J. Hero to Zero
graph matchings; math
There are no heroes in this problem. I guess we should have named it ""To Zero"".You are given two arrays \(a\) and \(b\), each of these arrays contains \(n\) non-negative integers.Let \(c\) be a matrix of size \(n \times n\) such that \(c_{i,j} = |a_i - b_j|\) for every \(i \in [1, n]\) and every \(j \in [1, n]\).Your goal is to transform the matrix \(c\) so that it becomes the zero matrix, i. e. a matrix where every element is exactly \(0\). In order to do so, you may perform the following operations any number of times, in any order: choose an integer \(i\), then decrease \(c_{i,j}\) by \(1\) for every \(j \in [1, n]\) (i. e. decrease all elements in the \(i\)-th row by \(1\)). In order to perform this operation, you pay \(1\) coin; choose an integer \(j\), then decrease \(c_{i,j}\) by \(1\) for every \(i \in [1, n]\) (i. e. decrease all elements in the \(j\)-th column by \(1\)). In order to perform this operation, you pay \(1\) coin; choose two integers \(i\) and \(j\), then decrease \(c_{i,j}\) by \(1\). In order to perform this operation, you pay \(1\) coin; choose an integer \(i\), then increase \(c_{i,j}\) by \(1\) for every \(j \in [1, n]\) (i. e. increase all elements in the \(i\)-th row by \(1\)). When you perform this operation, you receive \(1\) coin; choose an integer \(j\), then increase \(c_{i,j}\) by \(1\) for every \(i \in [1, n]\) (i. e. increase all elements in the \(j\)-th column by \(1\)). When you perform this operation, you receive \(1\) coin. You have to calculate the minimum number of coins required to transform the matrix \(c\) into the zero matrix. Note that all elements of \(c\) should be equal to \(0\) simultaneously after the operations.
The first line contains one integer \(n\) (\(2 \le n \le 2 \cdot 10^5\)).The second line contains \(n\) integers \(a_1, a_2, \dots, a_n\) (\(0 \le a_i \le 10^8\)).The third line contains \(n\) integers \(b_1, b_2, \dots, b_n\) (\(0 \le b_j \le 10^8\)).
Print one integer — the minimum number of coins required to transform the matrix \(c\) into the zero matrix.
In the first example, the matrix looks as follows: 111000111 You can turn it into a zero matrix using \(2\) coins as follows: subtract \(1\) from the first row, paying \(1\) coin; subtract \(1\) from the third row, paying \(1\) coin. In the second example, the matrix looks as follows: 221001221 You can turn it into a zero matrix using \(5\) coins as follows: subtract \(1\) from the first row, paying \(1\) coin; subtract \(1\) from the third row, paying \(1\) coin; subtract \(1\) from the third row, paying \(1\) coin; subtract \(1\) from \(a_{2,3}\), paying \(1\) coin; add \(1\) to the third column, receiving \(1\) coin; subtract \(1\) from the first row, paying \(1\) coin; subtract \(1\) from \(a_{2,3}\), paying \(1\) coin.