Suffix Array | Set 2 (nLogn Algorithm)

A suffix array is a sorted array of all suffixes of a given string. The definition is similar to Suffix Tree which is compressed trie of all suffixes of the given text.

Let the given string be "banana".

0 banana                          5 a
1 anana     Sort the Suffixes     3 ana
2 nana      ---------------->     1 anana  
3 ana        alphabetically       0 banana  
4 na                              4 na   
5 a                               2 nana

The suffix array for "banana" is {5, 3, 1, 0, 4, 2} 

We have discussed Naive algorithm for construction of suffix array. The Naive algorithm is to consider all suffixes, sort them using a O(nLogn) sorting algorithm and while sorting, maintain original indexes. Time complexity of the Naive algorithm is O(n2Logn) where n is the number of characters in the input string.

In this post, a O(nLogn) algorithm for suffix array construction is discussed. Let us first discuss a O(n * Logn * Logn) algorithm for simplicity. The idea is to use the fact that strings that are to be sorted are suffixes of a single string.
We first sort all suffixes according to first character, then according to first 2 characters, then first 4 characters and so on while the number of characters to be considered is smaller than 2n. The important point is, if we have sorted suffixes according to first 2i characters, then we can sort suffixes according to first 2i+1 characters in O(nLogn) time using a nLogn sorting algorithm like Merge Sort. This is possible as two suffixes can be compared in O(1) time (we need to compare only two values, see the below example and code).
The sort function is called O(Logn) times (Note that we increase number of characters to be considered in powers of 2). Therefore overall time complexity becomes O(nLognLogn). See http://www.stanford.edu/class/cs97si/suffix-array.pdf for more details.

Let us build suffix array the example string “banana” using above algorithm.

Sort according to first two characters Assign a rank to all suffixes using ASCII value of first character. A simple way to assign rank is to do “str[i] – ‘a'” for ith suffix of strp[]

Index     Suffix            Rank
 0        banana             1   
 1        anana              0 
 2        nana               13 
 3        ana                0
 4        na                 13
 5        a                  0 

For every character, we also store rank of next adjacent character, i.e., the rank of character at str[i + 1] (This is needed to sort the suffixes according to first 2 characters). If a character is last character, we store next rank as -1

Index    Suffix            Rank          Next Rank 
 0       banana             1              0
 1       anana              0              13    
 2       nana               13             0
 3       ana                0              13
 4       na                 13             0 
 5       a                  0             -1 

Sort all Suffixes according to rank and adjacent rank. Rank is considered as first digit or MSD, and adjacent rank is considered as second digit.

Index    Suffix            Rank          Next Rank 
 5        a                  0              -1
 1        anana              0               13    
 3        ana                0               13
 0        banana             1               0
 2        nana               13              0
 4        na                 13              0  

Sort according to first four character
Assign new ranks to all suffixes. To assign new ranks, we consider the sorted suffixes one by one. Assign 0 as new rank to first suffix. For assigning ranks to remaining suffixes, we consider rank pair of suffix just before the current suffix. If previous rank pair of a suffix is same as previous rank of suffix just before it, then assign it same rank. Otherwise assign rank of previous suffix plus one.

Index       Suffix          Rank       
  5          a               0     [Assign 0 to first]        
  1          anana           1     (0, 13) is different from previous
  3          ana             1     (0, 13) is same as previous     
  0          banana          2     (1, 0) is different from previous      
  2          nana            3     (13, 0) is different from previous      
  4          na              3     (13, 0) is same as previous      

For every suffix str[i], also store rank of next suffix at str[i + 2]. If there is no next suffix at i + 2, we store next rank as -1

Index       Suffix          Rank        Next Rank
  5          a               0             -1
  1          anana           1              1      
  3          ana             1              0 
  0          banana          2              3
  2          nana            3              3 
  4          na              3              -1       

Sort all Suffixes according to rank and next rank.

Index       Suffix          Rank        Next Rank
  5          a               0             -1
  3          ana             1              0 
  1          anana           1              1      
  0          banana          2              3
  4          na              3             -1
  2          nana            3              3       
// C++ program for building suffix array of a given text
#include <iostream>
#include <cstring>
#include <algorithm>
using namespace std;

// Structure to store information of a suffix
struct suffix
{
    int index; // To store original index
    int rank[2]; // To store ranks and next rank pair
};

// A comparison function used by sort() to compare two suffixes
// Compares two pairs, returns 1 if first pair is smaller
int cmp(struct suffix a, struct suffix b)
{
    return (a.rank[0] == b.rank[0])? (a.rank[1] < b.rank[1] ?1: 0):
               (a.rank[0] < b.rank[0] ?1: 0);
}

// This is the main function that takes a string 'txt' of size n as an
// argument, builds and return the suffix array for the given string
int *buildSuffixArray(char *txt, int n)
{
    // A structure to store suffixes and their indexes
    struct suffix suffixes[n];

    // Store suffixes and their indexes in an array of structures.
    // The structure is needed to sort the suffixes alphabatically
    // and maintain their old indexes while sorting
    for (int i = 0; i < n; i++)
    {
        suffixes[i].index = i;
        suffixes[i].rank[0] = txt[i] - 'a';
        suffixes[i].rank[1] = ((i+1) < n)? (txt[i + 1] - 'a'): -1;
    }

    // Sort the suffixes using the comparison function
    // defined above.
    sort(suffixes, suffixes+n, cmp);

    // At his point, all suffixes are sorted according to first
    // 2 characters.  Let us sort suffixes according to first 4
    // characters, then first 8 and so on
    int ind[n];  // This array is needed to get the index in suffixes[]
                 // from original index.  This mapping is needed to get
                 // next suffix.
    for (int k = 4; k < 2*n; k = k*2)
    {
        // Assigning rank and index values to first suffix
        int rank = 0;
        int prev_rank = suffixes[0].rank[0];
        suffixes[0].rank[0] = rank;
        ind[suffixes[0].index] = 0;

        // Assigning rank to suffixes
        for (int i = 1; i < n; i++)
        {
            // If first rank and next ranks are same as that of previous
            // suffix in array, assign the same new rank to this suffix
            if (suffixes[i].rank[0] == prev_rank &&
                    suffixes[i].rank[1] == suffixes[i-1].rank[1])
            {
                prev_rank = suffixes[i].rank[0];
                suffixes[i].rank[0] = rank;
            }
            else // Otherwise increment rank and assign
            {
                prev_rank = suffixes[i].rank[0];
                suffixes[i].rank[0] = ++rank;
            }
            ind[suffixes[i].index] = i;
        }

        // Assign next rank to every suffix
        for (int i = 0; i < n; i++)
        {
            int nextindex = suffixes[i].index + k/2;
            suffixes[i].rank[1] = (nextindex < n)?
                                  suffixes[ind[nextindex]].rank[0]: -1;
        }

        // Sort the suffixes according to first k characters
        sort(suffixes, suffixes+n, cmp);
    }

    // Store indexes of all sorted suffixes in the suffix array
    int *suffixArr = new int[n];
    for (int i = 0; i < n; i++)
        suffixArr[i] = suffixes[i].index;

    // Return the suffix array
    return  suffixArr;
}

// A utility function to print an array of given size
void printArr(int arr[], int n)
{
    for (int i = 0; i < n; i++)
        cout << arr[i] << " ";
    cout << endl;
}

// Driver program to test above functions
int main()
{
    char txt[] = "banana";
    int n = strlen(txt);
    int *suffixArr = buildSuffixArray(txt,  n);
    cout << "Following is suffix array for " << txt << endl;
    printArr(suffixArr, n);
    return 0;
}

Output:

Following is suffix array for banana
5 3 1 0 4 2

Note that the above algorithm uses standard sort function and therefore time complexity is O(nLognLogn). We can use Radix Sort here to reduce the time complexity to O(nLogn).

Please note that suffx arrays can be constructed in O(n) time also. We will soon be discussing O(n) algorithms.

References:
http://www.stanford.edu/class/cs97si/suffix-array.pdf
http://www.cbcb.umd.edu/confcour/Fall2012/lec14b.pdf

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