Minimum Spanning Tree

Description

Given a connected and undirected graph, a spanning tree of that graph is a sub-graph that is a tree and connects all the vertices together. A single graph can have many different spanning trees. A minimum spanning tree (MST) or minimum weight spanning tree for a weighted, connected and undirected graph is a spanning tree with weight less than or equal to the weight of every other spanning tree. The weight of a spanning tree is the sum of weights given to each edge of the spanning tree.

A minimum spanning tree has (V – 1) edges where V is the number of vertices in the given graph.

There are lots of applications of MST, while the most well-known is network design: telephone, electrical, hydraulic, TV cable, computer, road
The standard application is to a problem like phone network design. You have a business with several offices; you want to lease phone lines to connect them up with each other; and the phone company charges different amounts of money to connect different pairs of cities. You want a set of lines that connects all your offices with a minimum total cost. It should be a spanning tree, since if a network isn’t a tree you can always remove some edges and save money.

Solution

Prim’s MST

Like Kruskal’s algorithm, Prim’s algorithm is also a Greedy algorithm. It starts with an empty spanning tree. The idea is to maintain two sets of vertices. The first set contains the vertices already included in the MST, the other set contains the vertices not yet included. At every step, it considers all the edges that connect the two sets, and picks the minimum weight edge from these edges.

reference

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// A C / C++ program for Prim's Minimum Spanning Tree (MST) algorithm.
// The program is for adjacency matrix representation of the graph
#include <stdio.h>
#include <limits.h>
// Number of vertices in the graph
#define V 5
// A utility function to find the vertex with minimum key value, from
// the set of vertices not yet included in MST
int minKey(int key[], bool mstSet[]) {
// Initialize min value
int min = INT_MAX, min_index;
for (int v = 0; v < V; v++)
if (mstSet[v] == false && key[v] < min)
min = key[v], min_index = v;
return min_index;
}
// A utility function to print the constructed MST stored in parent[]
int printMST(int parent[], int n, int graph[V][V]) {
printf("Edge Weight\n");
for (int i = 1; i < V; i++)
printf("%d - %d %d \n", parent[i], i, graph[i][parent[i]]);
}
// Function to construct and print MST for a graph represented using adjacency
// matrix representation
void primMST(int graph[V][V]) {
int parent[V]; // Array to store constructed MST
int key[V]; // Key values used to pick minimum weight edge in cut
bool mstSet[V]; // To represent set of vertices not yet included in MST
// Initialize all keys as INFINITE
for (int i = 0; i < V; i++)
key[i] = INT_MAX, mstSet[i] = false;
// Always include first 1st vertex in MST.
key[0] = 0; // Make key 0 so that this vertex is picked as first vertex
parent[0] = -1; // First node is always root of MST
// The MST will have V vertices
for (int count = 0; count < V-1; count++) {
// Pick thd minimum key vertex from the set of vertices
// not yet included in MST
int u = minKey(key, mstSet);
// Add the picked vertex to the MST Set
mstSet[u] = true;
// Update key value and parent index of the adjacent vertices of
// the picked vertex. Consider only those vertices which are not yet
// included in MST
for (int v = 0; v < V; v++)
// graph[u][v] is non zero only for adjacent vertices of m
// mstSet[v] is false for vertices not yet included in MST
// Update the key only if graph[u][v] is smaller than key[v]
if (graph[u][v] && mstSet[v] == false && graph[u][v] < key[v])
parent[v] = u, key[v] = graph[u][v];
}
// print the constructed MST
printMST(parent, V, graph);
}
// 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 graph[V][V] = {{0, 2, 0, 6, 0},
{2, 0, 3, 8, 5},
{0, 3, 0, 0, 7},
{6, 8, 0, 0, 9},
{0, 5, 7, 9, 0},
};
// Print the solution
primMST(graph);
return 0;
}

Kruskal’s

Below are the steps for finding MST using Kruskal’s algorithm

  1. Sort all the edges in non-decreasing order of their weight.
  2. Pick the smallest edge. Check if it forms a cycle with the spanning tree formed so far. If cycle is not formed, include this edge. Else, discard it.
  3. Repeat step#2 until there are (V-1) edges in the spanning tree.

reference

Union-Find Algorithm | Set 1 (Detect Cycle in a Graph)
Union-Find Algorithm | Set 2 (Union By Rank and Path Compression)

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// 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\n");
for (i = 0; i < e; ++i)
printf("%d -- %d == %d\n", result[i].src, result[i].dest, result[i].weight);
return;
}
// Driver program to test above functions
int main() {
/* Let us create following weighted graph
10
0--------1
| \ |
6| 5\ |15
| \ |
2--------3
4 */
int V = 4; // Number of vertices in graph
int E = 5; // Number of edges in graph
struct Graph* graph = createGraph(V, E);
// add edge 0-1
graph->edge[0].src = 0;
graph->edge[0].dest = 1;
graph->edge[0].weight = 10;
// add edge 0-2
graph->edge[1].src = 0;
graph->edge[1].dest = 2;
graph->edge[1].weight = 6;
// add edge 0-3
graph->edge[2].src = 0;
graph->edge[2].dest = 3;
graph->edge[2].weight = 5;
// add edge 1-3
graph->edge[3].src = 1;
graph->edge[3].dest = 3;
graph->edge[3].weight = 15;
// add edge 2-3
graph->edge[4].src = 2;
graph->edge[4].dest = 3;
graph->edge[4].weight = 4;
KruskalMST(graph);
return 0;
}

Basics

  1. void qsort (void base, size_t num, size_t size, int (compar)(const void,const void));
    base: Pointer to the first object of the array to be sorted, converted to a void*.
    num: Number of elements in the array pointed to by base. size_t is an unsigned integral type.
    size: Size in bytes of each element in the array. size_t is an unsigned integral type.
    compar: Pointer to a function that compares two elements. This function is called repeatedly by qsort to compare two elements. It shall follow the following prototype:

    int compar (const void p1, const void p2);
    Taking two pointers as arguments (both converted to const void*). The function defines the order of the elements by returning (in a stable and transitive manner):
    return value meaning
    <0 0="" the="" element="" pointed="" to="" by="" p1="" goes="" before="" p2="" is="" equivalent="" \="">0 The element pointed to by p1 goes after the element pointed to by p2

  2. void bsearch (const void key, const void base, size_t num, size_t size, int (compar)(const void,const void));

    Searches the given key in the array pointed to by base (which is formed by num elements, each of size bytes), and returns a void pointer to a matching element, if found. The rule of the compar is quite the same as qsort*.

Note the elements that compare less than key using compar should precede those that compare equal, and these should precede those that compare greater.

Always welcome new ideas and practical tricks, just leave them in the comments!