Strings and the Burrows-Wheeler Transform

Author: Franco Saliola <saliola at gmail.com>

Sage/Python includes a builtin datastructure from strings.

There are several ways to input strings. You can input a string using single quotes (‘) or double quotes (”):

{{{id=0| s = "This is a string!" s /// 'This is a string!' }}}

{{{id=1| t = 'So is this!' print t /// So is this! }}}

You can also input a string using three quotes (“”” or ‘’‘). This is useful if you want to use both ” and ‘ in your string, or you want your string to span multiple lines:

{{{id=2| s = """ This is a multi-line string that includes 'single quotes' and "double quotes". """ print s /// This is a multi-line string that includes 'single quotes' and "double quotes". }}}

Exercises

1. Create and print the following string

     \ | ( | ) / /
   _________________
   |               |
   |               |
   |  I <3 Coffee! /--\
   |               |  |
    \             /\--/
     \___________/

2. Without using cut-and-paste(!) replace the substring I <3 Coffee! with the substring I <3 Tea!.

3. Print a copy of your string with all the letters capitalized (upercase).

Operations on strings

Strings behave very much like lists. The table below summarizes their common operations.

Operation	Syntax for lists	Syntax for strings
Accessing a letter	list[3]	string[3]
Slicing	list[3:17:2]	string[3:17:2]
Concatenation	list1 + list2	string1 + sting2
A copy	list[:]	string[:]
A reversed copy	list[::-1]	string[::-1]
Length	len(list)	len(string)

Exercises

1. The factors of length 2 of ‘rhubarb’ are

   rh

   hu

   ub

   ba

   ar

   rb

   Write a function called factors that returns a list of the factors of length l of s , and list all the factors of length 3 of ‘rhubarb’.

   {{{id=3| # edit here /// }}}

2. What happens if you apply your function factors to the list [0,1,1,0,1,0,0,1] ? If it doesn’t work for both lists and strings, go back and modify your function so that it does work for both.

   {{{id=4| # edit here /// }}}

Run-length encoding

The string

WWWWWWWWWWWWBWWWWWWWWWWWWBBBWWWWWWWWWWWWWWWWWWWWWWWWBWWWWWWWWWWWWWW

begins with W 12 times, then B once, then W 12 times, then B 3 times, then W 24 times, then B once and then W 14 times. Thus, it can be encoded by the tuples:

(W, 12), (B, 1), (W, 12), (B, 3), (W, 24), (B, 1), (W, 14)

This is called the run-length encoding of the string.

Exercises

1. Write a function that returns the run-length encoding of a string. Does your function work for lists as well as strings? If not, then can you make it so that it works for both strings and lists? Use your function to compute the run-length encoding of the list:

   [0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]

   {{{id=5| # edit here /// }}}

Rotations

The rotations of the string ‘bananas’ are:

bananas

ananasb

nanasba

anasban

nasbana

asbanan

sbanana

and if we sort these alphabetically, then we get:

ananasb

anasban

asbanan

bananas

nanasba

nasbana

sbanana

Exercises

1. Define a function print_sorted_rotations that sorts all the rotations of a string and prints them in an array as above. Print the sorted rotations of the strings ‘ananas’ and ‘cocomero’.

   {{{id=6| # edit here /// }}}

The Burrows-Wheeler Transform

The Burrows-Wheeler Transform (BWT) of a string s sorts all the rotations of s and then returns the last column.

For example, if we sort the rotations of ‘bananas’:

ananasb

anasban

asbanan

bananas

nanasba

nasbana

sbanana

then the last column is bnnsaaa , so the BWT of bananas is bnnsaaa.

Exercises

1. Write a function that returns the BWT of a string. Compute the BWT of bananas , ananas and cocomero . (Hint: You can return you answer as a list, but if you want to return a string, then you might want to use the join method for strings.)

   {{{id=7| # edit here /// }}}

2. Combine the functions you defined above to create an @interact object that takes a string s and prints:

   • the sorted rotations of s
   • the run-length encoding of s
   • the BWT of s
   • the run-length encoding of the BWT of s

   (Hint: String formatting can be done using the % operator. Here is an example:

   {{{id=8| print 'The sum of %s and %s is %s.' % (3,2,3+2) /// The sum of 3 and 2 is 5. }}}

   If you are familiar with C then you will notice that string formating is very similar to C ‘s sprintf statement.)

   {{{id=9| # edit here /// }}}

3. Use your interact object to explore this transformation, and to answer the following questions.

   1. Compute the BWT of the following.
      • xxyxyxyxyxyxyxyxyxxyxyxyxyxyxyxyxyxy
      • 01101001100101101001011001101001100101100110100101
      • cdccdcdccdccdcdccdcdccdccdcdccdccdcdccdcdccdccdcdc
   2. Do you notice any patterns in the BWT of a string?
   3. Can you think of an application for this transformation?
   4. Find 3 other strings that have a ‘nice’ image under the BWT.
   5. Is the Burrows-Wheeler transformation invertible? (That is, can you find two strings that have the same BWT?)
   {{{id=10| # edit here /// }}}

4. By comparing the BWT of a string with the first column of the array of sorted rotations of a string s , devise and implement an algorithm that reconstructs the first column of the array from the BWT of s .

   {{{id=11| # edit here /// }}}

5. By examining the first two columns of the array, devise and implement an algorithm that reconstructs the first two columns of the array from the BWT of a string. ( Hint: compare the last and first column with the first two columns.)

   {{{id=12| # edit here /// }}}

6. By examining the first three columns of the array, devise and implement an algorithm that reconstructs the first three columns of the array from the BWT of a string.

   {{{id=13| # edit here /// }}}

7. Write a function that reconstructs the entire array of sorted rotations of a string from the BWT of the string.

   {{{id=14| # edit here /// }}}

8. A Lyndon word is a word w that comes first in alphabetical order among all its rotations. Is the BWT invertible on Lyndon words?

   {{{id=15| # edit here /// }}}

9. Explain how one can modify the BWT to make it invertible on arbitrary words. Implement your modified transformation and the inverse transformation.

   {{{id=16| # edit here /// }}}