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++11 (judge verdict: Accepted)

#include <algorithm>
#include <iostream>
#include <set>
#include <iterator>
#include <cmath>
#include <tuple>
#include <stack>
#include <iomanip>
#include <string>
#include <vector>
#include <queue>
#include <limits>
#include <cassert>

using namespace std;

//#define endl '\n' //fastio
typedef long long ll;

template<class Flow=long long>
class FlowNetwork{
 private:
    constexpr static Flow PINF = std::numeric_limits<Flow>::max(); 
    class Edge{
     public:
        int to;
        Flow cap; //capacity
        int rev_edge_id; //id of reverse edge (rev_edge = graph[e.to][e.rev_edge_id];
    };

    int n;
    vector<vector<Edge>> graph;

    long long dfs(int from, int to, Flow max_flow){
        vector<bool> is_used(n, false);
        return dfs(from,to,max_flow,is_used);
    }

    long long dfs(int from, int to, Flow max_flow, vector<bool>& is_used) {
        if (from == to) return max_flow;
        is_used[from] = true;
        for(auto& e : graph[from]){
            if(is_used[e.to] || e.cap == 0) continue;

            Edge& rev = graph[e.to][e.rev_edge_id];
            Flow flow = min(e.cap, dfs(e.to, to, min(max_flow, e.cap), is_used));
            if(flow != 0){
                e.cap -= flow;
                rev.cap += flow;
                return flow;
            }
        }
        return 0;
    }
 public:
    FlowNetwork(int _n) : n(_n), graph(_n) {}
    void add_Edge(int from, int to, Flow capacity) {
        int edge_id = graph[from].size(), rev_edge_id = graph[to].size();
        graph[from].push_back({to, capacity, rev_edge_id});
        graph[to].push_back({from, 0, edge_id});
    }

    long long max_flow(int s, int t) {
        ll res = 0;
        for (;;) {
            ll f = dfs(s, t, PINF);
            if (f == 0) return res;
            res += f;
        }
    }
};

int main(int , char** ){
	cin.tie(nullptr); ios::sync_with_stdio(false);
	//cout << fixed << setprecision(10);

    int n,m;
    cin >> n >> m;
    FlowNetwork<long long> fn(n);
    for(int i = 0; i < m; ++i){
        int u,v,c;
        cin >> u >> v >> c;
        fn.add_Edge(u,v,c);
    }
    cout << fn.max_flow(0,n-1) << endl;

return 0;/*
    int n,m,x;
    cin >> n >> m >> x;

    vector<set<int>> Edges(n);
    for(int i = 0; i < m; ++i){

    }
 
 		cout << 1 << endl;
 */
  return 0;
}