En una red de flujo, un corte st es un corte que requiere que la fuente ‘s’ y la sumidero ‘t’ estén en diferentes subconjuntos, y consiste en bordes que van desde el lado de la fuente hasta el lado del sumidero. La capacidad de un corte de st se define por la suma de la capacidad de cada borde en el conjunto de cortes. (Fuente: Wiki ) El problema discutido aquí es encontrar la capacidad mínima de corte st de la red dada. La salida esperada es todos los bordes del corte mínimo. Por ejemplo, en la siguiente red de flujo, los cortes de st de ejemplo son {{0,1}, {0, 2}}, {{0, 2}, {1, 2}, {1, 3}}, etc. el corte de pt mínimo es {{1, 3}, {4, 3}, {4 5}} que tiene una capacidad de 12+7+4 = 23.
Recomendamos encarecidamente leer la siguiente publicación primero. Algoritmo de Ford-Fulkerson para el problema de flujo máximo
Corte Mínimo y Caudal Máximo:
Al igual que la coincidencia bipartita máxima , este es otro problema que se puede resolver utilizando el algoritmo Ford-Fulkerson . Esto se basa en el teorema de corte mínimo de flujo máximo.
El teorema de corte mínimo de flujo máximo establece que en una red de flujo, la cantidad de flujo máximo es igual a la capacidad del corte mínimo.
De Ford-Fulkerson, obtenemos capacidad de corte mínimo. ¿Cómo imprimir todos los bordes que forman el corte mínimo? La idea es utilizar gráfico residual .
Los siguientes son los pasos para imprimir todos los bordes del corte mínimo.
- Ejecute el algoritmo de Ford-Fulkerson y considere el gráfico residual final .
- Encuentre el conjunto de vértices que son accesibles desde la fuente en el gráfico residual.
- Todos los bordes que van desde un vértice alcanzable hasta un vértice no alcanzable son bordes de corte mínimo. Imprima todos esos bordes.
A continuación se muestra la implementación del enfoque anterior.
C++
// C++ program for finding minimum cut using Ford-Fulkerson
#include <iostream>
#include <limits.h>
#include <string.h>
#include <queue>
using namespace std;
// Number of vertices in given graph
#define V 6
/* Returns true if there is a path from source 's' to sink 't' in
residual graph. Also fills parent[] to store the path */
int bfs(int rGraph[V][V], int s, int t, int parent[])
{
// Create a visited array and mark all vertices as not visited
bool visited[V];
memset(visited, 0, sizeof(visited));
// Create a queue, enqueue source vertex and mark source vertex
// as visited
queue <int> q;
q.push(s);
visited[s] = true;
parent[s] = -1;
// Standard BFS Loop
while (!q.empty())
{
int u = q.front();
q.pop();
for (int v=0; v<V; v++)
{
if (visited[v]==false && rGraph[u][v] > 0)
{
q.push(v);
parent[v] = u;
visited[v] = true;
}
}
}
// If we reached sink in BFS starting from source, then return
// true, else false
return (visited[t] == true);
}
// A DFS based function to find all reachable vertices from s. The function
// marks visited[i] as true if i is reachable from s. The initial values in
// visited[] must be false. We can also use BFS to find reachable vertices
void dfs(int rGraph[V][V], int s, bool visited[])
{
visited[s] = true;
for (int i = 0; i < V; i++)
if (rGraph[s][i] && !visited[i])
dfs(rGraph, i, visited);
}
// Prints the minimum s-t cut
void minCut(int graph[V][V], int s, int t)
{
int u, v;
// Create a residual graph and fill the residual graph with
// given capacities in the original graph as residual capacities
// in residual graph
int rGraph[V][V]; // rGraph[i][j] indicates residual capacity of edge i-j
for (u = 0; u < V; u++)
for (v = 0; v < V; v++)
rGraph[u][v] = graph[u][v];
int parent[V]; // This array is filled by BFS and to store path
// Augment the flow while there is a path from source to sink
while (bfs(rGraph, s, t, parent))
{
// Find minimum residual capacity of the edhes along the
// path filled by BFS. Or we can say find the maximum flow
// through the path found.
int path_flow = INT_MAX;
for (v=t; v!=s; v=parent[v])
{
u = parent[v];
path_flow = min(path_flow, rGraph[u][v]);
}
// update residual capacities of the edges and reverse edges
// along the path
for (v=t; v != s; v=parent[v])
{
u = parent[v];
rGraph[u][v] -= path_flow;
rGraph[v][u] += path_flow;
}
}
// Flow is maximum now, find vertices reachable from s
bool visited[V];
memset(visited, false, sizeof(visited));
dfs(rGraph, s, visited);
// Print all edges that are from a reachable vertex to
// non-reachable vertex in the original graph
for (int i = 0; i < V; i++)
for (int j = 0; j < V; j++)
if (visited[i] && !visited[j] && graph[i][j])
cout << i << " - " << j << endl;
return;
}
// Driver program to test above functions
int main()
{
// Let us create a graph shown in the above example
int graph[V][V] = { {0, 16, 13, 0, 0, 0},
{0, 0, 10, 12, 0, 0},
{0, 4, 0, 0, 14, 0},
{0, 0, 9, 0, 0, 20},
{0, 0, 0, 7, 0, 4},
{0, 0, 0, 0, 0, 0}
};
minCut(graph, 0, 5);
return 0;
}
Java
// Java program for finding min-cut in the given graph
import java.util.LinkedList;
import java.util.Queue;
public class Graph {
// Returns true if there is a path
// from source 's' to sink 't' in residual
// graph. Also fills parent[] to store the path
private static boolean bfs(int[][] rGraph, int s,
int t, int[] parent) {
// Create a visited array and mark
// all vertices as not visited
boolean[] visited = new boolean[rGraph.length];
// Create a queue, enqueue source vertex
// and mark source vertex as visited
Queue<Integer> q = new LinkedList<Integer>();
q.add(s);
visited[s] = true;
parent[s] = -1;
// Standard BFS Loop
while (!q.isEmpty()) {
int v = q.poll();
for (int i = 0; i < rGraph.length; i++) {
if (rGraph[v][i] > 0 && !visited[i]) {
q.offer(i);
visited[i] = true;
parent[i] = v;
}
}
}
// If we reached sink in BFS starting
// from source, then return true, else false
return (visited[t] == true);
}
// A DFS based function to find all reachable
// vertices from s. The function marks visited[i]
// as true if i is reachable from s. The initial
// values in visited[] must be false. We can also
// use BFS to find reachable vertices
private static void dfs(int[][] rGraph, int s,
boolean[] visited) {
visited[s] = true;
for (int i = 0; i < rGraph.length; i++) {
if (rGraph[s][i] > 0 && !visited[i]) {
dfs(rGraph, i, visited);
}
}
}
// Prints the minimum s-t cut
private static void minCut(int[][] graph, int s, int t) {
int u,v;
// Create a residual graph and fill the residual
// graph with given capacities in the original
// graph as residual capacities in residual graph
// rGraph[i][j] indicates residual capacity of edge i-j
int[][] rGraph = new int[graph.length][graph.length];
for (int i = 0; i < graph.length; i++) {
for (int j = 0; j < graph.length; j++) {
rGraph[i][j] = graph[i][j];
}
}
// This array is filled by BFS and to store path
int[] parent = new int[graph.length];
// Augment the flow while tere is path from source to sink
while (bfs(rGraph, s, t, parent)) {
// Find minimum residual capacity of the edhes
// along the path filled by BFS. Or we can say
// find the maximum flow through the path found.
int pathFlow = Integer.MAX_VALUE;
for (v = t; v != s; v = parent[v]) {
u = parent[v];
pathFlow = Math.min(pathFlow, rGraph[u][v]);
}
// update residual capacities of the edges and
// reverse edges along the path
for (v = t; v != s; v = parent[v]) {
u = parent[v];
rGraph[u][v] = rGraph[u][v] - pathFlow;
rGraph[v][u] = rGraph[v][u] + pathFlow;
}
}
// Flow is maximum now, find vertices reachable from s
boolean[] isVisited = new boolean[graph.length];
dfs(rGraph, s, isVisited);
// Print all edges that are from a reachable vertex to
// non-reachable vertex in the original graph
for (int i = 0; i < graph.length; i++) {
for (int j = 0; j < graph.length; j++) {
if (graph[i][j] > 0 && isVisited[i] && !isVisited[j]) {
System.out.println(i + " - " + j);
}
}
}
}
//Driver Program
public static void main(String args[]) {
// Let us create a graph shown in the above example
int graph[][] = { {0, 16, 13, 0, 0, 0},
{0, 0, 10, 12, 0, 0},
{0, 4, 0, 0, 14, 0},
{0, 0, 9, 0, 0, 20},
{0, 0, 0, 7, 0, 4},
{0, 0, 0, 0, 0, 0}
};
minCut(graph, 0, 5);
}
}
// This code is contributed by Himanshu Shekhar
Python
# Python program for finding min-cut in the given graph
# Complexity : (E*(V^3))
# Total augmenting path = VE and BFS
# with adj matrix takes :V^2 times
from collections import defaultdict
# This class represents a directed graph
# using adjacency matrix representation
class Graph:
def __init__(self,graph):
self.graph = graph # residual graph
self.org_graph = [i[:] for i in graph]
self. ROW = len(graph)
self.COL = len(graph[0])
'''Returns true if there is a path from
source 's' to sink 't' in
residual graph. Also fills
parent[] to store the path '''
def BFS(self,s, t, parent):
# Mark all the vertices as not visited
visited =[False]*(self.ROW)
# Create a queue for BFS
queue=[]
# Mark the source node as visited and enqueue it
queue.append(s)
visited[s] = True
# Standard BFS Loop
while queue:
#Dequeue a vertex from queue and print it
u = queue.pop(0)
# Get all adjacent vertices of
# the dequeued vertex u
# If a adjacent has not been
# visited, then mark it
# visited and enqueue it
for ind, val in enumerate(self.graph[u]):
if visited[ind] == False and val > 0 :
queue.append(ind)
visited[ind] = True
parent[ind] = u
# If we reached sink in BFS starting
# from source, then return
# true, else false
return True if visited[t] else False
# Function for Depth first search
# Traversal of the graph
def dfs(self, graph,s,visited):
visited[s]=True
for i in range(len(graph)):
if graph[s][i]>0 and not visited[i]:
self.dfs(graph,i,visited)
# Returns the min-cut of the given graph
def minCut(self, source, sink):
# This array is filled by BFS and to store path
parent = [-1]*(self.ROW)
max_flow = 0 # There is no flow initially
# Augment the flow while there is path from source to sink
while self.BFS(source, sink, parent) :
# Find minimum residual capacity of the edges along the
# path filled by BFS. Or we can say find the maximum flow
# through the path found.
path_flow = float("Inf")
s = sink
while(s != source):
path_flow = min (path_flow, self.graph[parent[s]][s])
s = parent[s]
# Add path flow to overall flow
max_flow += path_flow
# update residual capacities of the edges and reverse edges
# along the path
v = sink
while(v != source):
u = parent[v]
self.graph[u][v] -= path_flow
self.graph[v][u] += path_flow
v = parent[v]
visited=len(self.graph)*[False]
self.dfs(self.graph,s,visited)
# print the edges which initially had weights
# but now have 0 weight
for i in range(self.ROW):
for j in range(self.COL):
if self.graph[i][j] == 0 and\
self.org_graph[i][j] > 0 and visited[i]:
print str(i) + " - " + str(j)
# Create a graph given in the above diagram
graph = [[0, 16, 13, 0, 0, 0],
[0, 0, 10, 12, 0, 0],
[0, 4, 0, 0, 14, 0],
[0, 0, 9, 0, 0, 20],
[0, 0, 0, 7, 0, 4],
[0, 0, 0, 0, 0, 0]]
g = Graph(graph)
source = 0; sink = 5
g.minCut(source, sink)
# This code is contributed by Neelam Yadav
C#
// C# program for finding min-cut in the given graph
using System;
using System.Collections.Generic;
class Graph
{
// Returns true if there is a path
// from source 's' to sink 't' in residual
// graph. Also fills parent[] to store the path
private static bool bfs(int[,] rGraph, int s,
int t, int[] parent)
{
// Create a visited array and mark
// all vertices as not visited
bool[] visited = new bool[rGraph.Length];
// Create a queue, enqueue source vertex
// and mark source vertex as visited
Queue<int> q = new Queue<int>();
q.Enqueue(s);
visited[s] = true;
parent[s] = -1;
// Standard BFS Loop
while (q.Count != 0)
{
int v = q.Dequeue();
for (int i = 0; i < rGraph.GetLength(0); i++)
{
if (rGraph[v,i] > 0 && !visited[i])
{
q.Enqueue(i);
visited[i] = true;
parent[i] = v;
}
}
}
// If we reached sink in BFS starting
// from source, then return true, else false
return (visited[t] == true);
}
// A DFS based function to find all reachable
// vertices from s. The function marks visited[i]
// as true if i is reachable from s. The initial
// values in visited[] must be false. We can also
// use BFS to find reachable vertices
private static void dfs(int[,] rGraph, int s,
bool[] visited)
{
visited[s] = true;
for (int i = 0; i < rGraph.GetLength(0); i++)
{
if (rGraph[s,i] > 0 && !visited[i])
{
dfs(rGraph, i, visited);
}
}
}
// Prints the minimum s-t cut
private static void minCut(int[,] graph, int s, int t)
{
int u, v;
// Create a residual graph and fill the residual
// graph with given capacities in the original
// graph as residual capacities in residual graph
// rGraph[i,j] indicates residual capacity of edge i-j
int[,] rGraph = new int[graph.Length,graph.Length];
for (int i = 0; i < graph.GetLength(0); i++)
{
for (int j = 0; j < graph.GetLength(1); j++)
{
rGraph[i, j] = graph[i, j];
}
}
// This array is filled by BFS and to store path
int[] parent = new int[graph.Length];
// Augment the flow while there is path
// from source to sink
while (bfs(rGraph, s, t, parent))
{
// Find minimum residual capacity of the edhes
// along the path filled by BFS. Or we can say
// find the maximum flow through the path found.
int pathFlow = int.MaxValue;
for (v = t; v != s; v = parent[v])
{
u = parent[v];
pathFlow = Math.Min(pathFlow, rGraph[u, v]);
}
// update residual capacities of the edges and
// reverse edges along the path
for (v = t; v != s; v = parent[v])
{
u = parent[v];
rGraph[u, v] = rGraph[u, v] - pathFlow;
rGraph[v, u] = rGraph[v, u] + pathFlow;
}
}
// Flow is maximum now, find vertices reachable from s
bool[] isVisited = new bool[graph.Length];
dfs(rGraph, s, isVisited);
// Print all edges that are from a reachable vertex to
// non-reachable vertex in the original graph
for (int i = 0; i < graph.GetLength(0); i++)
{
for (int j = 0; j < graph.GetLength(1); j++)
{
if (graph[i, j] > 0 &&
isVisited[i] && !isVisited[j])
{
Console.WriteLine(i + " - " + j);
}
}
}
}
// Driver Code
public static void Main(String []args)
{
// Let us create a graph shown
// in the above example
int [,]graph = {{0, 16, 13, 0, 0, 0},
{0, 0, 10, 12, 0, 0},
{0, 4, 0, 0, 14, 0},
{0, 0, 9, 0, 0, 20},
{0, 0, 0, 7, 0, 4},
{0, 0, 0, 0, 0, 0}};
minCut(graph, 0, 5);
}
}
// This code is contributed by PrinciRaj1992
Producción
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Publicación traducida automáticamente
Artículo escrito por GeeksforGeeks-1 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA