Dado un gráfico acíclico dirigido ponderado y un vértice fuente en el gráfico, encuentre las rutas más cortas desde la fuente dada a todos los demás vértices.
Para un gráfico ponderado general, podemos calcular las distancias más cortas de una sola fuente en tiempo O (VE) utilizando el algoritmo Bellman-Ford . Para un gráfico sin pesos negativos, podemos hacerlo mejor y calcular las distancias más cortas de fuente única en tiempo O(E + VLogV) usando el algoritmo de Dijkstra . ¿Podemos hacerlo aún mejor para el gráfico acíclico dirigido (DAG)? Podemos calcular las distancias más cortas de una sola fuente en tiempo O(V+E) para los DAG. La idea es utilizar Clasificación topológica .
Inicializamos las distancias a todos los vértices como infinito y la distancia a la fuente como 0, luego encontramos una clasificación topológica del gráfico. La clasificación topológica de un gráfico representa una ordenación lineal del gráfico (consulte a continuación, la figura (b) es una representación lineal de la figura (a)). Una vez que tenemos el orden topológico (o representación lineal), procesamos uno por uno todos los vértices en orden topológico. Para cada vértice que se procesa, actualizamos las distancias de sus adyacentes usando la distancia del vértice actual.
La siguiente figura está tomada de esta fuente. Muestra el proceso paso a paso para encontrar los caminos más cortos.
El siguiente es un algoritmo completo para encontrar las distancias más cortas.
- Inicialice dist[] = {INF, INF, ….} y dist[s] = 0 donde s es el vértice de origen.
- Crea un orden topológico de todos los vértices.
- Haz lo siguiente para cada vértice u en orden topológico.
………..Haga lo siguiente para cada vértice adyacente v de u
………………si (dist[v] > dist[u] + peso(u, v))
………………………dist[ v] = dist[u] + peso(u, v)
Implementación:
C++
// C++ program to find single source shortest
// paths for Directed Acyclic Graphs
#include<iostream>
#include <bits/stdc++.h>
#define INF INT_MAX
using namespace std;
// Graph is represented using adjacency list. Every node
// of adjacency list contains vertex number of the vertex
// to which edge connects. It also
// contains weight of the edge
class AdjListNode
{
int v;
int weight;
public:
AdjListNode(int _v, int _w) { v = _v; weight = _w;}
int getV() { return v; }
int getWeight() { return weight; }
};
// Class to represent a graph using adjacency
// list representation
class Graph
{
int V; // No. of vertices'
// Pointer to an array containing adjacency lists
list<AdjListNode> *adj;
// A function used by shortestPath
void topologicalSortUtil(int v, bool visited[], stack<int> &Stack);
public:
Graph(int V); // Constructor
// function to add an edge to graph
void addEdge(int u, int v, int weight);
// Finds shortest paths from given source vertex
void shortestPath(int s);
};
Graph::Graph(int V)
{
this->V = V;
adj = new list<AdjListNode>[V];
}
void Graph::addEdge(int u, int v, int weight)
{
AdjListNode node(v, weight);
adj[u].push_back(node); // Add v to u's list
}
// A recursive function used by shortestPath.
// See below link for details
// https://www.geeksforgeeks.org/topological-sorting/
void Graph::topologicalSortUtil(int v, bool visited[], stack<int> &Stack)
{
// Mark the current node as visited
visited[v] = true;
// Recur for all the vertices adjacent to this vertex
list<AdjListNode>::iterator i;
for (i = adj[v].begin(); i != adj[v].end(); ++i)
{
AdjListNode node = *i;
if (!visited[node.getV()])
topologicalSortUtil(node.getV(), visited, Stack);
}
// Push current vertex to stack which stores topological sort
Stack.push(v);
}
// The function to find shortest paths from given vertex.
// It uses recursive topologicalSortUtil() to get topological
// sorting of given graph.
void Graph::shortestPath(int s)
{
stack<int> Stack;
int dist[V];
// Mark all the vertices as not visited
bool *visited = new bool[V];
for (int i = 0; i < V; i++)
visited[i] = false;
// Call the recursive helper function to store
// Topological Sort starting from all vertices
// one by one
for (int i = 0; i < V; i++)
if (visited[i] == false)
topologicalSortUtil(i, visited, Stack);
// Initialize distances to all vertices as
// infinite and distance to source as 0
for (int i = 0; i < V; i++)
dist[i] = INF;
dist[s] = 0;
// Process vertices in topological order
while (Stack.empty() == false)
{
// Get the next vertex from topological order
int u = Stack.top();
Stack.pop();
// Update distances of all adjacent vertices
list<AdjListNode>::iterator i;
if (dist[u] != INF)
{
for (i = adj[u].begin(); i != adj[u].end(); ++i)
if (dist[i->getV()] > dist[u] + i->getWeight())
dist[i->getV()] = dist[u] + i->getWeight();
}
}
// Print the calculated shortest distances
for (int i = 0; i < V; i++)
(dist[i] == INF)? cout << "INF ": cout << dist[i] << " ";
}
// Driver program to test above functions
int main()
{
// Create a graph given in the above diagram.
// Here vertex numbers are 0, 1, 2, 3, 4, 5 with
// following mappings: 0=r, 1=s, 2=t, 3=x, 4=y, 5=z
Graph g(6);
g.addEdge(0, 1, 5);
g.addEdge(0, 2, 3);
g.addEdge(1, 3, 6);
g.addEdge(1, 2, 2);
g.addEdge(2, 4, 4);
g.addEdge(2, 5, 2);
g.addEdge(2, 3, 7);
g.addEdge(3, 4, -1);
g.addEdge(4, 5, -2);
int s = 1;
cout << "Following are shortest distances from source " << s <<" n";
g.shortestPath(s);
return 0;
}
Java
// Java program to find single source shortest paths in Directed Acyclic Graphs
import java.io.*;
import java.util.*;
class ShortestPath
{
static final int INF=Integer.MAX_VALUE;
class AdjListNode
{
private int v;
private int weight;
AdjListNode(int _v, int _w) { v = _v; weight = _w; }
int getV() { return v; }
int getWeight() { return weight; }
}
// Class to represent graph as an adjacency list of
// nodes of type AdjListNode
class Graph
{
private int V;
private LinkedList<AdjListNode>adj[];
Graph(int v)
{
V=v;
adj = new LinkedList[V];
for (int i=0; i<v; ++i)
adj[i] = new LinkedList<AdjListNode>();
}
void addEdge(int u, int v, int weight)
{
AdjListNode node = new AdjListNode(v,weight);
adj[u].add(node);// Add v to u's list
}
// A recursive function used by shortestPath.
// See below link for details
void topologicalSortUtil(int v, Boolean visited[], Stack stack)
{
// Mark the current node as visited.
visited[v] = true;
Integer i;
// Recur for all the vertices adjacent to this vertex
Iterator<AdjListNode> it = adj[v].iterator();
while (it.hasNext())
{
AdjListNode node =it.next();
if (!visited[node.getV()])
topologicalSortUtil(node.getV(), visited, stack);
}
// Push current vertex to stack which stores result
stack.push(new Integer(v));
}
// The function to find shortest paths from given vertex. It
// uses recursive topologicalSortUtil() to get topological
// sorting of given graph.
void shortestPath(int s)
{
Stack stack = new Stack();
int dist[] = new int[V];
// Mark all the vertices as not visited
Boolean visited[] = new Boolean[V];
for (int i = 0; i < V; i++)
visited[i] = false;
// Call the recursive helper function to store Topological
// Sort starting from all vertices one by one
for (int i = 0; i < V; i++)
if (visited[i] == false)
topologicalSortUtil(i, visited, stack);
// Initialize distances to all vertices as infinite and
// distance to source as 0
for (int i = 0; i < V; i++)
dist[i] = INF;
dist[s] = 0;
// Process vertices in topological order
while (stack.empty() == false)
{
// Get the next vertex from topological order
int u = (int)stack.pop();
// Update distances of all adjacent vertices
Iterator<AdjListNode> it;
if (dist[u] != INF)
{
it = adj[u].iterator();
while (it.hasNext())
{
AdjListNode i= it.next();
if (dist[i.getV()] > dist[u] + i.getWeight())
dist[i.getV()] = dist[u] + i.getWeight();
}
}
}
// Print the calculated shortest distances
for (int i = 0; i < V; i++)
{
if (dist[i] == INF)
System.out.print( "INF ");
else
System.out.print( dist[i] + " ");
}
}
}
// Method to create a new graph instance through an object
// of ShortestPath class.
Graph newGraph(int number)
{
return new Graph(number);
}
public static void main(String args[])
{
// Create a graph given in the above diagram. Here vertex
// numbers are 0, 1, 2, 3, 4, 5 with following mappings:
// 0=r, 1=s, 2=t, 3=x, 4=y, 5=z
ShortestPath t = new ShortestPath();
Graph g = t.newGraph(6);
g.addEdge(0, 1, 5);
g.addEdge(0, 2, 3);
g.addEdge(1, 3, 6);
g.addEdge(1, 2, 2);
g.addEdge(2, 4, 4);
g.addEdge(2, 5, 2);
g.addEdge(2, 3, 7);
g.addEdge(3, 4, -1);
g.addEdge(4, 5, -2);
int s = 1;
System.out.println("Following are shortest distances "+
"from source " + s );
g.shortestPath(s);
}
}
//This code is contributed by Aakash Hasija
Python3
# Python program to find single source shortest paths
# for Directed Acyclic Graphs Complexity :O(V+E)
from collections import defaultdict
# Graph is represented using adjacency list. Every
# node of adjacency list contains vertex number of
# the vertex to which edge connects. It also contains
# weight of the edge
class Graph:
def __init__(self,vertices):
self.V = vertices # No. of vertices
# dictionary containing adjacency List
self.graph = defaultdict(list)
# function to add an edge to graph
def addEdge(self,u,v,w):
self.graph[u].append((v,w))
# A recursive function used by shortestPath
def topologicalSortUtil(self,v,visited,stack):
# Mark the current node as visited.
visited[v] = True
# Recur for all the vertices adjacent to this vertex
if v in self.graph.keys():
for node,weight in self.graph[v]:
if visited[node] == False:
self.topologicalSortUtil(node,visited,stack)
# Push current vertex to stack which stores topological sort
stack.append(v)
''' The function to find shortest paths from given vertex.
It uses recursive topologicalSortUtil() to get topological
sorting of given graph.'''
def shortestPath(self, s):
# Mark all the vertices as not visited
visited = [False]*self.V
stack =[]
# Call the recursive helper function to store Topological
# Sort starting from source vertices
for i in range(self.V):
if visited[i] == False:
self.topologicalSortUtil(s,visited,stack)
# Initialize distances to all vertices as infinite and
# distance to source as 0
dist = [float("Inf")] * (self.V)
dist[s] = 0
# Process vertices in topological order
while stack:
# Get the next vertex from topological order
i = stack.pop()
# Update distances of all adjacent vertices
for node,weight in self.graph[i]:
if dist[node] > dist[i] + weight:
dist[node] = dist[i] + weight
# Print the calculated shortest distances
for i in range(self.V):
print (("%d" %dist[i]) if dist[i] != float("Inf") else "Inf" ,end=" ")
g = Graph(6)
g.addEdge(0, 1, 5)
g.addEdge(0, 2, 3)
g.addEdge(1, 3, 6)
g.addEdge(1, 2, 2)
g.addEdge(2, 4, 4)
g.addEdge(2, 5, 2)
g.addEdge(2, 3, 7)
g.addEdge(3, 4, -1)
g.addEdge(4, 5, -2)
# source = 1
s = 1
print ("Following are shortest distances from source %d " % s)
g.shortestPath(s)
# This code is contributed by Neelam Yadav
C#
// C# program to find single source shortest
// paths in Directed Acyclic Graphs
using System;
using System.Collections.Generic;
public class ShortestPath
{
static readonly int INF = int.MaxValue;
class AdjListNode
{
public int v;
public int weight;
public AdjListNode(int _v, int _w) { v = _v; weight = _w; }
public int getV() { return v; }
public int getWeight() { return weight; }
}
// Class to represent graph as an adjacency list of
// nodes of type AdjListNode
class Graph
{
public int V;
public List<AdjListNode>[]adj;
public Graph(int v)
{
V = v;
adj = new List<AdjListNode>[V];
for (int i = 0; i < v; ++i)
adj[i] = new List<AdjListNode>();
}
public void addEdge(int u, int v, int weight)
{
AdjListNode node = new AdjListNode(v,weight);
adj[u].Add(node);// Add v to u's list
}
// A recursive function used by shortestPath.
// See below link for details
public void topologicalSortUtil(int v, Boolean []visited,
Stack<int> stack)
{
// Mark the current node as visited.
visited[v] = true;
// Recur for all the vertices adjacent to this vertex
foreach(AdjListNode it in adj[v])
{
AdjListNode node = it;
if (!visited[node.getV()])
topologicalSortUtil(node.getV(), visited, stack);
}
// Push current vertex to stack which stores result
stack.Push(v);
}
// The function to find shortest paths from given vertex. It
// uses recursive topologicalSortUtil() to get topological
// sorting of given graph.
public void shortestPath(int s)
{
Stack<int> stack = new Stack<int>();
int []dist = new int[V];
// Mark all the vertices as not visited
Boolean []visited = new Boolean[V];
for (int i = 0; i < V; i++)
visited[i] = false;
// Call the recursive helper function to store Topological
// Sort starting from all vertices one by one
for (int i = 0; i < V; i++)
if (visited[i] == false)
topologicalSortUtil(i, visited, stack);
// Initialize distances to all vertices as infinite and
// distance to source as 0
for (int i = 0; i < V; i++)
dist[i] = INF;
dist[s] = 0;
// Process vertices in topological order
while (stack.Count != 0)
{
// Get the next vertex from topological order
int u = (int)stack.Pop();
// Update distances of all adjacent vertices
if (dist[u] != INF)
{
foreach(AdjListNode it in adj[u])
{
AdjListNode i= it;
if (dist[i.getV()] > dist[u] + i.getWeight())
dist[i.getV()] = dist[u] + i.getWeight();
}
}
}
// Print the calculated shortest distances
for (int i = 0; i < V; i++)
{
if (dist[i] == INF)
Console.Write( "INF ");
else
Console.Write( dist[i] + " ");
}
}
}
// Method to create a new graph instance through an object
// of ShortestPath class.
Graph newGraph(int number)
{
return new Graph(number);
}
// Driver code
public static void Main(String []args)
{
// Create a graph given in the above diagram. Here vertex
// numbers are 0, 1, 2, 3, 4, 5 with following mappings:
// 0=r, 1=s, 2=t, 3=x, 4=y, 5=z
ShortestPath t = new ShortestPath();
Graph g = t.newGraph(6);
g.addEdge(0, 1, 5);
g.addEdge(0, 2, 3);
g.addEdge(1, 3, 6);
g.addEdge(1, 2, 2);
g.addEdge(2, 4, 4);
g.addEdge(2, 5, 2);
g.addEdge(2, 3, 7);
g.addEdge(3, 4, -1);
g.addEdge(4, 5, -2);
int s = 1;
Console.WriteLine("Following are shortest distances "+
"from source " + s );
g.shortestPath(s);
}
}
// This code is contributed by Rajput-Ji
Producción
Following are shortest distances from source 1 nINF 0 2 6 5 3
Complejidad de tiempo: la complejidad de tiempo de la clasificación topológica es O (V + E). Después de encontrar el orden topológico, el algoritmo procesa todos los vértices y, para cada vértice, ejecuta un bucle para todos los vértices adyacentes. El total de vértices adyacentes en un gráfico es O(E). Entonces, el bucle interno se ejecuta O (V + E) veces. Por lo tanto, la complejidad temporal total de este algoritmo es O(V+E).
Publicación traducida automáticamente
Artículo escrito por GeeksforGeeks-1 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA