Problem:
Score : 100 points

Problem Statement

There is a simple directed graph G with N vertices, numbered 1, 2, \ldots, N.

For each i and j (1 \leq i, j \leq N), you are given an integer a_{i, j} that represents whether there is a directed edge from Vertex i to j.
If a_{i, j} = 1, there is a directed edge from Vertex i to j; if a_{i, j} = 0, there is not.

Find the number of different directed paths of length K in G, modulo 10^9 + 7.
We will also count a path that traverses the same edge multiple times.

Constraints

All values in input are integers.

1 \leq N \leq 50

1 \leq K \leq 10^{18}

a_{i, j} is 0 or 1.

a_{i, i} = 0

Input

Input is given from Standard Input in the following format:

N K
a_{1, 1} \ldots a_{1, N}
:
a_{N, 1} \ldots a_{N, N}

Output

Print the number of different directed paths of length K in G, modulo 10^9 + 7.

Sample Input 1

4 2
0 1 0 0
0 0 1 1
0 0 0 1
1 0 0 0

Sample Output 1

6

G is drawn in the figure below:

There are six directed paths of length 2:

1 → 2 → 3

1 → 2 → 4

2 → 3 → 4

2 → 4 → 1

3 → 4 → 1

4 → 1 → 2

Sample Input 2

3 3
0 1 0
1 0 1
0 0 0

Sample Output 2

3

G is drawn in the figure below:

There are three directed paths of length 3:

1 → 2 → 1 → 2

2 → 1 → 2 → 1

2 → 1 → 2 → 3

Sample Input 3

6 2
0 0 0 0 0 0
0 0 1 0 0 0
0 0 0 0 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 0 0 0 0

Sample Output 3

1

G is drawn in the figure below:

There is one directed path of length 2:

4 → 5 → 6

Sample Input 4

1 1
0

Sample Output 4

0

Sample Input 5

10 1000000000000000000
0 0 1 1 0 0 0 1 1 0
0 0 0 0 0 1 1 1 0 0
0 1 0 0 0 1 0 1 0 1
1 1 1 0 1 1 0 1 1 0
0 1 1 1 0 1 0 1 1 1
0 0 0 1 0 0 1 0 1 0
0 0 0 1 1 0 0 1 0 1
1 0 0 0 1 0 1 0 0 0
0 0 0 0 0 1 0 0 0 0
1 0 1 1 1 0 1 1 1 0

Sample Output 5

957538352

Be sure to print the count modulo 10^9 + 7.

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <stdio.h>
#include <cstdio>
#include <iostream>
#include <queue>
#include <set>
#include <vector>
#include <cstring>

const long long int MOD = 1e9 + 7;

long long int fast_power(std::vector<std::vector<long long int> > &X, long long int N, long long int n, std::vector<std::vector<long long int> > &ANS);
long long int mat_prod(std::vector<std::vector<long long int> > &M1, std::vector<std::vector<long long int> > &M2, long long int N, std::vector<std::vector<long long int> > &ANS);
int mat_copy(std::vector<std::vector<long long int> > &Orig, long long int size, std::vector<std::vector<long long int> > &Copy);


int main(){
  
  // read problem
  long long int N, K, tmp;
  scanf("%lld%lld", &N, &K);
  
  std::vector<std::vector<long long int> > a(N);
  for(long long int i = 0; i < N; i++){
    for(long long int j = 0; j < N; j++){
      scanf("%lld", &tmp);
      a[i].push_back(tmp);
    }
  }
  
  std::vector<std::vector<long long int> > ANS(N);
  for(long long int i = 0; i < N; i++){
    for(long long int j = 0; j < N; j++){
      ANS[i].push_back(0);
    }
  }
  
  fast_power(a, N, K, ANS);
  
  long long int ans = 0;
  for(long long int i = 0; i < N; i++){
    for(long long int j = 0; j < N; j++){
      ans += ANS[i][j];
      ans %= MOD;
    }
  }

  std::cout << ans << std::endl;
  
  return 0;
}

long long int fast_power(std::vector<std::vector<long long int> > &X, long long int N, long long int n, std::vector<std::vector<long long int> > &ANS){
  
  // prepare ANS as unit matrix
  std::vector<std::vector<long long int> > Ans_tmp(N), X_new(N);
  for(long long int i = 0; i < N; i++){
    for(long long int j = 0; j < N; j++){
      ANS[i][j] = 0;
      Ans_tmp[i].push_back(0);
      X_new[i].push_back(0);
    }
  }
  for(long long int i = 0; i < N; i++){
    ANS[i][i] = 1;
  }

  // solve problem
  while(n > 0){
    if(n&1){
      mat_prod(ANS, X, N, Ans_tmp);
      mat_copy(Ans_tmp, N, ANS);
      //ANS = Ans_tmp;
    }
    mat_prod(X, X, N, X_new);
    mat_copy(X_new, N, X);
    //X = X_new;
    n = n>>1;
  }
  
  return 0;
}

long long int mat_prod(std::vector<std::vector<long long int> > &M1, std::vector<std::vector<long long int> > &M2, long long int size, std::vector<std::vector<long long int> > &ANS){
  for(long long int i = 0; i < size; i++){
    for(long long int j = 0; j < size; j++){
      ANS[i][j] = 0;
    }
  }
  for(long long int i = 0; i < size; i++){
    for(long long int k = 0; k < size; k++){
      for(long long int j = 0; j < size; j++){
        ANS[i][j] += M1[i][k] * M2[k][j];
        ANS[i][j] %= MOD;
      }
    }
  }
  
  return 0;
}

int mat_copy(std::vector<std::vector<long long int> > &Orig, long long int size, std::vector<std::vector<long long int> > &Copy){
  for(long long int i = 0; i < size; i++){
    for(long long int j = 0; j < size; j++){
      Copy[i][j] = Orig[i][j];
    }
  }
  return 0;
}