Problem:
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. $|V|$, $|E|$ 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 (Python):



"""
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] * v_num
    queue = deque()
    queue.appendleft(source)
    visited[source] = True

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

    return True if visited[target] else False


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

    while graph_bfs(source, sink, parent):
        path_flow = float('inf')

        bk_1 = sink
        while bk_1 is not source:
            path_flow = min(path_flow, adj_table[parent[bk_1]][bk_1])
            bk_1 = parent[bk_1]

        max_flow += path_flow

        bk_2 = sink
        while bk_2 is not source:
            parent_bk_2 = parent[bk_2]
            assert parent_bk_2 != -1
            adj_table[parent_bk_2].setdefault(bk_2, 0)
            adj_table[bk_2].setdefault(parent_bk_2, 0)

            
            adj_table[parent_bk_2][bk_2] -= path_flow
            adj_table[bk_2][parent_bk_2] += path_flow
            bk_2 = parent[bk_2]

    return max_flow


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

    return init_adj_table


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

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

Solution (Python):



"""
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] * v_num
    queue = deque()
    queue.appendleft(source)
    visited[source] = True

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

    return True if visited[target] else False


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

    while graph_bfs(source, sink, parent):
        path_flow = float('inf')

        bk_1 = sink
        while bk_1 is not source:
            path_flow = min(path_flow, adj_table[parent[bk_1]][bk_1])
            bk_1 = parent[bk_1]

        max_flow += path_flow

        bk_2 = sink
        while bk_2 is not source:
            parent_bk_2 = parent[bk_2]
            assert parent_bk_2 != -1
            
            adj_table[bk_2].setdefault(parent_bk_2, 0)

            
            adj_table[parent_bk_2][bk_2] -= path_flow
            adj_table[bk_2][parent_bk_2] += path_flow
            bk_2 = parent[bk_2]

    return max_flow


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

    return init_adj_table


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

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

Solution (Python):



"""
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] * v_num
    queue = deque()
    queue.appendleft(source)
    visited[source] = True

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

    return True if visited[target] else False


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

    while graph_bfs(source, sink, parent):
        path_flow = float('inf')

        bk_1 = sink
        while bk_1 is not source:
            path_flow = min(path_flow, adj_table[parent[bk_1]][bk_1])
            bk_1 = parent[bk_1]

        max_flow += path_flow

        bk_2 = sink
        while bk_2 is not source:
            parent_bk_2 = parent[bk_2]
            assert parent_bk_2 != -1
            
            adj_table[bk_2].setdefault(parent_bk_2, 0)

            
            adj_table[parent_bk_2][bk_2] -= path_flow
            adj_table[bk_2][parent_bk_2] += path_flow
            bk_2 = parent[bk_2]

    return max_flow


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

    return init_adj_table


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

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

Solution (Python):



"""
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] * v_num
    queue = deque()
    queue.appendleft(source)
    visited[source] = True

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

    return True if visited[target] else False


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

    while graph_bfs(source, sink, parent):
        path_flow = float('inf')

        bk_1 = sink
        while bk_1 is not source:
            path_flow = min(path_flow, adj_table[parent[bk_1]][bk_1])
            bk_1 = parent[bk_1]

        max_flow += path_flow

        bk_2 = sink
        while bk_2 is not source:
            parent_bk_2 = parent[bk_2]
            assert parent_bk_2 != -1
            
            adj_table[bk_2].setdefault(parent_bk_2, 0)

            
            adj_table[parent_bk_2][bk_2] -= path_flow
            adj_table[bk_2][parent_bk_2] += path_flow
            bk_2 = parent[bk_2]

    return max_flow


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

    return init_adj_table


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

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

Solution (Python):



"""
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] * v_num
    queue = deque()
    queue.appendleft(source)
    visited[source] = True

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

    return True if visited[target] else False


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

    while graph_bfs(source, sink, parent):
        path_flow = float('inf')

        bk_1 = sink
        while bk_1 is not source:
            path_flow = min(path_flow, adj_table[parent[bk_1]][bk_1])
            bk_1 = parent[bk_1]

        max_flow += path_flow

        bk_2 = sink
        while bk_2 is not source:
            parent_bk_2 = parent[bk_2]
            assert parent_bk_2 != -1
            
            adj_table[bk_2].setdefault(parent_bk_2, 0)

            
            adj_table[parent_bk_2][bk_2] -= path_flow
            adj_table[bk_2][parent_bk_2] += path_flow
            bk_2 = parent[bk_2]

    return max_flow


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

    return init_adj_table


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

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

Solution (Python):



"""
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] * v_num
    queue = deque()
    queue.appendleft(source)
    visited[source] = True

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

    return True if visited[target] else False


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

    while graph_bfs(source, sink, parent):
        path_flow = float('inf')

        bk_1 = sink
        while bk_1 is not source:
            parent_bk_1 = parent[bk_1]
            assert parent_bk_1 != -1
            path_flow = min(path_flow, adj_table[parent_bk_1][bk_1])
            bk_1 = parent[bk_1]

        max_flow += path_flow

        bk_2 = sink
        while bk_2 is not source:
            parent_bk_2 = parent[bk_2]
            assert parent_bk_2 != -1
            
            adj_table[bk_2].setdefault(parent_bk_2, 0)

            
            adj_table[parent_bk_2][bk_2] -= path_flow
            adj_table[bk_2][parent_bk_2] += path_flow
            bk_2 = parent[bk_2]

    return max_flow


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

    return init_adj_table


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

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

Solution (Python):



"""
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] * v_num
    queue = deque()
    queue.appendleft(source)
    visited[source] = True

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

    return True if visited[target] else False


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

    while graph_bfs(source, sink, parent):
        path_flow = float('inf')

        bk_1 = sink
        while bk_1 is not source:
            parent_bk_1 = parent[bk_1]
            assert parent_bk_1 != -1
            path_flow = min(path_flow, adj_table[parent_bk_1][bk_1])
            bk_1 = parent[bk_1]

        max_flow += path_flow

        bk_2 = sink
        while bk_2 is not source:
            parent_bk_2 = parent[bk_2]
            assert parent_bk_2 != -1
            
            adj_table[bk_2].setdefault(parent_bk_2, 0)

            
            adj_table[parent_bk_2][bk_2] -= path_flow
            adj_table[bk_2][parent_bk_2] += path_flow
            bk_2 = parent[bk_2]

    return max_flow


def generate_adj_table(_edges):
    for edge in _edges:
        source, target, cp = map(int, edge)
        init_adj_table[source].setdefault(target, cp)

    return init_adj_table


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

    init_adj_table = [dict() for _ in range(v_num)]
    adj_table = generate_adj_table(edges)
    ans = graph_FordFulkerson(source=0, sink=v_num - 1)
    print(ans)

Solution (Python):



"""
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 bfs(source, target, parent):
    queue = deque()
    queue.appendleft(source)
    visited = [False] * v_num
    visited[source] = True

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

    return True if visited[target] else False


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

    while bfs(source, sink, parent):
        path_flow = float('inf')

        bk_1 = sink
        while bk_1 != source:
            parent_bk_1 = parent[bk_1]
            assert parent_bk_1 != -1
            path_flow = min(path_flow, adj_table[parent_bk_1][bk_1])
            bk_1 = parent[bk_1]

        max_flow += path_flow

        bk_2 = sink
        while bk_2 != source:
            parent_bk_2 = parent[bk_2]
            assert parent_bk_2 != -1
            adj_table[bk_2].setdefault(parent_bk_2, 0)

            adj_table[parent_bk_2][bk_2] -= path_flow
            adj_table[bk_2][parent_bk_2] += path_flow
            bk_2 = parent[bk_2]

    return max_flow


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

    adj_table = tuple(dict() for _ in range(v_num))
    for edge in edges:
        s, t, c = map(int, edge)
        adj_table[s][t] = c

    print(graph_FordFulkerson(source=0, sink=v_num - 1))

Solution (Python):



"""
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 bfs(source, target, parent):
    queue = deque()
    queue.appendleft(source)
    visited = [False] * v_num
    visited[source] = True

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

    return True if visited[target] else False


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

    while bfs(source, sink, parent):
        path_flow = float('inf')

        bk_1 = sink
        while bk_1 != source:
            parent_bk_1 = parent[bk_1]
            assert parent_bk_1 != -1
            path_flow = min(path_flow, adj_table[parent_bk_1][bk_1])
            bk_1 = parent[bk_1]

        max_flow += path_flow

        bk_2 = sink
        while bk_2 != source:
            parent_bk_2 = parent[bk_2]
            assert parent_bk_2 != -1
            adj_table[bk_2].setdefault(parent_bk_2, 0)

            adj_table[parent_bk_2][bk_2] -= path_flow
            adj_table[bk_2][parent_bk_2] += path_flow
            bk_2 = parent[bk_2]

    return max_flow


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

    adj_table = tuple(dict() for _ in range(v_num))
    for edge in edges:
        s, t, cp = map(int, edge)
        adj_table[s][t] = cp

    ans = graph_FordFulkerson(source=0, sink=v_num - 1)
    print(ans)

Solution (Python):



"""
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 bfs(source, target, parent):
    visited = [False] * v_num
    queue = deque()
    queue.appendleft(source)
    visited[source] = True

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

    return True if visited[target] else False


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

    while bfs(source, sink, parent):
        path_flow = float('inf')
        
        
        bk_1 = sink
        while bk_1 != source:
            parent_bk_1 = parent[bk_1]
            assert parent_bk_1 != -1
            path_flow = min(path_flow, adj_table[parent_bk_1][bk_1])
            bk_1 = parent[bk_1]

        max_flow += path_flow

        
        bk_2 = sink
        while bk_2 != source:
            parent_bk_2 = parent[bk_2]
            assert parent_bk_2 != -1
            adj_table[bk_2].setdefault(parent_bk_2, 0)
            adj_table[parent_bk_2][bk_2] -= path_flow
            adj_table[bk_2][parent_bk_2] += path_flow
            bk_2 = parent[bk_2]

    return max_flow


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

    adj_table = tuple(dict() for _ in range(v_num))
    for edge in edges:
        s, t, c = map(int, edge)
        adj_table[s][t] = c

    print(graph_FordFulkerson(source=0, sink=v_num - 1))

Solution (Python):


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



import collections


def bfs(start, goal, parent):
    unvisited = [True] * V
    queue = collections.deque()
    queue.append(start)
    unvisited[start] = False
    while queue:
        u = queue.popleft()
        for v, flow in enumerate(adj_mat[u]):
            if unvisited[v] and (flow > 0):
                queue.append(v)
                unvisited[v] = False
                parent[v] = u
    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


print(ford_fulkerson(0, V - 1))

Solution (Python):


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



import collections


def bfs(start, goal, parent):
    unvisited = [True] * V
    queue = collections.deque()
    queue.append(start)
    unvisited[start] = False
    while queue:
        u = queue.popleft()
        for v, flow in enumerate(adj_mat[u]):
            if unvisited[v] and (flow > 0):
                queue.append(v)
                unvisited[v] = False
                parent[v] = u
                if v == goal:
                    return True

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


print(ford_fulkerson(0, V - 1))

Solution (Python):


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



import collections


def bfs(start, goal, parent):
    flow = [0] * V
    queue = collections.deque()
    queue.append(start)
    flow[start] = 10000
    while queue:
        u = queue.popleft()
        for v, r_capacity in enumerate(adj_mat[u]):
            if not flow[v] and (r_capacity > 0):
                queue.append(v)
                flow[v] = min(flow[u], r_capacity)
                parent[v] = u
    return flow[goal]

def ford_fulkerson(source, sink):
    max_flow = 0
    while True:
        parent = [None] * V
        aug_path_flow = bfs(source, sink, parent)
        if aug_path_flow:
            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
        else:
            break
    return max_flow


print(ford_fulkerson(0, V - 1))

Solution (Python):


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



import collections


def bfs(start, goal, parent):
    flow = [0] * V
    queue = collections.deque()
    queue.append(start)
    flow[start] = 10000
    while queue:
        u = queue.popleft()
        for v, r_capacity in enumerate(adj_mat[u]):
            if not flow[v] and (r_capacity > 0):
                queue.append(v)
                flow[v] = min(flow[u], r_capacity)
                parent[v] = u
                if v == goal:
                    return flow[goal]

def ford_fulkerson(source, sink):
    max_flow = 0
    while True:
        parent = [None] * V
        aug_path_flow = bfs(source, sink, parent)
        if aug_path_flow:
            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
        else:
            break
    return max_flow


print(ford_fulkerson(0, V - 1))

Solution (Python):


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



import collections


def bfs(start, goal, parent):
    unvisited = [True] * V
    queue = collections.deque()
    queue.append(start)
    unvisited[start] = False
    while queue:
        u = queue.popleft()
        for v, flow in enumerate(adj_mat[u]):
            if unvisited[v] and (flow > 0):
                queue.append(v)
                unvisited[v] = False
                parent[v] = u
    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


print(ford_fulkerson(0, V - 1))

Solution (Python):


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



import collections


def bfs(start, goal, parent):
    flow = [0] * V
    queue = collections.deque()
    queue.append(start)
    flow[start] = 10000
    while queue:
        u = queue.popleft()
        for v, r_capacity in enumerate(adj_mat[u]):
            if not flow[v] and (r_capacity > 0):
                queue.append(v)
                flow[v] = min(flow[u], r_capacity)
                parent[v] = u
                if v == goal:
                    return flow[goal]

def ford_fulkerson(source, sink):
    max_flow = 0
    while True:
        parent = [None] * V
        aug_path_flow = bfs(source, sink, parent)
        if aug_path_flow:
            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
        else:
            break
    return max_flow


print(ford_fulkerson(0, V - 1))

Solution (Python):


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



import collections


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


print(ford_fulkerson(0, V - 1))

Solution (Python):


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



import collections


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


print(ford_fulkerson(0, V - 1))

Solution (Python):


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



import collections


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


print(ford_fulkerson(0, V - 1))

Solution (Python):


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



import collections


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


print(ford_fulkerson(0, V - 1))