Educational DP Contest AtCoder R - Walk

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Points: 15
Time limit: 0.6s
Memory limit: 1G

Problem types

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

For each i and j (1 \le i, j \le 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 \le N \le 50
  • 1 \le K \le 10^{18}
  • a_{i,j} is 0 or 1.
  • a_{i,i} = 0

Input Specification

The first line will contain 2 space separated integers N, K.

The next N lines will each contain N space separated integers, a_{i,j}.

Output Specification

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

Explanation For Sample 1

G is drawn in the figure below:

There are six directed paths of length 2:

  • 1 \to 2 \to 3
  • 1 \to 2 \to 4
  • 2 \to 3 \to 4
  • 2 \to 4 \to 1
  • 3 \to 4 \to 1
  • 4 \to 1 \to 2

Sample Input 2

3 3
0 1 0
1 0 1
0 0 0

Sample Output 2

3

Explanation For Sample 2

G is drawn in the figure below:

There are three directed paths of length 3:

  • 1 \to 2 \to 1 \to 2
  • 2 \to 1 \to 2 \to 1
  • 2 \to 1 \to 2 \to 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

Explanation For Sample 3

G is drawn in the figure below:

There is one directed path of length 2:

  • 4 \to 5 \to 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

Explanation For Sample 5

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


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