Tutorial 7 - Strings and the Burrows-Wheeler Transform.sagews

Strings and the Burrows-Wheeler Transform

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 ("):

s = "This is a string!"
s

'This is a string!'

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:

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".

Exercise: Create and print the following string.

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

 

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

 

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

 

Strings behave very much like lists. For example,

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

 

 

Exercise: 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'.

 

Exercise: 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.

Run-length encoding

The string$$WWWWWWWWWWWWBWWWWWWWWWWWWBBBWWWWWWWWWWWWWWWWWWWWWWWWBWWWWWWWWWWWWWW$$begins with $12$ copies of $W$, one copy of $B$, then $12$ copies of $W$, then $3$ copies of $B$, then $24$ copies of $W$, then 1 copy of $B$ and $14$ copies of $W$. Thus, it can be encoded as $$(W, 12), (B, 1), (W, 12), (B, 3), (W, 24), (B, 1), (W, 14).$$ This is called the run-length encoding of the string.

Exercise:

• 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 following list.
$$[0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]$$

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

 

Exercise: 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'.

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.

Exercise: 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.)

Exercise: 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 are some examples:

print 'The sum of %s and %s is %s.' % (3,2,3+2)

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

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

Exercise: What is the BWT of the following?

• xxyxyxyxyxyxyxyxyxxyxyxyxyxyxyxyxyxy

• 01101001100101101001011001101001100101100110100101

• cdccdcdccdccdcdccdcdccdccdcdccdccdcdccdcdccdccdcdc

Do you notice any patterns in the BWT of a string?

Can you think of an application for this transformation?

Find 3 other strings that have a 'nice' image under the BWT.

 

Exercise: Is the Burrows-Wheeler transformation invertible? (That is, can you find two strings that have the same BWT?)

 

Exercise: 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.

 

Exercise: 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.)

 

Exercise: 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.

 

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

 

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

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