# Inplace (Fixed space) M x N size matrix transpose

Given an M x N matrix, transpose the matrix without auxiliary memory.It is easy to transpose matrix using an auxiliary array. If the matrix is symmetric in size, we can transpose the matrix inplace by mirroring the 2D array across it’s diagonal (try yourself). How to transpose an arbitrary size matrix inplace? See the following matrix,

a b c       a d g j
d e f  ==>  b e h k
g h i       c f i l
j k l

As per 2D numbering in C/C++, corresponding location mapping looks like,

Org element New
0     a     0
1     b     4
2     c     8
3     d     1
4     e     5
5     f     9
6     g     2
7     h     6
8     i    10
9     j     3
10    k     7
11    l    11

Note that the first and last elements stay in their original location. We can easily see the transformation forms few permutation cycles.

• 1->4->5->9->3->1  – Total 5 elements form the cycle
• 2->8->10->7->6->2 – Another 5 elements form the cycle
• 0  – Self cycle
• 11 – Self cycle

From the above example, we can easily devise an algorithm to move the elements along these cycles. How can we generate permutation cycles? Number of elements in both the matrices are constant, given by N = R * C, where R is row count and C is column count. An element at location ol (old location in R x C matrix), moved to nl (new location in C x R matrix). We need to establish relation between ol, nl, R and C. Assume ol = A[or][oc]. In C/C++ we can calculate the element address as,

ol = or x C + oc (ignore base reference for simplicity)

It is to be moved to new location nl in the transposed matrix, say nl = A[nr][nc], or in C/C++ terms

nl = nr x R + nc (R - column count, C is row count as the matrix is transposed)

Observe, nr = oc and nc = or, so replacing these for nl,

nl = oc x R + or -----> [eq 1]

after solving for relation between ol and nl, we get

ol     = or x C     + oc
ol x R = or x C x R + oc x R
= or x N     + oc x R    (from the fact R * C = N)
= or x N     + (nl - or) --- from [eq 1]
= or x (N-1) + nl

OR,

nl = ol x R - or x (N-1)

Note that the values of nl and ol never go beyond N-1, so considering modulo division on both the sides by (N-1), we get the following based on properties of congruence,

nl mod (N-1) = (ol x R - or x (N-1)) mod (N-1)
= (ol x R) mod (N-1) - or x (N-1) mod(N-1)
= ol x R mod (N-1), since second term evaluates to zero
nl = (ol x R) mod (N-1), since nl is always less than N-1

A curious reader might have observed the significance of above relation. Every location is scaled by a factor of R (row size). It is obvious from the matrix that every location is displaced by scaled factor of R. The actual multiplier depends on congruence class of (N-1), i.e. the multiplier can be both -ve and +ve value of the congruent class.Hence every location transformation is simple modulo division. These modulo divisions form cyclic permutations. We need some book keeping information to keep track of already moved elements. Here is code for inplace matrix transformation,

Implementation:

## C++

 // C++ program for in-place matrix transpose #include #define HASH_SIZE 128   using namespace std;   // A utility function to print a 2D array  // of size nr x nc and base address A void Print2DArray(int *A, int nr, int nc) {     for(int r = 0; r < nr; r++)     {         for(int c = 0; c < nc; c++)         {             cout< b; // hash to mark moved elements       b.reset();     b[0] = b[size] = 1;     i = 1; // Note that A[0] and A[size-1] won't move     while (i < size)     {         cycleBegin = i;         t = A[i];         do        {             // Input matrix [r x c]             // Output matrix             // i_new = (i*r)%(N-1)             next = (i*r)%size;             swap(A[next], t);             b[i] = 1;             i = next;         }         while (i != cycleBegin);           // Get Next Move (what about querying random location?)         for (i = 1; i < size && b[i]; i++)             ;         cout << endl;     } }   // Driver program to test above function int main() {     int r = 5, c = 6;     int size = r*c;     int *A = new int[size];       for(int i = 0; i < size; i++)         A[i] = i+1;       Print2DArray(A, r, c);     MatrixInplaceTranspose(A, r, c);     Print2DArray(A, c, r);       delete[] A;       return 0; }   // This code is contributed by rrrtnx.

## C

 // Program for in-place matrix transpose #include #include #include #define HASH_SIZE 128   using namespace std;   // A utility function to print a 2D array of size nr x nc and base address A void Print2DArray(int *A, int nr, int nc) {     for(int r = 0; r < nr; r++)     {         for(int c = 0; c < nc; c++)             printf("%4d", *(A + r*nc + c));           printf("\n");     }       printf("\n\n"); }   // Non-square matrix transpose of matrix of size r x c and base address A void MatrixInplaceTranspose(int *A, int r, int c) {     int size = r*c - 1;     int t; // holds element to be replaced, eventually becomes next element to move     int next; // location of 't' to be moved     int cycleBegin; // holds start of cycle     int i; // iterator     bitset b; // hash to mark moved elements       b.reset();     b[0] = b[size] = 1;     i = 1; // Note that A[0] and A[size-1] won't move     while (i < size)     {         cycleBegin = i;         t = A[i];         do        {             // Input matrix [r x c]             // Output matrix             // i_new = (i*r)%(N-1)             next = (i*r)%size;             swap(A[next], t);             b[i] = 1;             i = next;         }         while (i != cycleBegin);           // Get Next Move (what about querying random location?)         for (i = 1; i < size && b[i]; i++)             ;         cout << endl;     } }   // Driver program to test above function int main(void) {     int r = 5, c = 6;     int size = r*c;     int *A = new int[size];       for(int i = 0; i < size; i++)         A[i] = i+1;       Print2DArray(A, r, c);     MatrixInplaceTranspose(A, r, c);     Print2DArray(A, c, r);       delete[] A;       return 0; }

## Java

 // Java program for in-place matrix transpose import java.util.*;   class gfg {     static int HASH_SIZE = 128;       // A utility function to pr a 2D array     // of size nr x nc and base address A     static void Pr2DArray(int[] A, int nr, int nc)     {         for (int r = 0; r < nr; r++) {             for (int c = 0; c < nc; c++) {                 System.out.print(String.format(                     "%1\$" + 4 + "s", (A[r * nc + c] + "")));             }               System.out.println();         }           System.out.println();     }       // Non-square matrix transpose of     // matrix of size r x c and base address A     static int[] MatrixInplaceTranspose(int[] A, int r,                                         int c)     {         int size = r * c - 1;         int t; // holds element to be replaced,                // eventually becomes next element to move         int next; // location of 't' to be moved         int cycleBegin; // holds start of cycle         int i; // iterator         int b = 1; // hash to mark moved elements           b |= (1 << size);           i = 1; // Note that A[0] and A[size-1] won't move         while (i < size) {             cycleBegin = i;             t = A[i];             do {                 // Input matrix [r x c]                 // Output matrix                 // i_new = (i*r)%(N-1)                   next = (i * r) % size;                 int temp = A[next];                 A[next] = t;                 t = temp;                 b |= (1 << i);                 i = next;             } while (i != cycleBegin);               // Get Next Move (what about querying random             // location?)             for (i = 1; i < size && ((b & (1 << i)) != 0);                  i++)                 ;             System.out.println();         }         return A;     }       // Driver program to test above function     public static void main(String[] args)     {         int r = 5, c = 6;         int size = r * c;         int[] A = new int[size];           for (int i = 0; i < size; i++)             A[i] = i + 1;           Pr2DArray(A, r, c);         A = MatrixInplaceTranspose(A, r, c);         Pr2DArray(A, c, r);     } } // This code is contributed by karandeep1234

## Python3

 #  Python3 program for in-place matrix transpose HASH_SIZE = 128  #  A utility function to pr a 2D array  #  of size nr x nc and base address A def Pr2DArray( A,  nr,  nc):           for r in range(nr):          for c in range(nc):               print('{0: >4}'.format(str(A[r * nc + c])), end = "")         print()     print()   #  Non-square matrix transpose of #  matrix of size r x c and base address A def MatrixInplaceTranspose( A,  r,  c):     size = r * c - 1;           b = 1; #  hash to mark moved elements           b |= (1 << size);       i = 1; #  Note that A[0] and A[size-1] won't move     while (i < size):               cycleBegin = i;         t = A[i];         while True:                       #  Input matrix [r x c]             #  Output matrix             #  i_new = (i*r)%(N-1)             next1 = (i*r)%size;             temp = A[next1]             A[next1] = t             t = temp             b |= (1 << i)              i = next1;             if i == cycleBegin:                 break                  #  Get next1 Move (what about querying random location?)         i = 1         while i < size and ( (b & (1 << i)) != 0):             i += 1                  print()     return A   #  Driver program to test above function r = 5c = 6; size = r*c; A = [i + 1 for i in range(size)]   Pr2DArray(A, r, c); A = MatrixInplaceTranspose(A, r, c); Pr2DArray(A, c, r);   #  This code is contributed by phasing17.

## C#

 // C# program for in-place matrix transpose using System; using System.Collections.Generic;   class GFg {   static int HASH_SIZE = 128;     // A utility function to pr a 2D array    // of size nr x nc and base address A   static void Pr2DArray(int[] A, int  nr, int  nc)   {     for(int r = 0; r < nr; r++)     {       for(int c = 0; c < nc; c++)       {         Console.Write(Convert.ToString(A[r * nc + c]).PadLeft(4));       }         Console.WriteLine();     }       Console.WriteLine();   }     // Non-square matrix transpose of   // matrix of size r x c and base address A   static int[] MatrixInplaceTranspose(int[] A, int  r, int  c)   {     int size = r * c - 1;     int t; // holds element to be replaced,      // eventually becomes next element to move     int next; // location of 't' to be moved     int cycleBegin; // holds start of cycle     int i; // iterator     int b = 1; // hash to mark moved elements       b |= (1 << size);       i = 1; // Note that A[0] and A[size-1] won't move     while (i < size)     {       cycleBegin = i;       t = A[i];       do      {         // Input matrix [r x c]         // Output matrix         // i_new = (i*r)%(N-1)           next = (i*r)%size;         int temp = A[next];         A[next] = t;         t = temp;         b |= (1 << i);          i = next;       }       while (i != cycleBegin);         // Get Next Move (what about querying random location?)       for (i = 1; i < size && ( (b & (1 << i)) != 0); i++)         ;       Console.WriteLine();     }     return A;   }     // Driver program to test above function   public static void Main(string[] args)   {     int r = 5, c = 6;     int size = r*c;     int[] A = new int[size];       for(int i = 0; i < size; i++)       A[i] = i+1;       Pr2DArray(A, r, c);     A = MatrixInplaceTranspose(A, r, c);     Pr2DArray(A, c, r);   } }   // This code is contributed by phasing17.

## Javascript

 // JS program for in-place matrix transpose   let HASH_SIZE = 128     // A utility function to pr a 2D array  // of size nr x nc and base address A  function Pr2DArray( A,  nr,  nc) {     for(let r = 0; r < nr; r++)     {         for(let c = 0; c < nc; c++)         {             process.stdout.write((A[r * nc + c].toString()).padStart(4))         }           console.log()     }       console.log() }   // Non-square matrix transpose of // matrix of size r x c and base address A function MatrixInplaceTranspose( A,  r,  c) {     let size = r * c - 1;     let t; // holds element to be replaced,             // eventually becomes next element to move     let next; // location of 't' to be moved     let cycleBegin; // holds start of cycle     let i; // iterator     let b = 1; // hash to mark moved elements           b |= (1 << size);       i = 1; // Note that A[0] and A[size-1] won't move     while (i < size)     {         cycleBegin = i;         t = A[i];         do        {             // Input matrix [r x c]             // Output matrix             // i_new = (i*r)%(N-1)                           next = (i*r)%size;             let temp = A[next]             A[next] = t             t = temp             b |= (1 << i)              i = next;         }         while (i != cycleBegin);           // Get Next Move (what about querying random location?)         for (i = 1; i < size && ( (b & (1 << i)) != 0); i++)             ;         console.log()     }     return A }   // Driver program to test above function let r = 5, c = 6; let size = r*c; let A = new Array(size);       for( i = 0; i < size; i++)         A[i] = i+1;       Pr2DArray(A, r, c);     A = MatrixInplaceTranspose(A, r, c);     Pr2DArray(A, c, r);       // This code is contributed by phasing17.

Output
1   2   3   4   5   6
7   8   9  10  11  12
13  14  15  16  17  18
19  20  21  22  23  24
25  26  27  28  29  30

1   7  13  19  25
2   8  14  20  26
3   9  15  21  27
4  10  16  22  28
5  11  17  23  29
6  12  18  24  30

Time Complexity: O(R*C), here R & C are number number of Rows and Columns respectively.

Auxiliary Space : O(size of the bitset)

Extension: 17 – March – 2013 Some readers identified similarity between the matrix transpose and string transformation. Without much theory I am presenting the problem and solution. In given array of elements like [a1b2c3d4e5f6g7h8i9j1k2l3m4]. Convert it to [abcdefghijklm1234567891234]. The program should run inplace. What we need is an inplace transpose. Given below is code.

Implementation:

## C++

 #include #define HASH_SIZE 128   using namespace std;   typedef char data_t;   void Print2DArray(char A[], int nr, int nc) {    int size = nr*nc;    for(int i = 0; i < size; i++)         cout< b; // hash to mark moved elements      b.reset();    b[0] = b[size] = 1;    i = 1; // Note that A[0] and A[size-1] won't move    while( i < size ) {       cycleBegin = i;       t = A[i];       do {          // Input matrix [r x c]          // Output matrix          // i_new = (i*r)%size          next = (i*r)%size;          swap(A[next], t);          b[i] = 1;          i = next;       } while( i != cycleBegin );         // Get Next Move (what about querying random location?)       for(i = 1; i < size && b[i]; i++)          ;       cout << endl;    } }   void Fill(data_t buf[], int size) {    // Fill abcd ...    for(int i = 0; i < size; i++)    buf[i] = 'a'+i;      // Fill 0123 ...    buf += size;    for(int i = 0; i < size; i++)       buf[i] = '0'+i; }   void TestCase_01(void) {    int r = 2, c = 10;    int size = r*c;    data_t *A = new data_t[size];      Fill(A, c);      Print2DArray(A, r, c), cout << endl;    MatrixTransposeInplaceArrangement(A, r, c);    Print2DArray(A, c, r), cout << endl;      delete[] A; }   int main() {    TestCase_01();      return 0; }   // This code is contributed by rutvik_56.

## C

 #include #include #include #define HASH_SIZE 128   using namespace std;   typedef char data_t;   void Print2DArray(char A[], int nr, int nc) {    int size = nr*nc;    for(int i = 0; i < size; i++)       printf("%4c", *(A + i));      printf("\n"); }   void MatrixTransposeInplaceArrangement(data_t A[], int r, int c) {    int size = r*c - 1;    data_t t; // holds element to be replaced, eventually becomes next element to move    int next; // location of 't' to be moved    int cycleBegin; // holds start of cycle    int i; // iterator    bitset b; // hash to mark moved elements      b.reset();    b[0] = b[size] = 1;    i = 1; // Note that A[0] and A[size-1] won't move    while( i < size ) {       cycleBegin = i;       t = A[i];       do {          // Input matrix [r x c]          // Output matrix          // i_new = (i*r)%size          next = (i*r)%size;          swap(A[next], t);          b[i] = 1;          i = next;       } while( i != cycleBegin );         // Get Next Move (what about querying random location?)       for(i = 1; i < size && b[i]; i++)          ;       cout << endl;    } }   void Fill(data_t buf[], int size) {    // Fill abcd ...    for(int i = 0; i < size; i++)    buf[i] = 'a'+i;      // Fill 0123 ...    buf += size;    for(int i = 0; i < size; i++)       buf[i] = '0'+i; }   void TestCase_01(void) {    int r = 2, c = 10;    int size = r*c;    data_t *A = new data_t[size];      Fill(A, c);      Print2DArray(A, r, c), cout << endl;    MatrixTransposeInplaceArrangement(A, r, c);    Print2DArray(A, c, r), cout << endl;      delete[] A; }   int main() {    TestCase_01();      return 0; }

## Java

 import java.util.*;   class GFG {   static int HASH_SIZE = 128;     public static String padLeft(String s, int n)   {     return String.format("%" + n + "s", s);   }     static void Print2DArray(char[] A, int nr, int nc)   {     int size = nr * nc;     for (int i = 0; i < size; i++)       System.out.print(       padLeft(String.valueOf(A[i]), 4));     System.out.print("\n");   }     static void MatrixTransposeInplaceArrangement(char[] A,                                                 int r,                                                 int c)   {     int size = r * c - 1;     char t; // holds element to be replaced, eventually     // becomes next element to move     int next; // location of 't' to be moved     int cycleBegin; // holds start of cycle     int i; // iterator     int b = 1; // hash to mark moved elements       b |= (1 << size);     i = 1; // Note that A[0] and A[size-1] won't move     while (i < size) {       cycleBegin = i;       t = A[i];       do {         // Input matrix [r x c]         // Output matrix         // i_new = (i*r)%size         next = (i * r) % size;         char temp = t;         t = A[next];         A[next] = temp;         b |= (1 << i);         i = next;       } while (i != cycleBegin);         // Get Next Move (what about querying random       // location?)       for (i = 1; i < size && ((b & (1 << i)) != 0);            i++)         ;       System.out.print("\n");     }   }     static void Fill(char[] buf, int size)   {     // Fill abcd ...     char[] intters       = { 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i',          'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r',          's', 't', 'u', 'v', 'w', 'x', 'y', 'z' };     for (var i = 0; i < size; i++)       buf[i] = intters[i];       // Fill 0123 ...     for (var i = size; i < 2 * size; i++)       buf[i] = (char)(48 + i - size);   }     static void TestCase_01()   {     int r = 2, c = 10;     int size = r * c;     char[] A = new char[size];       Fill(A, c);       Print2DArray(A, r, c);     MatrixTransposeInplaceArrangement(A, r, c);     Print2DArray(A, c, r);     System.out.print("\n");   }     public static void main(String[] args)   {     TestCase_01();   } }   // This code is contributed by phasing17.

## Python3

 HASH_SIZE = 128    def Print2DArray(A, nr, nc):     size = nr*nc     for i in range(size):         print(str(A[i]).rjust(4), end="")     def MatrixTransposeInplaceArrangement(A, r, c):     size = r*c - 1    b = 1  # hash to mark moved elements       b |= (1 << size)     i = 1  # Note that A[0] and A[size-1] won't move     while(i < size):         cycleBegin = i         t = A[i]         while True:             # Input matrix [r x c]             # Output matrix             # i_new = (i*r)%size             next1 = (i*r) % size             t, A[next1] = A[next1], t             b |= (1 << i)             i = next1             if (i == cycleBegin):                 break          # Get next1 Move (what about querying random location?)         i = 1        while (i < size and (b & (1 << i)) != 0):             i += 1      print()     def Fill(buf, size):     # Fill abcd ...     letters = ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l',                'm', 'n', 'o', 'p', 'q', 'r', 's', 't', 'u', 'v', 'w', 'x', 'y', 'z']     for i in range(size):         buf[i] = letters[i]       # Fill 0123 ..     for i in range(size, 2 * size):         buf[i] = str(i - size)     def TestCase_01():     r = 2    c = 10    size = r*c     A = [0] * size       Fill(A, c)       Print2DArray(A, r, c)     MatrixTransposeInplaceArrangement(A, r, c)     Print2DArray(A, c, r)     print()     TestCase_01()

## C#

 using System; using System.Collections.Generic;   class GFG {   static int HASH_SIZE = 128;     static void Print2DArray(char[] A, int nr, int nc)   {     int size = nr * nc;     for (int i = 0; i < size; i++)       Console.Write(       Convert.ToString(A[i]).PadLeft(4));     Console.Write("\n");   }     static void MatrixTransposeInplaceArrangement(char[] A,                                                 int r,                                                 int c)   {     int size = r * c - 1;     char t; // holds element to be replaced, eventually     // becomes next element to move     int next; // location of 't' to be moved     int cycleBegin; // holds start of cycle     int i; // iterator     int b = 1; // hash to mark moved elements       b |= (1 << size);     i = 1; // Note that A[0] and A[size-1] won't move     while (i < size) {       cycleBegin = i;       t = A[i];       do {         // Input matrix [r x c]         // Output matrix         // i_new = (i*r)%size         next = (i * r) % size;         char temp = t;         t = A[next];         A[next] = temp;         b |= (1 << i);         i = next;       } while (i != cycleBegin);         // Get Next Move (what about querying random       // location?)       for (i = 1; i < size && ((b & (1 << i)) != 0);            i++)         ;       Console.Write("\n");     }   }     static void Fill(char[] buf, int size)   {     // Fill abcd ...     char[] intters       = { 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i',          'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r',          's', 't', 'u', 'v', 'w', 'x', 'y', 'z' };     for (var i = 0; i < size; i++)       buf[i] = intters[i];       // Fill 0123 ...     for (var i = size; i < 2 * size; i++)       buf[i] = Convert.ToChar(48 + i - size);   }     static void TestCase_01()   {     int r = 2, c = 10;     int size = r * c;     char[] A = new char[size];       Fill(A, c);       Print2DArray(A, r, c);     MatrixTransposeInplaceArrangement(A, r, c);     Print2DArray(A, c, r);     Console.Write("\n");   }     public static void Main(string[] args)   {     TestCase_01();   } }   // This code is contributed by phasing17.

## Javascript

 let HASH_SIZE = 128   function Print2DArray(A, nr, nc) {    let size = nr*nc;    for(var i = 0; i < size; i++)         process.stdout.write(String(A[i]).padStart(4))     console.log() }   function MatrixTransposeInplaceArrangement(A, r, c) {    let size = r*c - 1;    let t; // holds element to be replaced, eventually becomes next element to move    let next; // location of 't' to be moved    let cycleBegin; // holds start of cycle    let i; // iterator    let b = 1; // hash to mark moved elements           b |= (1 << size);     i = 1; // Note that A[0] and A[size-1] won't move    while( i < size ) {       cycleBegin = i;       t =A[i]       do {          // Input matrix [r x c]          // Output matrix          // i_new = (i*r)%size          next = (i*r)%size;          let temp = t          t = A[next]          A[next] = temp          b |= (1 << i);          i = next;       } while( i != cycleBegin );         // Get Next Move (what about querying random location?)       for(i = 1; i < size && (b & (1 << i)); i++)          ;       console.log()    } }   function Fill(buf, size) {    // Fill abcd ...    let letters = ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r', 's', 't', 'u', 'v', 'w', 'x', 'y', 'z']    for(var i = 0; i < size; i++)    buf[i] = letters[i];      // Fill 0123 ...    for(var i = size; i < 2 * size; i++)       buf[i] = String(i - size); }   function TestCase_01() {    let r = 2, c = 10;    let size = r*c;    let A = new Array(size);      Fill(A, c);      Print2DArray(A, r, c);    MatrixTransposeInplaceArrangement(A, r, c);    Print2DArray(A, c, r)     console.log() }   TestCase_01();     // This code is contributed by phasing17.

Output
a   b   c   d   e   f   g   h   i   j   0   1   2   3   4   5   6   7   8   9

a   0   b   1   c   2   d   3   e   4   f   5   g   6   h   7   i   8   j   9

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Update 09-July-2016: Notes on space complexity and storage order.

After long time, it happened to review this post. Some readers pointed valid questions on how can it be in-place (?) when we are using bitset as marker (hash in code). Apologies for incorrect perception by looking at the article heading or content. While preparing the initial content, I was thinking of naive implementation using auxiliary space of atleast O(MN) needed to transpose rectangular matrix. The program presented above is using constant space as bitset size is fixed at compile time. However, to support arbitrary size of matrices we need bitset size atleast O(MN) size. One can use a HashMap (amortized O(1) complexity) for marking finished locations, yet HashMap’s worst case complexity can be O(N) or O(log N) based on implementation. HashMap space cost also increases based on items inserted. Please note that in-place was used w.r.t. matrix space.

Also, it was assumed that the matrix will be stored in row major ordering (contigueous locations in memory). The reader can derive the formulae, if the matrix is represented in column major order by the programming language (e.g. Fortran/Julia).
Thanks to the readers who pointed these two gaps.
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The post is incomplete without mentioning two links.

1. Aashish covered good theory behind cycle leader algorithm. See his post on string transformation.
2. As usual, Sambasiva demonstrated his exceptional skills in recursion to the problem. Ensure to understand his solution.

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