Example Suffix Array, A[ ], of “$banana”

An example BWT of “^BANANA|”

An FM-index is created by first taking the Burrows-Wheeler transform (BWT) of the input text. For example, the BWT of the string T = “abracadabra” is “ard$rcaaaabb”, and here it is represented by the matrix M where each row is a rotation of the text that has been sorted. The transform corresponds to the last column labeled L.

C[c] is a table that, for each character c in the alphabet, contains the number of occurrences of lexically smaller characters in the text.

The function Occ(c, k) is the number of occurrences of character c in the prefix L[1..k]

The program Bowtie uses an FM index technique

• Ultrafast and memory-efficient alignment of short DNA sequences to the human genome

Charles M. - BWT Notes

BWA datastructure lecture
# Suffix array
# Burrows Wheeler Transform
	- variation on suffix array, useful for data compression and indexing
# FM index
	- uses both suffix arrays and BWT for fast lookup

suffix array definition: X[0:n] with a '$' character spliced in the range of [0,n].

e.g. string S: z a b c d e
	    index: 0,1,2,3,4,5

# BWT definition:
	B[i] = X[S[i]-1]

1) look at suffixes of a string
a
de
cde
bcde
abcde
zabcde

2) sort order of suffixes:
Pos	Suffix
6	$
1	abcde$
2	bcde$
3	cde$
4	de$
5	e$
0	zabcde$

3) rotate the string:
zabcde[$]
abcde$[z]
bcde$z[a]
.
.
.
$zabcd[e]
       ^
    (last column is the burrows wheeler transform)

left column (since it is in alphabet order) just record where a character starts since there is
repetition of a character.
->  $
    $
    $
->  A
    A
    A
    A
    ...
->  C
    C
    ...


4) Record Occupancy:
        Definitions:
	C(a)	  # Where char's start.
	Occ(a,i)  # Array of number of char's before i in B. (e.g. # of a's in B[0,i])
		       - Can store part of the Occupancy array to obtain greater
                         compression at the cost of slightly greater cpu time.
	B 	  # BWT array

Use last_to_first(i) function which allows you to take the BWT string and find the 1st column
row index of a BWT character.
Use that row index to find the previous character in the BWT string (It's the last column
character of the same row index).
Occupancy function: C(B[i]) + Occ(B[i],i)

# Search algorithm
Given Pattern P
R(P): where pattern starts
Rbar(P): where pattern ends

Rbar(aP) = C(a) + Occ(a, Rbar(P))
R(aP) = C(a) + Occ(a,R(P)-1)+1

Additional resources and better examples can be found below:

BWT and FM Index tutorial

BWT paper