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 Python3 (judge verdict: Time Limit Exceeded)

#!/usr/bin/env python
# -*- coding: utf-8 -*-

"""
input:
4 5
0 1 2
0 2 1
1 2 1
1 3 1
2 3 2

output:
3
"""

import sys
from collections import deque


def graph_bfs(source, target, parent):
    visited = [False] * vertices
    queue = deque()
    queue.append(source)
    visited[source] = True

    while queue:
        current = queue.popleft()
        for adj, cp in adj_table[current].items():
            if not visited[adj] and cp:
                queue.append(adj)
                visited[adj] = True
                parent[adj] = current

    return True if visited[target] else False


def graphFordFulkerson(source, sink):
    parent = [-1] * vertices
    max_flow = 0

    while graph_bfs(source, sink, parent):
        path_flow = float('inf')
        end = sink
        while end is not source:
            path_flow = min(path_flow, adj_table[parent[end]][end])
            end = parent[end]

        max_flow += path_flow
        cursor = sink
        while cursor is not source:
            c_parent = parent[cursor]
            # print(c_parent, cursor, path_flow)
            adj_table[c_parent].setdefault(cursor, -1)
            adj_table[cursor].setdefault(c_parent, -1)
            
            adj_table[c_parent][cursor] -= path_flow
            adj_table[cursor][c_parent] += path_flow
            cursor = parent[c_parent]

    return max_flow


def generate_adj_table(v_table):
    for each in v_table:
        source, target, cp = map(int, each)
        init_adj_table[source][target] = cp

    return init_adj_table


if __name__ == '__main__':
    _input = sys.stdin.readlines()
    vertices, edges = map(int, _input[0].split())
    v_info = map(lambda x: x.split(), _input[1:])

    init_adj_table = [dict() for _ in range(vertices)]
    adj_table = generate_adj_table(v_info)
    ans = graphFordFulkerson(source=0, sink=vertices - 1)
    print(ans)