Problem:
MathJax.Hub.Config({ tex2jax: { inlineMath: [["$","$"], ["\\(","\\)"]], processEscapes: true }});

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 Python3 (judge verdict: Accepted)

import sys
from collections import deque


class MaxFlow:
    class Edge:
        def __init__(self, to, cap, rev):
            self.to, self.cap, self.rev = to, cap, rev

    def __init__(self, node_size, inf):
        self._node = node_size
        self._inf = inf
        self._level = [-1]*self._node
        self._iter = [0]*self._node
        self._graph = [[] for _ in range(self._node)]

    def add_edge(self, from_, to, cap):
        self._graph[from_].append(self.Edge(to, cap, len(self._graph[to])))
        self._graph[to].append(self.Edge(from_, 0, len(self._graph[from_])-1))

    def bfs(self, start):
        for i in range(self._node):
            self._level[i] = -1
        que = deque()
        self._level[start] = 0
        que.append(start)
        while que:
            cur_vertex = que.popleft()
            for e in self._graph[cur_vertex]:
                if self._level[e.to] < 0 < e.cap:
                    self._level[e.to] = self._level[cur_vertex] + 1
                    que.append(e.to)

    def dfs(self, cur_vertex, end_vertex, flow):
        if cur_vertex == end_vertex:
            return flow
        for e in self._graph[cur_vertex][self._iter[cur_vertex]:len(self._graph[cur_vertex])]:
            if e.cap > 0 and self._level[cur_vertex] < self._level[e.to]:
                flowed = self.dfs(e.to, end_vertex, min(flow, e.cap))
                if flowed > 0:
                    e.cap -= flowed
                    self._graph[e.to][e.rev].cap += flowed
                    return flowed
        return 0

    def solve(self, source, sink):
        flow = 0
        while True:
            self.bfs(source)
            if self._level[sink] < 0:
                return flow
            for i in range(self._node):
                self._iter[i] = 0
            while True:
                f = self.dfs(source, sink, self._inf)
                if f == 0:
                    break
                flow += f


def main():
    n, m = map(int, sys.stdin.readline().split())
    mf = MaxFlow(n, 10 ** 4)
    for _ in range(m):
        u, v, c = map(int, sys.stdin.readline().split())
        mf.add_edge(u, v, c)
    print(mf.solve(0, n - 1))

if __name__ == '__main__':
    main()