Hemos discutido el algoritmo de Prim y su implementación para la representación de gráficos de array de adyacencia .
Como se discutió en la publicación anterior, en el algoritmo de Prim , se mantienen dos conjuntos, un conjunto contiene una lista de vértices ya incluidos en MST, otro conjunto contiene vértices aún no incluidos. En cada iteración, consideramos la arista de peso mínimo entre las aristas que conectan los dos conjuntos.
La implementación discutida en la publicación anterior.utiliza dos arrays para encontrar el borde de peso mínimo que conecta los dos conjuntos. Aquí usamos uno en MST[V]. El valor de MST[i] será verdadero si el vértice i se incluye en el MST. En cada paso, consideramos solo aquellos bordes tales que un vértice del borde está incluido en MST y el otro no. Después de elegir un borde, marcamos ambos vértices como incluidos en MST.
C++
// A simple C++ implementation to find minimum
// spanning tree for adjacency representation.
#include <bits/stdc++.h>
using namespace std;
#define V 5
// Returns true if edge u-v is a valid edge to be
// include in MST. An edge is valid if one end is
// already included in MST and other is not in MST.
bool isValidEdge(int u, int v, vector<bool> inMST)
{
if (u == v)
return false;
if (inMST[u] == false && inMST[v] == false)
return false;
else if (inMST[u] == true && inMST[v] == true)
return false;
return true;
}
void primMST(int cost[][V])
{
vector<bool> inMST(V, false);
// Include first vertex in MST
inMST[0] = true;
// Keep adding edges while number of included
// edges does not become V-1.
int edge_count = 0, mincost = 0;
while (edge_count < V - 1) {
// Find minimum weight valid edge.
int min = INT_MAX, a = -1, b = -1;
for (int i = 0; i < V; i++) {
for (int j = 0; j < V; j++) {
if (cost[i][j] < min) {
if (isValidEdge(i, j, inMST)) {
min = cost[i][j];
a = i;
b = j;
}
}
}
}
if (a != -1 && b != -1) {
printf("Edge %d:(%d, %d) cost: %d \n",
edge_count++, a, b, min);
mincost = mincost + min;
inMST[b] = inMST[a] = true;
}
}
printf("\n Minimum cost= %d \n", mincost);
}
// driver program to test above function
int main()
{
/* Let us create the following graph
2 3
(0)--(1)--(2)
| / \ |
6| 8/ \5 |7
| / \ |
(3)-------(4)
9 */
int cost[][V] = {
{ INT_MAX, 2, INT_MAX, 6, INT_MAX },
{ 2, INT_MAX, 3, 8, 5 },
{ INT_MAX, 3, INT_MAX, INT_MAX, 7 },
{ 6, 8, INT_MAX, INT_MAX, 9 },
{ INT_MAX, 5, 7, 9, INT_MAX },
};
// Print the solution
primMST(cost);
return 0;
}
Java
// A simple Java implementation to find minimum
// spanning tree for adjacency representation.
import java.util.*;
class GFG
{
static int V = 5;
static int INT_MAX = Integer.MAX_VALUE;
// Returns true if edge u-v is a valid edge to be
// include in MST. An edge is valid if one end is
// already included in MST and other is not in MST.
static boolean isValidEdge(int u, int v,
boolean[] inMST)
{
if (u == v)
return false;
if (inMST[u] == false && inMST[v] == false)
return false;
else if (inMST[u] == true && inMST[v] == true)
return false;
return true;
}
static void primMST(int cost[][])
{
boolean []inMST = new boolean[V];
// Include first vertex in MST
inMST[0] = true;
// Keep adding edges while number of included
// edges does not become V-1.
int edge_count = 0, mincost = 0;
while (edge_count < V - 1)
{
// Find minimum weight valid edge.
int min = INT_MAX, a = -1, b = -1;
for (int i = 0; i < V; i++)
{
for (int j = 0; j < V; j++)
{
if (cost[i][j] < min)
{
if (isValidEdge(i, j, inMST))
{
min = cost[i][j];
a = i;
b = j;
}
}
}
}
if (a != -1 && b != -1)
{
System.out.printf("Edge %d:(%d, %d) cost: %d \n",
edge_count++, a, b, min);
mincost = mincost + min;
inMST[b] = inMST[a] = true;
}
}
System.out.printf("\n Minimum cost = %d \n", mincost);
}
// Driver Code
public static void main(String[] args)
{
/* Let us create the following graph
2 3
(0)--(1)--(2)
| / \ |
6| 8/ \5 |7
| / \ |
(3)-------(4)
9 */
int cost[][] = {{ INT_MAX, 2, INT_MAX, 6, INT_MAX },
{ 2, INT_MAX, 3, 8, 5 },
{ INT_MAX, 3, INT_MAX, INT_MAX, 7 },
{ 6, 8, INT_MAX, INT_MAX, 9 },
{ INT_MAX, 5, 7, 9, INT_MAX }};
// Print the solution
primMST(cost);
}
}
// This code is contributed by Rajput-Ji
Python3
# Python3 implementation to find minimum
# spanning tree for adjacency representation.
from sys import maxsize
INT_MAX = maxsize
V = 5
# Returns true if edge u-v is a valid edge to be
# include in MST. An edge is valid if one end is
# already included in MST and other is not in MST.
def isValidEdge(u, v, inMST):
if u == v:
return False
if inMST[u] == False and inMST[v] == False:
return False
elif inMST[u] == True and inMST[v] == True:
return False
return True
def primMST(cost):
inMST = [False] * V
# Include first vertex in MST
inMST[0] = True
# Keep adding edges while number of included
# edges does not become V-1.
edge_count = 0
mincost = 0
while edge_count < V - 1:
# Find minimum weight valid edge.
minn = INT_MAX
a = -1
b = -1
for i in range(V):
for j in range(V):
if cost[i][j] < minn:
if isValidEdge(i, j, inMST):
minn = cost[i][j]
a = i
b = j
if a != -1 and b != -1:
print("Edge %d: (%d, %d) cost: %d" %
(edge_count, a, b, minn))
edge_count += 1
mincost += minn
inMST[b] = inMST[a] = True
print("Minimum cost = %d" % mincost)
# Driver Code
if __name__ == "__main__":
''' Let us create the following graph
2 3
(0)--(1)--(2)
| / \ |
6| 8/ \5 |7
| / \ |
(3)-------(4)
9 '''
cost = [[INT_MAX, 2, INT_MAX, 6, INT_MAX],
[2, INT_MAX, 3, 8, 5],
[INT_MAX, 3, INT_MAX, INT_MAX, 7],
[6, 8, INT_MAX, INT_MAX, 9],
[INT_MAX, 5, 7, 9, INT_MAX]]
# Print the solution
primMST(cost)
# This code is contributed by
# sanjeev2552
C#
// A simple C# implementation to find minimum
// spanning tree for adjacency representation.
using System;
class GFG
{
static int V = 5;
static int INT_MAX = int.MaxValue;
// Returns true if edge u-v is a valid edge to be
// include in MST. An edge is valid if one end is
// already included in MST and other is not in MST.
static bool isValidEdge(int u, int v,
bool[] inMST)
{
if (u == v)
return false;
if (inMST[u] == false && inMST[v] == false)
return false;
else if (inMST[u] == true &&
inMST[v] == true)
return false;
return true;
}
static void primMST(int [,]cost)
{
bool []inMST = new bool[V];
// Include first vertex in MST
inMST[0] = true;
// Keep adding edges while number of
// included edges does not become V-1.
int edge_count = 0, mincost = 0;
while (edge_count < V - 1)
{
// Find minimum weight valid edge.
int min = INT_MAX, a = -1, b = -1;
for (int i = 0; i < V; i++)
{
for (int j = 0; j < V; j++)
{
if (cost[i, j] < min)
{
if (isValidEdge(i, j, inMST))
{
min = cost[i, j];
a = i;
b = j;
}
}
}
}
if (a != -1 && b != -1)
{
Console.Write("Edge {0}:({1}, {2}) cost: {3} \n",
edge_count++, a, b, min);
mincost = mincost + min;
inMST[b] = inMST[a] = true;
}
}
Console.Write("\n Minimum cost = {0} \n", mincost);
}
// Driver Code
public static void Main(String[] args)
{
/* Let us create the following graph
2 3
(0)--(1)--(2)
| / \ |
6| 8/ \5 |7
| / \ |
(3)-------(4)
9 */
int [,]cost = {{ INT_MAX, 2, INT_MAX, 6, INT_MAX },
{ 2, INT_MAX, 3, 8, 5 },
{ INT_MAX, 3, INT_MAX, INT_MAX, 7 },
{ 6, 8, INT_MAX, INT_MAX, 9 },
{ INT_MAX, 5, 7, 9, INT_MAX }};
// Print the solution
primMST(cost);
}
}
// This code is contributed by PrinciRaj1992
Javascript
<script>
// JavaScript implementation to find minimum
// spanning tree for adjacency representation.
let V = 5;
let let_MAX = Number.MAX_VALUE;
// Returns true if edge u-v is a valid edge to be
// include in MST. An edge is valid if one end is
// already included in MST and other is not in MST.
function isValidEdge(u, v, inMST)
{
if (u == v)
return false;
if (inMST[u] == false && inMST[v] == false)
return false;
else if (inMST[u] == true && inMST[v] == true)
return false;
return true;
}
function primMST(cost)
{
let inMST = Array(V).fill(false);
// Include first vertex in MST
inMST[0] = true;
// Keep adding edges while number of included
// edges does not become V-1.
let edge_count = 0, mincost = 0;
while (edge_count < V - 1)
{
// Find minimum weight valid edge.
let min = let_MAX, a = -1, b = -1;
for (let i = 0; i < V; i++)
{
for (let j = 0; j < V; j++)
{
if (cost[i][j] < min)
{
if (isValidEdge(i, j, inMST))
{
min = cost[i][j];
a = i;
b = j;
}
}
}
}
if (a != -1 && b != -1)
{
document.write("Edge " +
edge_count++ + ": (" + a +"," + b +
") " + "cost: " + min + "<br/>");
mincost = mincost + min;
inMST[b] = inMST[a] = true;
}
}
document.write("<br>");
document.write(" ");
document.write("Minimum cost = " + mincost);
}
// driver code
/* Let us create the following graph
2 3
(0)--(1)--(2)
| / \ |
6| 8/ \5 |7
| / \ |
(3)-------(4)
9 */
let cost = [[ let_MAX, 2, let_MAX, 6, let_MAX ],
[ 2, let_MAX, 3, 8, 5 ],
[ let_MAX, 3, let_MAX, let_MAX, 7 ],
[ 6, 8, let_MAX, let_MAX, 9 ],
[ let_MAX, 5, 7, 9, let_MAX ]];
// Print the solution
primMST(cost);
</script>
Producción:
Edge 0:(0, 1) cost: 2 Edge 1:(1, 2) cost: 3 Edge 2:(1, 4) cost: 5 Edge 3:(0, 3) cost: 6 Minimum cost= 16
Complejidad de tiempo: O(V 3 )
Tenga en cuenta que la complejidad de tiempo del enfoque anterior que usa array de adyacencia es O(V 2 ) y la complejidad de tiempo de la implementación de representación de lista de adyacencia es O((E+V)LogV).
Espacio Auxiliar: O(E + V)
Publicación traducida automáticamente
Artículo escrito por AnkurKarmakar y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA