Dado un gráfico y un vértice fuente en el gráfico, encuentra los caminos más cortos desde la fuente hasta todos los vértices en el gráfico dado.
El algoritmo de Dijkstra es muy similar al algoritmo de Prim para el árbol de expansión mínimo . Al igual que el MST de Prim, generamos un SPT (árbol de ruta más corta) con una fuente determinada como raíz. Mantenemos dos conjuntos, un conjunto contiene vértices incluidos en el árbol de ruta más corta, otro conjunto incluye vértices que aún no están incluidos en el árbol de ruta más corta. En cada paso del algoritmo, encontramos un vértice que está en el otro conjunto (conjunto de aún no incluido) y tiene una distancia mínima de la fuente.
A continuación se detallan los pasos utilizados en el algoritmo de Dijkstra para encontrar el camino más corto desde un único vértice fuente a todos los demás vértices en el gráfico dado.
Algoritmo
1) Cree un conjunto sptSet (conjunto de árboles de ruta más corta) que realiza un seguimiento de los vértices incluidos en el árbol de ruta más corta, es decir, cuya distancia mínima desde la fuente se calcula y finaliza. Inicialmente, este conjunto está vacío.
2) Asigne un valor de distancia a todos los vértices en el gráfico de entrada. Inicialice todos los valores de distancia como INFINITOS. Asigne el valor de distancia como 0 para el vértice de origen para que se elija primero.
3) Mientras que sptSet no incluye todos los vértices
…. a) Elija un vértice u que no esté en sptSet y tenga un valor de distancia mínimo.
…. b) Incluya u en sptSet .
….c) Actualice el valor de la distancia de todos los vértices adyacentes de u. Para actualizar los valores de distancia, itere a través de todos los vértices adyacentes. Para cada vértice adyacente v, si la suma del valor de la distancia de u (desde la fuente) y el peso del borde uv es menor que el valor de la distancia de v, entonces actualice el valor de la distancia de v.
C++
// A C++ program for Dijkstra's single source shortest path algorithm.
// The program is for adjacency matrix representation of the graph
#include <iostream>
using namespace std;
#include <limits.h>
// Number of vertices in the graph
#define V 9
// A utility function to find the vertex with minimum distance value, from
// the set of vertices not yet included in shortest path tree
int minDistance(int dist[], bool sptSet[])
{
// Initialize min value
int min = INT_MAX, min_index;
for (int v = 0; v < V; v++)
if (sptSet[v] == false && dist[v] <= min)
min = dist[v], min_index = v;
return min_index;
}
// A utility function to print the constructed distance array
void printSolution(int dist[])
{
cout <<"Vertex \t Distance from Source" << endl;
for (int i = 0; i < V; i++)
cout << i << " \t\t"<<dist[i]<< endl;
}
// Function that implements Dijkstra's single source shortest path algorithm
// for a graph represented using adjacency matrix representation
void dijkstra(int graph[V][V], int src)
{
int dist[V]; // The output array. dist[i] will hold the shortest
// distance from src to i
bool sptSet[V]; // sptSet[i] will be true if vertex i is included in shortest
// path tree or shortest distance from src to i is finalized
// Initialize all distances as INFINITE and stpSet[] as false
for (int i = 0; i < V; i++)
dist[i] = INT_MAX, sptSet[i] = false;
// Distance of source vertex from itself is always 0
dist[src] = 0;
// Find shortest path for all vertices
for (int count = 0; count < V - 1; count++) {
// Pick the minimum distance vertex from the set of vertices not
// yet processed. u is always equal to src in the first iteration.
int u = minDistance(dist, sptSet);
// Mark the picked vertex as processed
sptSet[u] = true;
// Update dist value of the adjacent vertices of the picked vertex.
for (int v = 0; v < V; v++)
// Update dist[v] only if is not in sptSet, there is an edge from
// u to v, and total weight of path from src to v through u is
// smaller than current value of dist[v]
if (!sptSet[v] && graph[u][v] && dist[u] != INT_MAX
&& dist[u] + graph[u][v] < dist[v])
dist[v] = dist[u] + graph[u][v];
}
// print the constructed distance array
printSolution(dist);
}
// driver program to test above function
int main()
{
/* Let us create the example graph discussed above */
int graph[V][V] = { { 0, 4, 0, 0, 0, 0, 0, 8, 0 },
{ 4, 0, 8, 0, 0, 0, 0, 11, 0 },
{ 0, 8, 0, 7, 0, 4, 0, 0, 2 },
{ 0, 0, 7, 0, 9, 14, 0, 0, 0 },
{ 0, 0, 0, 9, 0, 10, 0, 0, 0 },
{ 0, 0, 4, 14, 10, 0, 2, 0, 0 },
{ 0, 0, 0, 0, 0, 2, 0, 1, 6 },
{ 8, 11, 0, 0, 0, 0, 1, 0, 7 },
{ 0, 0, 2, 0, 0, 0, 6, 7, 0 } };
dijkstra(graph, 0);
return 0;
}
// This code is contributed by shivanisinghss2110
C
// A C program for Dijkstra's single source shortest path algorithm.
// The program is for adjacency matrix representation of the graph
#include <limits.h>
#include <stdio.h>
#include <stdbool.h>
// Number of vertices in the graph
#define V 9
// A utility function to find the vertex with minimum distance value, from
// the set of vertices not yet included in shortest path tree
int minDistance(int dist[], bool sptSet[])
{
// Initialize min value
int min = INT_MAX, min_index;
for (int v = 0; v < V; v++)
if (sptSet[v] == false && dist[v] <= min)
min = dist[v], min_index = v;
return min_index;
}
// A utility function to print the constructed distance array
void printSolution(int dist[])
{
printf("Vertex \t\t Distance from Source\n");
for (int i = 0; i < V; i++)
printf("%d \t\t %d\n", i, dist[i]);
}
// Function that implements Dijkstra's single source shortest path algorithm
// for a graph represented using adjacency matrix representation
void dijkstra(int graph[V][V], int src)
{
int dist[V]; // The output array. dist[i] will hold the shortest
// distance from src to i
bool sptSet[V]; // sptSet[i] will be true if vertex i is included in shortest
// path tree or shortest distance from src to i is finalized
// Initialize all distances as INFINITE and stpSet[] as false
for (int i = 0; i < V; i++)
dist[i] = INT_MAX, sptSet[i] = false;
// Distance of source vertex from itself is always 0
dist[src] = 0;
// Find shortest path for all vertices
for (int count = 0; count < V - 1; count++) {
// Pick the minimum distance vertex from the set of vertices not
// yet processed. u is always equal to src in the first iteration.
int u = minDistance(dist, sptSet);
// Mark the picked vertex as processed
sptSet[u] = true;
// Update dist value of the adjacent vertices of the picked vertex.
for (int v = 0; v < V; v++)
// Update dist[v] only if is not in sptSet, there is an edge from
// u to v, and total weight of path from src to v through u is
// smaller than current value of dist[v]
if (!sptSet[v] && graph[u][v] && dist[u] != INT_MAX
&& dist[u] + graph[u][v] < dist[v])
dist[v] = dist[u] + graph[u][v];
}
// print the constructed distance array
printSolution(dist);
}
// driver program to test above function
int main()
{
/* Let us create the example graph discussed above */
int graph[V][V] = { { 0, 4, 0, 0, 0, 0, 0, 8, 0 },
{ 4, 0, 8, 0, 0, 0, 0, 11, 0 },
{ 0, 8, 0, 7, 0, 4, 0, 0, 2 },
{ 0, 0, 7, 0, 9, 14, 0, 0, 0 },
{ 0, 0, 0, 9, 0, 10, 0, 0, 0 },
{ 0, 0, 4, 14, 10, 0, 2, 0, 0 },
{ 0, 0, 0, 0, 0, 2, 0, 1, 6 },
{ 8, 11, 0, 0, 0, 0, 1, 0, 7 },
{ 0, 0, 2, 0, 0, 0, 6, 7, 0 } };
dijkstra(graph, 0);
return 0;
}
Java
// A Java program for Dijkstra's single source shortest path algorithm.
// The program is for adjacency matrix representation of the graph
import java.util.*;
import java.lang.*;
import java.io.*;
class ShortestPath {
// A utility function to find the vertex with minimum distance value,
// from the set of vertices not yet included in shortest path tree
static final int V = 9;
int minDistance(int dist[], Boolean sptSet[])
{
// Initialize min value
int min = Integer.MAX_VALUE, min_index = -1;
for (int v = 0; v < V; v++)
if (sptSet[v] == false && dist[v] <= min) {
min = dist[v];
min_index = v;
}
return min_index;
}
// A utility function to print the constructed distance array
void printSolution(int dist[])
{
System.out.println("Vertex \t\t Distance from Source");
for (int i = 0; i < V; i++)
System.out.println(i + " \t\t " + dist[i]);
}
// Function that implements Dijkstra's single source shortest path
// algorithm for a graph represented using adjacency matrix
// representation
void dijkstra(int graph[][], int src)
{
int dist[] = new int[V]; // The output array. dist[i] will hold
// the shortest distance from src to i
// sptSet[i] will true if vertex i is included in shortest
// path tree or shortest distance from src to i is finalized
Boolean sptSet[] = new Boolean[V];
// Initialize all distances as INFINITE and stpSet[] as false
for (int i = 0; i < V; i++) {
dist[i] = Integer.MAX_VALUE;
sptSet[i] = false;
}
// Distance of source vertex from itself is always 0
dist[src] = 0;
// Find shortest path for all vertices
for (int count = 0; count < V - 1; count++) {
// Pick the minimum distance vertex from the set of vertices
// not yet processed. u is always equal to src in first
// iteration.
int u = minDistance(dist, sptSet);
// Mark the picked vertex as processed
sptSet[u] = true;
// Update dist value of the adjacent vertices of the
// picked vertex.
for (int v = 0; v < V; v++)
// Update dist[v] only if is not in sptSet, there is an
// edge from u to v, and total weight of path from src to
// v through u is smaller than current value of dist[v]
if (!sptSet[v] && graph[u][v] != 0 && dist[u] != Integer.MAX_VALUE && dist[u] + graph[u][v] < dist[v])
dist[v] = dist[u] + graph[u][v];
}
// print the constructed distance array
printSolution(dist);
}
// Driver method
public static void main(String[] args)
{
/* Let us create the example graph discussed above */
int graph[][] = new int[][] { { 0, 4, 0, 0, 0, 0, 0, 8, 0 },
{ 4, 0, 8, 0, 0, 0, 0, 11, 0 },
{ 0, 8, 0, 7, 0, 4, 0, 0, 2 },
{ 0, 0, 7, 0, 9, 14, 0, 0, 0 },
{ 0, 0, 0, 9, 0, 10, 0, 0, 0 },
{ 0, 0, 4, 14, 10, 0, 2, 0, 0 },
{ 0, 0, 0, 0, 0, 2, 0, 1, 6 },
{ 8, 11, 0, 0, 0, 0, 1, 0, 7 },
{ 0, 0, 2, 0, 0, 0, 6, 7, 0 } };
ShortestPath t = new ShortestPath();
t.dijkstra(graph, 0);
}
}
// This code is contributed by Aakash Hasija
Python
# Python program for Dijkstra's single
# source shortest path algorithm. The program is
# for adjacency matrix representation of the graph
# Library for INT_MAX
import sys
class Graph():
def __init__(self, vertices):
self.V = vertices
self.graph = [[0 for column in range(vertices)]
for row in range(vertices)]
def printSolution(self, dist):
print("Vertex \tDistance from Source")
for node in range(self.V):
print(node, "\t", dist[node])
# A utility function to find the vertex with
# minimum distance value, from the set of vertices
# not yet included in shortest path tree
def minDistance(self, dist, sptSet):
# Initialize minimum distance for next node
min = sys.maxsize
# Search not nearest vertex not in the
# shortest path tree
for u in range(self.V):
if dist[u] < min and sptSet[u] == False:
min = dist[u]
min_index = u
return min_index
# Function that implements Dijkstra's single source
# shortest path algorithm for a graph represented
# using adjacency matrix representation
def dijkstra(self, src):
dist = [sys.maxsize] * self.V
dist[src] = 0
sptSet = [False] * self.V
for cout in range(self.V):
# Pick the minimum distance vertex from
# the set of vertices not yet processed.
# x is always equal to src in first iteration
x = self.minDistance(dist, sptSet)
# Put the minimum distance vertex in the
# shortest path tree
sptSet[x] = True
# Update dist value of the adjacent vertices
# of the picked vertex only if the current
# distance is greater than new distance and
# the vertex in not in the shortest path tree
for y in range(self.V):
if self.graph[x][y] > 0 and sptSet[y] == False and \
dist[y] > dist[x] + self.graph[x][y]:
dist[y] = dist[x] + self.graph[x][y]
self.printSolution(dist)
# Driver program
g = Graph(9)
g.graph = [[0, 4, 0, 0, 0, 0, 0, 8, 0],
[4, 0, 8, 0, 0, 0, 0, 11, 0],
[0, 8, 0, 7, 0, 4, 0, 0, 2],
[0, 0, 7, 0, 9, 14, 0, 0, 0],
[0, 0, 0, 9, 0, 10, 0, 0, 0],
[0, 0, 4, 14, 10, 0, 2, 0, 0],
[0, 0, 0, 0, 0, 2, 0, 1, 6],
[8, 11, 0, 0, 0, 0, 1, 0, 7],
[0, 0, 2, 0, 0, 0, 6, 7, 0]
];
g.dijkstra(0);
# This code is contributed by Divyanshu Mehta and Updated by Pranav Singh Sambyal
C#
// A C# program for Dijkstra's single
// source shortest path algorithm.
// The program is for adjacency matrix
// representation of the graph
using System;
class GFG {
// A utility function to find the
// vertex with minimum distance
// value, from the set of vertices
// not yet included in shortest
// path tree
static int V = 9;
int minDistance(int[] dist,
bool[] sptSet)
{
// Initialize min value
int min = int.MaxValue, min_index = -1;
for (int v = 0; v < V; v++)
if (sptSet[v] == false && dist[v] <= min) {
min = dist[v];
min_index = v;
}
return min_index;
}
// A utility function to print
// the constructed distance array
void printSolution(int[] dist)
{
Console.Write("Vertex \t\t Distance "
+ "from Source\n");
for (int i = 0; i < V; i++)
Console.Write(i + " \t\t " + dist[i] + "\n");
}
// Function that implements Dijkstra's
// single source shortest path algorithm
// for a graph represented using adjacency
// matrix representation
void dijkstra(int[, ] graph, int src)
{
int[] dist = new int[V]; // The output array. dist[i]
// will hold the shortest
// distance from src to i
// sptSet[i] will true if vertex
// i is included in shortest path
// tree or shortest distance from
// src to i is finalized
bool[] sptSet = new bool[V];
// Initialize all distances as
// INFINITE and stpSet[] as false
for (int i = 0; i < V; i++) {
dist[i] = int.MaxValue;
sptSet[i] = false;
}
// Distance of source vertex
// from itself is always 0
dist[src] = 0;
// Find shortest path for all vertices
for (int count = 0; count < V - 1; count++) {
// Pick the minimum distance vertex
// from the set of vertices not yet
// processed. u is always equal to
// src in first iteration.
int u = minDistance(dist, sptSet);
// Mark the picked vertex as processed
sptSet[u] = true;
// Update dist value of the adjacent
// vertices of the picked vertex.
for (int v = 0; v < V; v++)
// Update dist[v] only if is not in
// sptSet, there is an edge from u
// to v, and total weight of path
// from src to v through u is smaller
// than current value of dist[v]
if (!sptSet[v] && graph[u, v] != 0 && dist[u] != int.MaxValue && dist[u] + graph[u, v] < dist[v])
dist[v] = dist[u] + graph[u, v];
}
// print the constructed distance array
printSolution(dist);
}
// Driver Code
public static void Main()
{
/* Let us create the example
graph discussed above */
int[, ] graph = new int[, ] { { 0, 4, 0, 0, 0, 0, 0, 8, 0 },
{ 4, 0, 8, 0, 0, 0, 0, 11, 0 },
{ 0, 8, 0, 7, 0, 4, 0, 0, 2 },
{ 0, 0, 7, 0, 9, 14, 0, 0, 0 },
{ 0, 0, 0, 9, 0, 10, 0, 0, 0 },
{ 0, 0, 4, 14, 10, 0, 2, 0, 0 },
{ 0, 0, 0, 0, 0, 2, 0, 1, 6 },
{ 8, 11, 0, 0, 0, 0, 1, 0, 7 },
{ 0, 0, 2, 0, 0, 0, 6, 7, 0 } };
GFG t = new GFG();
t.dijkstra(graph, 0);
}
}
// This code is contributed by ChitraNayal
Javascript
<script>
// A Javascript program for Dijkstra's single
// source shortest path algorithm.
// The program is for adjacency matrix
// representation of the graph
let V = 9;
// A utility function to find the
// vertex with minimum distance
// value, from the set of vertices
// not yet included in shortest
// path tree
function minDistance(dist,sptSet)
{
// Initialize min value
let min = Number.MAX_VALUE;
let min_index = -1;
for(let v = 0; v < V; v++)
{
if (sptSet[v] == false && dist[v] <= min)
{
min = dist[v];
min_index = v;
}
}
return min_index;
}
// A utility function to print
// the constructed distance array
function printSolution(dist)
{
document.write("Vertex \t\t Distance from Source<br>");
for(let i = 0; i < V; i++)
{
document.write(i + " \t\t " +
dist[i] + "<br>");
}
}
// Function that implements Dijkstra's
// single source shortest path algorithm
// for a graph represented using adjacency
// matrix representation
function dijkstra(graph, src)
{
let dist = new Array(V);
let sptSet = new Array(V);
// Initialize all distances as
// INFINITE and stpSet[] as false
for(let i = 0; i < V; i++)
{
dist[i] = Number.MAX_VALUE;
sptSet[i] = false;
}
// Distance of source vertex
// from itself is always 0
dist[src] = 0;
// Find shortest path for all vertices
for(let count = 0; count < V - 1; count++)
{
// Pick the minimum distance vertex
// from the set of vertices not yet
// processed. u is always equal to
// src in first iteration.
let u = minDistance(dist, sptSet);
// Mark the picked vertex as processed
sptSet[u] = true;
// Update dist value of the adjacent
// vertices of the picked vertex.
for(let v = 0; v < V; v++)
{
// Update dist[v] only if is not in
// sptSet, there is an edge from u
// to v, and total weight of path
// from src to v through u is smaller
// than current value of dist[v]
if (!sptSet[v] && graph[u][v] != 0 &&
dist[u] != Number.MAX_VALUE &&
dist[u] + graph[u][v] < dist[v])
{
dist[v] = dist[u] + graph[u][v];
}
}
}
// Print the constructed distance array
printSolution(dist);
}
// Driver code
let graph = [ [ 0, 4, 0, 0, 0, 0, 0, 8, 0 ],
[ 4, 0, 8, 0, 0, 0, 0, 11, 0 ],
[ 0, 8, 0, 7, 0, 4, 0, 0, 2 ],
[ 0, 0, 7, 0, 9, 14, 0, 0, 0],
[ 0, 0, 0, 9, 0, 10, 0, 0, 0 ],
[ 0, 0, 4, 14, 10, 0, 2, 0, 0],
[ 0, 0, 0, 0, 0, 2, 0, 1, 6 ],
[ 8, 11, 0, 0, 0, 0, 1, 0, 7 ],
[ 0, 0, 2, 0, 0, 0, 6, 7, 0 ] ]
dijkstra(graph, 0);
// This code is contributed by rag2127
</script>
Publicación traducida automáticamente
Artículo escrito por GeeksforGeeks-1 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA