Inverting the Burrows – Wheeler Transform

Prerequisite: Burrows – Wheeler Data Transform Algorithm

Why inverse of BWT? The main idea behind it:

1. The remarkable thing about BWT algorithm is that this particular transform is invertible with minimal data overhead.



2. To compute inverse of BWT is to undo the BWT and recover the original string. The naive method of implementing this algorithm can be studied from here. The naive approach is speed and memory intensive and requires us to store |text| cyclic rotations of the string |text|.

3. Let’s discuss a faster algorithm where we have with us only two things:
i. bwt_arr[] which is the last column of sorted rotations list given as “annb$aa”.
ii. ‘x’ which is the row index at which our original string “banana$” appears in the sorted rotations list. We can see that ‘x’ is 4 in the example below.

 Row Index    Original Rotations    Sorted Rotations
 ~~~~~~~~~    ~~~~~~~~~~~~~~~~~~    ~~~~~~~~~~~~~~~~
    0             banana$               $banana
    1             anana$b               a$banan
    2             nana$ba               ana$ban
    3             ana$ban               anana$b
   *4             na$bana               banana$
    5             a$banan               na$bana
    6             $banana               nana$ba

4. An important observation: If the jth original rotation (which is original rotation shifted j characters to the left) is the ith row in the sorted order, then l_shift[i] records in the sorted order where (j+1)st original rotation appears. For example, the 0th original rotation “banana$” is row 4 of sorted order, and since l_shift[4] is 3, the next original rotation “anana$b” is row 3 of the sorted order.

Row Index  Original Rotations  Sorted Rotations l_shift 
~~~~~~~~~ ~~~~~~~~~~~~~~~~~~  ~~~~~~~~~~~~~~~~  ~~~~~~~
   0           banana$         $banana           4
   1           anana$b         a$banan           0
   2           nana$ba         ana$ban           5
   3           ana$ban         anana$b           6
  *4           na$bana         banana$           3
   5           a$banan         na$bana           1
   6           $banana         nana$ba           2

5. Our job is to deduce l_shift[] from the information available to us which is bwt_arr[] and ‘x’ and with its help compute the inverse of BWT.

How to compute l_shift[] ?
1. We know BWT which is “annb$aa”. This implies that we know all the characters of our original string, even though they’re permuted in wrong order.

2. By sorting bwt_arr[], we can reconstruct first column of sorted rotations list and we call it sorted_bwt[].

 Row Index    Sorted Rotations   bwt_arr    l_shift  
 ~~~~~~~~~    ~~~~~~~~~~~~~~~~~~~~~~~~~~    ~~~~~~~   
     0         $  ?  ?  ?  ?  ?  a             4
     1         a  ?  ?  ?  ?  ?  n
     2         a  ?  ?  ?  ?  ?  n
     3         a  ?  ?  ?  ?  ?  b
    *4         b  ?  ?  ?  ?  ?  $             3
     5         n  ?  ?  ?  ?  ?  a
     6         n  ?  ?  ?  ?  ?  a

3. Since ‘$’ occurs only once in the string ‘sorted_bwt[]’ and rotations are formed using cyclic wrap around, we can deduce that l_shift[0] = 4. Similarly, ‘b’ occurs once, so we can deduce that l_shift[4] = 3.

4. But, because ‘n’ appears twice, it seems ambiguous whether l_shift[5] = 1 and l_shift[6] = 2 or whether l_shift[5] = 2 and l_shift[6] = 1.

5. Rule to solve this ambiguity is that if rows i and j both start with the same letter and i<j, then l_shift[i] < l_shift[j]. This implies l_shift[5] = 1 and l_shift[6] =2. Continuing in a similar fashion, l_shift[] gets computed to the following.

 Row Index    Sorted Rotations   bwt_arr    l_shift  
 ~~~~~~~~~    ~~~~~~~~~~~~~~~~~~~~~~~~~~    ~~~~~~~   
     0         $  ?  ?  ?  ?  ?  a             4
     1         a  ?  ?  ?  ?  ?  n             0
     2         a  ?  ?  ?  ?  ?  n             5
     3         a  ?  ?  ?  ?  ?  b             6
    *4         b  ?  ?  ?  ?  ?  $             3
     5         n  ?  ?  ?  ?  ?  a             1
     6         n  ?  ?  ?  ?  ?  a             2

Why is the ambiguity resolving rule valid?


1. The rotations are sorted in such a way that row 5 is lexicographically less than row 6.

2. Thus, the five unknown characters in row 5 must be less than the five unknown characters in row 6 (as both start with ‘n’).

3. We also know that between the two rows than end with ‘n’, row 1 is lower than row 2.

4. But, the five unknown characters in rows 5 and 6 are precisely the first five characters in rows 1 and 2 or this would contradict the fact that rotations were sorted.

5. Thus, l_shift[5] = 1 and l_shift[6] = 2.

Way of implementation:

1. Sort BWT: Using qsort(), we arrange characters of bwt_arr[] in sorted order and store it in sorted_arr[].

2. Compute l_shift[]:
i. We take an array of pointers struct node *arr[], each of which points to a linked list.

ii. Making each distinct character of bwt_arr[] a head node of a linked list, we append nodes to the linked list whose data part contains index at which that character occurs in bwt_arr[].


   i        *arr[128]           Linked Lists
~~~~~~~~~    ~~~~~~~~~      ~~~~~~~~~~~~~~~~~~~~~~  
   37          $     ----->    4 ->  NULL
   97          a     ----->    0 -> 5 -> 6 -> NULL
   110         n     ----->    1 -> 2 -> NULL
   98          b     ----->    3 -> NULL

iii. Making distinct characters of sorted_bwt[] heads of linked lists, we traverse linked lists and get corresponding l_shift[] values.

     int[] l_shift = { 4, 0, 5, 6, 3, 1, 2 };

3. Iterating string length times, we decode BWT with x = l_shift[x] and output bwt_arr[x].

     x = l_shift[4] 
     x = 3
     bwt_arr[3] = 'b'

     x = l_shift[3] 
     x = 6
     bwt_arr[6] = 'a'

Examples:

Input : annb$aa // Burrows - Wheeler Transform
        4 // Row index at which original message 
          // appears in sorted rotations list 
Output : banana$

Input : ard$rcaaaabb
        3
Output : abracadabra$

Following is the C code for way of implementation explained above:

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// C program to find inverse of Burrows
// Wheeler transform
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
  
// Structure to store info of a node of
// linked list
struct node {
    int data;
    struct node* next;
};
  
// Compares the characters of bwt_arr[]
// and sorts them alphabetically
int cmpfunc(const void* a, const void* b)
{
    const char* ia = (const char*)a;
    const char* ib = (const char*)b;
    return strcmp(ia, ib);
}
  
// Creates the new node
struct node* getNode(int i)
{
    struct node* nn = 
        (struct node*)malloc(sizeof(struct node));
    nn->data = i;
    nn->next = NULL;
    return nn;
}
  
// Does insertion at end in the linked list
void addAtLast(struct node** head, struct node* nn)
{
    if (*head == NULL) {
        *head = nn;
        return;
    }
    struct node* temp = *head;
    while (temp->next != NULL)
        temp = temp->next;
    temp->next = nn;
}
  
// Computes l_shift[]
void* computeLShift(struct node** head, int index,
                    int* l_shift)
{
    l_shift[index] = (*head)->data;
    (*head) = (*head)->next;
}
  
void invert(char bwt_arr[])
{
    int i,len_bwt = strlen(bwt_arr);
    char* sorted_bwt = (char*)malloc(len_bwt * sizeof(char));
    strcpy(sorted_bwt, bwt_arr);
    int* l_shift = (int*)malloc(len_bwt * sizeof(int));
  
    // Index at which original string appears
    // in the sorted rotations list
    int x = 4;
  
    // Sorts the characters of bwt_arr[] alphabetically
    qsort(sorted_bwt, len_bwt, sizeof(char), cmpfunc);
  
    // Array of pointers that act as head nodes
    // to linked lists created to compute l_shift[]
    struct node* arr[128] = { NULL };
  
    // Takes each distinct character of bwt_arr[] as head
    // of a linked list and appends to it the new node
    // whose data part contains index at which
    // character occurs in bwt_arr[]
    for (i = 0; i < len_bwt; i++) {
        struct node* nn = getNode(i);
        addAtLast(&arr[bwt_arr[i]], nn);
    }
  
    // Takes each distinct character of sorted_arr[] as head
    // of a linked list and finds l_shift[]
    for (i = 0; i < len_bwt; i++)
        computeLShift(&arr[sorted_bwt[i]], i, l_shift);
  
    printf("Burrows - Wheeler Transform: %s\n", bwt_arr);
    printf("Inverse of Burrows - Wheeler Transform: ");
    // Decodes the bwt
    for (i = 0; i < len_bwt; i++) {
        x = l_shift[x];
        printf("%c", bwt_arr[x]);
    }
}
  
// Driver program to test functions above
int  main()
{
    char bwt_arr[] = "annb$aa";
    invert(bwt_arr);
    return 0;
}

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Output:


Burrows - Wheeler Transform: annb$aa
Inverse of Burrows - Wheeler Transform: banana$

Time Complexity: O(nLogn) as qsort() takes O(nLogn) time.

Exercise: Implement inverse of Inverse of Burrows – Wheeler Transform in O(n) time.

Source:
http://www.cs.princeton.edu/courses/archive/fall07/cos226/assignments/burrows.html

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