Problem:
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Maximum Flow

A flow network is a directed graph which has a $source$ and a $sink$. In a flow network, each edge $(u, v)$ has a capacity $c(u, v)$. Each edge receives a flow, but the amount of flow on the edge can not exceed the corresponding capacity. Find the maximum flow from the $source$ to the $sink$.

Input

A flow network is given in the following format.

$|V|\;|E|$

$u_0\;v_0\;c_0$

$u_1\;v_1\;c_1$

:

$u_{|E|-1}\;v_{|E|-1}\;c_{|E|-1}$

$|V|$, $|E|$ is the number of vertices and edges of the flow network respectively. The vertices in $G$ are named with the numbers 0, 1,..., $|V|-1$. The source is 0 and the sink is $|V|-1$.

$u_i$, $v_i$, $c_i$ represent $i$-th edge of the flow network.
A pair of $u_i$ and $v_i$ denotes that there is an edge from $u_i$ to $v_i$ and $c_i$ is the capacity of $i$-th edge.

Output

Print the maximum flow.

Constraints

$2 \leq |V| \leq 100$

$1 \leq |E| \leq 1000$

$0 \leq c_i \leq 10000$

Sample Input 1

4 5
0 1 2
0 2 1
1 2 1
1 3 1
2 3 2

Sample Output 1

3

Solution in C++ (judge verdict: Accepted)



#include <iostream>
#include <vector>
#include <string>
using namespace std;

const int MAX_V = 105, INF = 10005;

struct edge {
    int to, cap, rev;
};

vector<edge> G[MAX_V];
vector<int> used(MAX_V, 0);

void add_edge (int from, int to, int cap) {
    G[from].push_back((edge){to, cap, (int)(G[to].size())});
    G[to].push_back((edge){from, 0, (int)G[from].size()-1});
}

int dfs (int v, int t, int f) {
    if (v == t) return f;
    used[v] = 1;
    for (int i = 0; i < G[v].size(); i++) {
        edge &e = G[v][i];
        if (!used[e.to] && e.cap > 0) {
            int d = dfs(e.to, t, min(f, e.cap));
            if (d > 0) {
                e.cap -= d;
                G[e.to][e.rev].cap += d;
                return d;
            }
        }
    }
    return 0;
}

int max_flow (int s, int t) {
    int ans = 0;
    for (; ; ) {
        for (int i = 0; i < MAX_V; i++) {
            used[i] = 0;
        }
        int f = dfs(s, t, INF);
        if (f == 0) {
            return ans;
        }
        ans += f;
    }
}


int main() {
    int v, n;
    cin >> v >> n;
    for (int i = 0; i < n; i++) {
        int u, v, c;
        cin >> u >> v >> c;
        add_edge(u, v, c);
    }/*
    for (int i = 0; i < v; i ++) {
        for (int j = 0; j < G[i].size(); j++) {
            edge &e = G[i][j];
            cout << "{"<<e.to <<", "<<e.cap <<", " << e.rev <<"}" << " | ";
        }
        cout << endl;
    }*/
    cout << max_flow(0, v-1) << endl;
    return 0;
}