -> str:
    """Apply Burrows–Wheeler transform to input string."""
    assert "\002" not in s and "\003" not in s, "Input string cannot contain STX and ETX characters"
    s = "\002" + s + "\003"  # Add start and end of text marker
    table = sorted(s[i:] + s[:i] for i in range(len(s)))  # Table of rotations of string
    last_column = [row[-1:] for row in table]  # Last characters of each row
    return "".join(last_column)  # Convert list of characters into string

The inverse transform repeatedly inserts r as the left column of the table and sorts the table. After the whole table is built, it returns the row that ends with ETX, minus the STX and ETX.

def ibwt(r: str) -> str:
    """Apply inverse Burrows–Wheeler transform."""
    table = [""] * len(r)  # Make empty table
    for i in range(len(r)):
        table = sorted(r[i] + table[i] for i in range(len(r)))  # Add a column of r
    s = [row for row in table if row.endswith("\003")][0]  # Find the correct row (ending in ETX)
    return s.rstrip("\003").strip("\002")  # Get rid of start and end markers

Following implementation notes from Manzini, it is equivalent to use a simple null character suffix instead. The sorting should be done in colexicographic order (string read right-to-left), i.e. sorted(..., key=lambda s: s[::-1]) in Python.[4] (The above control codes actually fail to satisfy EOF being the last character; the two codes are actually the first. The rotation holds nevertheless.)

The advent of next-generation sequencing (NGS) techniques at the end of the 2000s decade has led to another application of the Burrows–Wheeler transformation. In NGS, DNA is fragmented into small pieces, of which the first few bases are sequenced, yielding several millions of "reads", each 30 to 500 base pairs ("DNA characters") long. In many experiments, e.g., in ChIP-Seq, the task is now to align these reads to a reference genome, i.e., to the known, nearly complete sequence of the organism in question (which may be up to several billion base pairs long). A number of alignment programs, specialized for this task, were published, which initially relied on hashing (e.g., Eland, SOAP,[10] or Maq[11]). In an effort to reduce the memory requirement for sequence alignment, several alignment programs were developed (Bowtie,[12] BWA,[13] and SOAP2[14]) that use the Burrows–Wheeler transform.

• Article by Mark Nelson on the BWT
• A Bijective String-Sorting Transform, by Gil and Scott
• Yuta's openbwt-v1.5.zip contains source code for various BWT routines including BWTS for bijective version
• On Bijective Variants of the Burrows–Wheeler Transform, by Kufleitner
• Blog post and project page for an open-source compression program and library based on the Burrows–Wheeler algorithm
• MIT open courseware lecture on BWT (Foundations of Computational and Systems Biology)
• League Table Sort (LTS) or The Weighting algorithm to BWT by Abderrahim Hechachena (claims to be faster, but correctness is not proven)