p

```
INF = float("inf")
class Dinic:
def __init__(self, n):
self.lvl = [0] * n
self.ptr = [0] * n
self.q = [0] * n
self.adj = [[] for _ in range(n)]
"""
Here we will add our edges containing with the following parameters:
vertex closest to source, vertex closest to sink and flow capacity
through that edge ...
"""
def add_edge(self, a, b, c, rcap=0):
self.adj[a].append([b, len(self.adj[b]), c, 0])
self.adj[b].append([a, len(self.adj[a]) - 1, rcap, 0])
# This is a sample depth first search to be used at max_flow
def depth_first_search(self, vertex, sink, flow):
if vertex == sink or not flow:
return flow
for i in range(self.ptr[vertex], len(self.adj[vertex])):
e = self.adj[vertex][i]
if self.lvl[e[0]] == self.lvl[vertex] + 1:
p = self.depth_first_search(e[0], sink, min(flow, e[2] - e[3]))
if p:
self.adj[vertex][i][3] += p
self.adj[e[0]][e[1]][3] -= p
return p
self.ptr[vertex] = self.ptr[vertex] + 1
return 0
# Here we calculate the flow that reaches the sink
def max_flow(self, source, sink):
flow, self.q[0] = 0, source
for l in range(31): # l = 30 maybe faster for random data # noqa: E741
while True:
self.lvl, self.ptr = [0] * len(self.q), [0] * len(self.q)
qi, qe, self.lvl[source] = 0, 1, 1
while qi < qe and not self.lvl[sink]:
v = self.q[qi]
qi += 1
for e in self.adj[v]:
if not self.lvl[e[0]] and (e[2] - e[3]) >> (30 - l):
self.q[qe] = e[0]
qe += 1
self.lvl[e[0]] = self.lvl[v] + 1
p = self.depth_first_search(source, sink, INF)
while p:
flow += p
p = self.depth_first_search(source, sink, INF)
if not self.lvl[sink]:
break
return flow
# Example to use
"""
Will be a bipartite graph, than it has the vertices near the source(4)
and the vertices near the sink(4)
"""
# Here we make a graphs with 10 vertex(source and sink includes)
graph = Dinic(10)
source = 0
sink = 9
"""
Now we add the vertices next to the font in the font with 1 capacity in this edge
(source -> source vertices)
"""
for vertex in range(1, 5):
graph.add_edge(source, vertex, 1)
"""
We will do the same thing for the vertices near the sink, but from vertex to sink
(sink vertices -> sink)
"""
for vertex in range(5, 9):
graph.add_edge(vertex, sink, 1)
"""
Finally we add the verices near the sink to the vertices near the source.
(source vertices -> sink vertices)
"""
for vertex in range(1, 5):
graph.add_edge(vertex, vertex + 4, 1)
# Now we can know that is the maximum flow(source -> sink)
print(graph.max_flow(source, sink))
```