Disgust Minimization Codechef Solution

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Disgust Minimization Codechef Solution

Disgust Minimization Codechef Solution

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Problem

You have a matrix A with dimensions N \times N. Let the disgust of the matrix be defined as:

\sum_{i=1}^{N} \sum_{j=1}^{N} (A_{(i, j)} – A_{(j, i)})^2

You also have M tuples with you. Each tuple consists of 3 integers X_i, Y_i, ans Z_i (1\le i \le M). In one operation, you can:

  • Choose an element (p, q) (1\le p,q \le N) such that the value of the element A_{(p,q)} = X_i.
  • Change the value of the chosen element to Y_i.

The cost of applying the above operation is Z_i. Note that (1 \le i \le M) and thus, the operation might correspond to any one of the given tuples.

Find the minimum value of (totalCost + disgust) after applying the given operation any (possibly zero) number of times.

Input Format

  • The first line contains one integer T – the number of test cases. Then T test cases follow.
  • The first line of each test case consists of two integer N and M, the row size and the number of tuples.
  • N lines follow, each consisting of N space-separated integers – the matrix A.
  • M lines follow, each consisting of 3 space-separated integers – each tuple X_i, Y_i, Z_i.

Output Format

For each test case, output the minimum value of (totalCost + disgust) after applying the given operation any (possibly zero) number of times.

Constraints

  • 1 \leq T \leq 100
  • 1 \leq N \leq 500
  • 1 \leq M \leq 10^{5}
  • 1 \leq A_{(i, j)} \leq N for (1\le i, j \le N)
  • 1 \leq X_i, Y_i \leq N for (1 \le i \le M)
  • 1 \leq Z_i \leq 10^9 for (1 \le i \le M)
  • It is guaranteed that the sum of N over all test cases does not exceed 500.
  • It is guaranteed that the sum of M over all test cases does not exceed 10^5.

Sample 1:

Input

Output

3
3 2
3 3 1
1 2 1
3 2 1
3 2 2
1 2 3
2 1
2 2
2 2
1 1 3
3 2
3 3 1
1 2 1
3 2 1
2 1 1
3 2 1
10
0
5

 

 

Explanation:

Test case 1: We convert A_{(1, 2)} to 2 for a cost of 2 and we convert A_{(3, 1)} to 2 for a cost of 2. The total cost is 2+2=4 and disgust is (3-3)^2 + (2-1)^2 + (1-2)^2 + (1-2)^2 + (2-2)^2 + (1-2)^2 + (2-1)^2 + (2-1)^2 + (1-1)^2 = 6. Thus, the value (totalCost + disgust) for the new matrix is 4+6=10. We can show that this is the minimum value possible.

Test case 2: We do not need to perform any operations. Thus, the value:
(totalCost + disgust) = 0 + (2-2)^2 + (2-2)^2 + (2-2)^2 + (2-2)^2 = 0.

Test case 3: We perform the following operations:

  • We choose A_{(1, 2)} and change it to 2 for a cost of 1. Then, we further change it to 1 for a cost of 1.
  • We choose A_{(3, 1)} and change it to 2 for a cost of 1. Then, we further change it to 1 for a cost of 1.
  • We choose A_{(3, 2)} and change it to 1 for a cost of 1. The new disgust is 0 while the cost is 5. The value (totalCost + disgust) for the new matrix is 5. We can verify that this is the minimum possible value.
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SOLUTION

Disgust Minimization Codechef Solution

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