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: Accepted)

# Acceptance of input

import sys

file_input = sys.stdin

V, E = map(int, file_input.readline().split())

adj_mat = [[0] * V for i in range(V)]

for line in file_input:
    u, v, c = map(int, line.split())
    adj_mat[u][v] = c

# Ford???Fulkerson method

import collections

# BFS for residual capacity network
def bfs(start, goal, parent):
    unvisited = [True] * V
    queue = collections.deque()
    queue.append(start)
    unvisited[start] = False
    while queue:
        u = queue.popleft()
        for v, r_capacity in enumerate(adj_mat[u]):
            if unvisited[v] and (r_capacity > 0):
                queue.append(v)
                unvisited[v] = False
                parent[v] = u
                if v == goal:
                    return not unvisited[goal]

def ford_fulkerson(source, sink):
    parent = [None] * V
    max_flow = 0
    while bfs(source, sink, parent):
        aug_path_flow = 10000
        v = sink
        while (v != source):
            aug_path_flow = min(aug_path_flow, adj_mat[parent[v]][v])
            v = parent[v]
        max_flow += aug_path_flow
        v = sink
        while (v != source):
            u = parent[v]
            adj_mat[u][v] -= aug_path_flow
            adj_mat[v][u] += aug_path_flow
            v = u
    return max_flow

# output
print(ford_fulkerson(0, V - 1))