Show an Algorithm for solving the minimum spanning tree algorithm. Assume that w
ID: 3673083 • Letter: S
Question
Show an Algorithm for solving the minimum spanning tree algorithm.
Assume that we have an algorithm A for solving the minimum spanning problem when all weights are positive. The input to A is the cost matrix. We want to solve the maximum spanning tree problem, that is. we are looking for a spanning tree whose total cost is as large as possible. Outline an algorithm for solving the maximum spanning tree algorithm, using A. You can change the cost matrix, but you cannot change A Justify that your algorithm in part (a) solves the maximum spanning tree algorithm.Explanation / Answer
*Try this program for the minimun spanning tree*
// C++ program for Kruskal's algorithm to find Minimum Spanning Tree
// of a given connected, undirected and weighted graph
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
// a structure to represent a weighted edge in graph
struct Edge
{
int src, dest, weight;
};
// a structure to represent a connected, undirected and weighted graph
struct Graph
{
// V-> Number of vertices, E-> Number of edges
int V, E;
// graph is represented as an array of edges. Since the graph is
// undirected, the edge from src to dest is also edge from dest
// to src. Both are counted as 1 edge here.
struct Edge* edge;
};
// Creates a graph with V vertices and E edges
struct Graph* createGraph(int V, int E)
{
struct Graph* graph = (struct Graph*) malloc( sizeof(struct Graph) );
graph->V = V;
graph->E = E;
graph->edge = (struct Edge*) malloc( graph->E * sizeof( struct Edge ) );
return graph;
}
// A structure to represent a subset for union-find
struct subset
{
int parent;
int rank;
};
// A utility function to find set of an element i
// (uses path compression technique)
int find(struct subset subsets[], int i)
{
// find root and make root as parent of i (path compression)
if (subsets[i].parent != i)
subsets[i].parent = find(subsets, subsets[i].parent);
return subsets[i].parent;
}
// A function that does union of two sets of x and y
// (uses union by rank)
void Union(struct subset subsets[], int x, int y)
{
int xroot = find(subsets, x);
int yroot = find(subsets, y);
// Attach smaller rank tree under root of high rank tree
// (Union by Rank)
if (subsets[xroot].rank < subsets[yroot].rank)
subsets[xroot].parent = yroot;
else if (subsets[xroot].rank > subsets[yroot].rank)
subsets[yroot].parent = xroot;
// If ranks are same, then make one as root and increment
// its rank by one
else
{
subsets[yroot].parent = xroot;
subsets[xroot].rank++;
}
}
// Compare two edges according to their weights.
// Used in qsort() for sorting an array of edges
int myComp(const void* a, const void* b)
{
struct Edge* a1 = (struct Edge*)a;
struct Edge* b1 = (struct Edge*)b;
return a1->weight > b1->weight;
}
// The main function to construct MST using Kruskal's algorithm
void KruskalMST(struct Graph* graph)
{
int V = graph->V;
struct Edge result[V]; // Tnis will store the resultant MST
int e = 0; // An index variable, used for result[]
int i = 0; // An index variable, used for sorted edges
// Step 1: Sort all the edges in non-decreasing order of their weight
// If we are not allowed to change the given graph, we can create a copy of
// array of edges
qsort(graph->edge, graph->E, sizeof(graph->edge[0]), myComp);
// Allocate memory for creating V ssubsets
struct subset *subsets =
(struct subset*) malloc( V * sizeof(struct subset) );
// Create V subsets with single elements
for (int v = 0; v < V; ++v)
{
subsets[v].parent = v;
subsets[v].rank = 0;
}
// Number of edges to be taken is equal to V-1
while (e < V - 1)
{
// Step 2: Pick the smallest edge. And increment the index
// for next iteration
struct Edge next_edge = graph->edge[i++];
int x = find(subsets, next_edge.src);
int y = find(subsets, next_edge.dest);
// If including this edge does't cause cycle, include it
// in result and increment the index of result for next edge
if (x != y)
{
result[e++] = next_edge;
Union(subsets, x, y);
}
// Else discard the next_edge
}
// print the contents of result[] to display the built MST
printf("Following are the edges in the constructed MST ");
for (i = 0; i < e; ++i)
printf("%d -- %d == %d ", result[i].src, result[i].dest,
result[i].weight);
return;
}