Problem 17: Hungarian Algorithm for Assignment Problem

Problem Statement: Given a cost matrix for n workers and n tasks, compute the minimum cost assignment using the Hungarian algorithm.

Input

First line contains n. The following n lines each contain n integers representing the cost matrix.

Output

Print the minimum cost assignment.

Examples


Input:
3
9 2 7
6 4 3
5 8 1

Output:
10

Solution Explanation

Method 1: Use the Hungarian algorithm by iteratively adjusting labels and finding augmenting paths to minimize the overall assignment cost. Method 2: Utilize a primal-dual approach that maintains dual variables for rows and columns and adjusts them as you build the matching.

Solution in C++


#include 
#include 
#include 
using namespace std;

int main(){
    int n; cin >> n;
    vector> cost(n, vector(n));
    for(int i=0; i> cost[i][j];
    }
    vector u(n+1), v(n+1), p(n+1), way(n+1);
    for(int i=1;i<=n;i++){
        p[0] = i;
        int j0 = 0;
        vector minv(n+1, 1e9);
        vector used(n+1, false);
        do{
            used[j0] = true;
            int i0 = p[j0], delta = 1e9, j1 = 0;
            for (int j=1;j<=n;j++){
                if(!used[j]){
                    int cur = cost[i0-1][j-1] - u[i0] - v[j];
                    if(cur < minv[j]) { minv[j] = cur; way[j] = j0; }
                    if(minv[j] < delta) { delta = minv[j]; j1 = j; }
                }
            }
            for(int j=0;j<=n;j++){
                if(used[j]) { u[p[j]] += delta; v[j] -= delta; } else { minv[j] -= delta; }
            }
            j0 = j1;
        } while(p[j0]!=0);
        do{
            int j1 = way[j0];
            p[j0] = p[j1];
            j0 = j1;
        } while(j0);
    }
    cout << -v[0];
    return 0;
}