Question

Can anyone please explain or suggest some good tutorial for the method of matrix exponentiation in order to optimize the solution of the problem :
great wall of byteland

The solution which I uploaded is based dynamic programming with this underlying equation :
[ f(n) = f(n-1) + 4*f(n-2) + 2*f(n-3) ]
but the solution is giving me Time Limit Exceeded Error.

This is the code I built :

    #include<iostream>
    #define num 1000000007
    using namespace std;
    int main(){
        int t;
        cin>>t;
        while(t--){
            int n;
            cin>>n;
            if(n<=3){
                switch(n){
                    case 1:
                        cout<<1<<endl;
                        break;
                    case 2:
                        cout<<5<<endl;
                        break;
                    case 3:
                        cout<<11<<endl;
                        break;
                }
            }
            else{
                int a=1 , b=5 , c=11 ;
                int next;
                for(int i=4;i<=n;i++){
                    next = (c + 4*b + 2*a)%num ;
                    a = b;
                    b = c;
                    c = next;
                }
                cout<<next<<endl;
            }
        }
        return 0;
    }

Please suggest the matrix exponentiation method for optimizing the run time of the solution.

Answer 1

If you have a sequence defined by:

u(0) to u(d-1) are given
for n > d u(n)=a(1)*u(n-1)+…+a(d)*u(n-d)

then let A be the companion matrix defined by:

A(i,j) = a(d+1-j) if i = d
         1 if i+1 = j
         0 otherwise

and let uinit = transpose(u(0) … u(d-1))

You have A^n*uinit = transpose(u(n) … u(n+d-1)) (you can verify by yourself that A*transpose(u(n) … u(n+d-1)) = transpose(u(n+1) … u(n+d))).

Then you can compute A^n in O(Log(n)) and use it to compute u(n).