Maximum sum of a Matrix where each value is from a unique row and column
Given a matrix of size N X N, the task is to find maximum sum of this Matrix where each value picked is from a unique column for every row.
Examples:
Input: matrix = [[3, 4, 4, 4],
[1, 3, 4, 4],
[3, 2, 3, 4],
[4, 4, 4, 4]]
Output: 16
Explanation:
Selecting (0, 1) from row 1 = 4
Selecting (1, 2) from row 2 = 4
Selecting (2, 3) from row 3 = 4
Selecting (3, 0) from row 4 = 4
Therefore, max sum = 4 + 4 + 4 + 4 = 16
Input: matrix = [[0, 1, 0, 1],
[3, 0, 0, 2],
[1, 0, 2, 0],
[0, 2, 0, 0]]
Output: 8
Explanation:
Selecting (0, 3) from row 1 = 1
Selecting (1, 0) from row 2 = 3
Selecting (2, 2) from row 3 = 2
Selecting (3, 1) from row 4 = 2
Therefore, max sum = 1 + 3 + 2 + 2 = 8
Approach:
- Genrate a numeric string of size N containing numbers from 1 to N
- Find the permutation of this string (N!).
- Now pairing is done between the permutations, such that each N! pairing has a unique column for every row.
- Then calculate the sum of values for all the pairs.
Below is the implementation of the above approach:
# Python code for maximum sum of # a Matrix where each value is # from a unique row and column # Permutations using library function from itertools import permutations # Function MaxSum to find # maximum sum in matrix def MaxSum(side, matrix): s = '' # Generating the string for i in range(0, side): s = s + str(i) # Permutations of s string permlist = permutations(s) # List all possible case cases = [] # Append all possible case in cases list for perm in list(permlist): cases.append(''.join(perm)) # List to store all Sums sum = [] # Iterate over all case for c in cases: p = [] tot = 0 for i in range(0, side): p.append(matrix[int(s[i])][int(c[i])]) p.sort() for i in range(side-1, -1, -1): tot = tot + p[i] sum.append(tot) # Maximum of sum list is # the max sum print(max(sum)) # Driver code if __name__ == '__main__': side = 4 matrix = [[3, 4, 4, 4], [1, 3, 4, 4], [3, 2, 3, 4], [4, 4, 4, 4]] MaxSum(side, matrix) side = 3 matrix = [[1, 2, 3], [6, 5, 4], [7, 9, 8]] MaxSum(side, matrix) |
Output:
16 18
Time complexity: O(K), where K = N!
Auxiliary Space complexity: O(K), where K = N!
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