Sudo Placement 2 | Matrix Series

A Matrix series is defined as follows:

M, M^T, M(M^T), M(M^T)², M²(M^T)³, M³(M^T)⁵, M⁵(M^T)⁸ . . . . . . . ., where M is a binary square matrix of size K x K (Binary Matrix is a special type of Matrix where each element of the matrix is either 0 or 1) and M^T represents the transpose of Matrix M.

Given N and K,  find the N^th term of the series.  Prerequisites : Modular Exponentiation Examples:

Input : N = 2, K = 4
 M = {
 {1, 1},
 {0, 1}
 }
Output : [ 3 1]
 [ 2 1]
Explanation:
The 4th term of the series is M2(MT)3 and the value of M2(MT)3  
is {{3, 1}, {2, 1}}.

Input : N = 2, K = 5
 M = {
 {1, 1},
 {0, 1}
 }
Output : [7 2]
 [3 1]
Explanation:
The 4th term of the series is M3(MT)5 and the value of M3(MT)5 
is {{7, 2}, {3, 1}}.

Approach : 

It can be observed that the powers of M^T are 0, 1, 1, 2, 3, 5, 8….. for the 1^st, 2^nd, 3^rd….. terms respectively. This pattern for the powers of M^T is nothing but the Fibonacci series. Except for the first term, it can be seen that the powers of M also have the same pattern but, here the power of M is the same as the power of M^T for the previous term. 

Since in K^th term M^T has a power of fib_K, M has the power of fib_{K – 1}. Where fib_i represents the i^th fibonacci number. Thus, for the K^th term (for K ? 1) of the series can be calculated as:

Sk = Mfib(k - 1)(MT) fib(K) 

As Fibonacci values increase pretty fast the 45^th Fibonacci number is close to 10¹⁰. So the K^th power cant be calculated by repeated multiplication of the matrices K times. To do this, efficiently we can calculate the K^th power of the matrix using an idea similar to Modular Exponentiation. 

As in Modular Exponentiation, the power is divided by 2 at every step, here also we follow the same Divide and conquer strategy except the fact that here we don't multiply numbers, instead, here multiplication of matrices is required which can be done in O(N³), where N is the size of the square matrix. Below program illustrate the above approach: 

Implementation:

3 1 
2 1 
7 2 
3 1 

Time Complexity: O(N³K), where N is the size of the square matrix, and K is the power to which it is raised.

Auxiliary Space: O(K), where K is the power to which the matrix is raised.