# Scottification of Burrows Wheeler Transform

The story of David Scott’s discovery of bijective improvement to Burrows Wheeler transform is quite fascinating. His name will probably not ring a bell in the vaunted world of academia, (where ‘big talkers’ like Ewan Birney get to become Fellow of Royal Society), he is nobody. His own website gives the following bio-

David A. Scott is not a writter and this is his first attempt at writting an article of this type. I graduated from Arizona State University with a BSEE in Electrical Engineering with a major in Fields and Waves. This was the closest thing to a Math Degree without all the English requirements. My main carreer was writting and debugging algorithms for the NAVY at CHINA LAKE CA. form 1970 to 1996. While at CHINA LAKE I was sent to The University of Southern California in 1972 to recieve my MSEE in Electrical Engineering the major field of study was control theroy. I am currently retired living in El Paso, Texas; in the near future I may pursue more education in Jaurez to become a medical doctor.

Yet, in the real world of intellectual contributions, his discovery will last a lot longer that ENCODE. Naturally, his discovery was not announced through a paper, but a comment in a discussion forum. As the story goes, around 2006 or 2007, he posted a message in the encryption-related mailing lists describing his improvement to BWT.

This page looks at a modified BWT compression package and suggests that the mods make for a better transform than the standard BWT. I called this bijective version a BWT that has been Scottified. It is what the BWT should have been. If one looks at only the transformed data without the index the old original BWT most likely leads to a better compression. But when you consider the need for an index to be passed along the BWTS transform will lead on the average to better compression.

Soon some mathematicians started to poke around and found that he indeed had a new way of computing BWT-like transform with two useful features -

(i) It is bijective without the added $ sign at the end.

(ii) Through Lyndon factorization, his method can be parallelized easily. We discussed about Lyndon word and Lyndon factorization here.

A Google engineer joined him to put together a formal paper, which can be found at the following link.

A Bijective String Sorting Transform

Given a string of characters, the Burrows-Wheeler Transform rearranges the characters in it so as to produce another string of the same length which is more amenable to compression techniques such as move to front, run-length encoding, and entropy encoders. We present a variant of the transform which gives rise to similar or better compression value, but, unlike the original, the transform we present is bijective, in that the inverse transformation exists for all strings. Our experiments indicate that using our variant of the transform gives rise to better compression ratio than the original Burrows- Wheeler transform. We also show that both the transform and its inverse can be computed in linear time and consuming linear storage.

Apparently, the above proof has several errors, whereas the following paper is mathematically correct.

On Bijective Variants of the Burrows-Wheeler Transform - Manfred Kufleitner

The sort transform (ST) is a modification of the Burrows-Wheeler transform (BWT). Both transformations map an arbitrary word of length n to a pair consisting of a word of length n and an index between 1 and n. The BWT sorts all rotation conjugates of the input word, whereas the ST of order k only uses the first k letters for sorting all such conjugates. If two conjugates start with the same prefix of length k, then the indices of the rotations are used for tie-breaking. Both transforms output the sequence of the last letters of the sorted list and the index of the input within the sorted list. In this paper, we discuss a bijective variant of the BWT (due to Scott), proving its correctness and relations to other results due to Gessel and Reutenauer (1993) and Crochemore, Desarmenien, and Perrin (2005). Further, we present a novel bijective variant of the ST.

Those, who do not like formal mathematics may enjoy the really simple introduction in zephyrtronium’s github page. It comes with the added benefit of code that you can download.

The Burrows-Wheeler-Scott transform (called the Burrows-Wheeler transform “Scottified” in existing literature, but that sounds silly) sorts together all infinitely repeated cycles of each Lyndon word of the input, then takes the last character of each rotation of each Lyndon word in the overall sorted order. Of course, this description is about as intelligible as any of the existing literature on it for someone not intimately familiar with the concepts involved, and incomplete for someone who is.

Lyndon words are sequences which are less than any of the rotations of that sequence. “Less than”, that is, the order, is defined in “the usual lexicographical way”: ‘a’ < ‘b’ by definition; ‘aa’ < ‘ab’ because the first positions are the same and the second position is lesser in ‘aa’; ‘ab’ < ‘ba’ because ‘ab’ is lesser in the first position. For now, the order of sequences whose lengths are not equal is left undefined.

A Lyndon word is a sequence such that no matter how many times you take the rightmost (or leftmost) element and attach it to the left (or right, respectively), the result is never less than the word. By the Chen-Fox-Lyndon theorem, every ordered sequence has a unique “Lyndon factorization” of Lyndon words, such that each word in the factorization is never greater than its predecessor, where order is here (not for the BWST algorithm) defined for words of unequal length such that if a is a prefix of b, then a < b.