MinCostDijkstra.cpp

/************************************************************************************

    Min Cost Flow (or Min Cost Max Flow) algorithm with
    Dijkstra algorithm (with potentials) as shortest path search method. 
    (Dijkstra for dense graphs running in O(N^2))

    Works O(N ^ 5). Less on practice. 
    Runs in O(N ^ 3) for bipartite matching case.

    Based on problem 394 from informatics.mccme.ru 
    http://informatics.mccme.ru//mod/statements/view3.php?chapterid=394

************************************************************************************/

#include <iostream>
#include <fstream>
#include <cmath>
#include <algorithm>
#include <vector>
#include <set>
#include <map>
#include <stack>
#include <queue>
#include <cstdlib>
#include <cstdio>
#include <string>
#include <cstring>
#include <cassert>
#include <utility>
#include <iomanip>

using namespace std;

const int MAXN = 1050;
const long long INF = (long long) 1e15;

struct edge {
    int from, to;
    int flow, cap;
    long long cost;
};

int n;
int cost[MAXN][MAXN];
vector <edge> e;
vector <int> g[MAXN];
long long phi[MAXN];
long long dist[MAXN];
bool used[MAXN];
int par[MAXN];
int edge_num;
int s = 0, t = MAXN - 1;

void fordBellman() {
    for (int i = s; i <= t; i++) 
        dist[i] = INF;
    dist[s] = 0;
    while (true) {
        bool change = false;
        for (int i = 0; i < edge_num; i++) {
            int from = e[i].from, to = e[i].to;
            if (e[i].flow == e[i].cap)
                continue;
            if (dist[from] == INF)
                continue;
            if (dist[to] > dist[from] + e[i].cost) {
                dist[to] = dist[from] + e[i].cost;
                change = true;
            }
        }
        if (!change)
            break;
    }
}

void dijkstra() {
    for (int i = s; i <= t; i++) {
        dist[i] = INF;
        used[i] = false;
    }
    dist[s] = 0;

    for (int i = s; i <= t; i++) {
        int cur = -1;
        for (int j = s; j <= t; j++)    
            if (!used[j])
                if (cur == -1 || dist[j] < dist[cur])
                    cur = j;     

        used[cur] = true;

        for (int j = 0; j < (int) g[cur].size(); j++) {
            int ind = g[cur][j];
            if (e[ind].flow == e[ind].cap)
                continue;
            int to = e[ind].to;
            int w = e[ind].cost + phi[cur] - phi[to];
            if (dist[cur] + w < dist[to]) {
                dist[to] = dist[cur] + w;
                par[to] = ind;
            }
        }
    }
}

long long minCost(int flow) {
    long long result = 0;

    fordBellman();  
    for (int i = s; i <= t; i++)
        phi[i] = dist[i];

    while (true) {

        dijkstra();

        if (dist[t] == INF)
            return result;

        for (int i = s; i <= t; i++)
            phi[i] = min(phi[i] + dist[i], INF);

        int push = flow;
        int cur = t;
        while (cur != s) {
            edge tmp = e[par[cur]];
            int from = tmp.from, can_push = tmp.cap - tmp.flow;
            push = min(push, can_push);
            cur = from;
        }

        flow -= push;
        cur = t;
        while (cur != s) {
            edge tmp = e[par[cur]];
            int from = tmp.from;
            e[par[cur]].flow += push;
            e[par[cur] ^ 1].flow -= push;
            result += 1ll * push * tmp.cost;
            cur = from;
        }

        if (flow == 0)
            break;
    }
    return result;
}   

void addEdge(int from, int to, int cap, long long cost) {
    edge tmp;
    tmp.from = from; tmp.to = to; tmp.flow = 0; tmp.cap = cap; tmp.cost = cost;
    e.push_back(tmp);
    g[from].push_back(edge_num);

    tmp.from = to; tmp.to = from; tmp.flow = cap; tmp.cap = cap; tmp.cost = -cost;
    e.push_back(tmp);
    g[to].push_back(edge_num + 1);

    edge_num += 2;
}

int main() {
    //assert(freopen("input.txt","r",stdin));
    //assert(freopen("output.txt","w",stdout));

    scanf("%d", &n);
    for (int i = 1; i <= n; i++)
        for (int j = 1; j <= n; j++)
            scanf("%d", &cost[i][j]);

    s = 0; t = 2 * n + 1;

    for (int i = 1; i <= n; i++) 
        addEdge(s, i, 1, 0);

    for (int i = n + 1; i <= 2 * n; i++) 
        addEdge(i, t, 1, 0);

    for (int i = 1; i <= n; i++)
        for (int j = 1; j <= n; j++)
            addEdge(i, n + j, 1, cost[i][j]);

    cout << minCost(n) << endl;

    return 0;
}