Prim's Algorithm | Algorithms - Computer Science Engineering (CSE) PDF Download

Introduction to Prim’s algorithm

We have discussed Kruskal’s algorithm for Minimum Spanning Tree. Like Kruskal’s algorithm, Prim’s algorithm is also a Greedy algorithm. Prim’s algorithm always starts with a single node and it moves through several adjacent nodes, in order to explore all of the connected edges along the way.

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. After picking the edge, it moves the other endpoint of the edge to the set containing MST. 

A group of edges that connects two sets of vertices in a graph is called cut in graph theory. So, at every step of Prim’s algorithm, find a cut (of two sets, one contains the vertices already included in MST and the other contains the rest of the vertices), pick the minimum weight edge from the cut, and include this vertex to MST Set (the set that contains already included vertices).

How does Prim’s Algorithm Work? 

The idea behind Prim’s algorithm is simple, a spanning tree means all vertices must be connected. So the two disjoint subsets (discussed above) of vertices must be connected to make a Spanning Tree. And they must be connected with the minimum weight edge to make it a Minimum Spanning Tree.
Follow the given steps to find MST using Prim’s Algorithm:

• Create a set mstSet that keeps track of vertices already included in MST. 
• Assign a key value to all vertices in the input graph. Initialize all key values as INFINITE. Assign the key value as 0 for the first vertex so that it is picked first. 
• While mstSet doesn’t include all vertices 
  • Pick a vertex u which is not there in mstSet and has a minimum key value. 
  • Include u in the mstSet. 
  • Update the key value of all adjacent vertices of u. To update the key values, iterate through all adjacent vertices. For every adjacent vertex v, if the weight of edge u-v is less than the previous key value of v, update the key value as the weight of u-v

The idea of using key values is to pick the minimum weight edge from the cut. The key values are used only for vertices that are not yet included in MST, the key value for these vertices indicates the minimum weight edges connecting them to the set of vertices included in MST. 

Let us understand with the following illustration:

Input graph:

Step 1: The set mstSet is initially empty and keys assigned to vertices are {0, INF, INF, INF, INF, INF, INF, INF} where INF indicates infinite. Now pick the vertex with the minimum key value. The vertex 0 is picked, include it in mstSet. So mstSet becomes {0}. After including it to mstSet, update key values of adjacent vertices. Adjacent vertices of 0 are 1 and 7. The key values of 1 and 7 are updated as 4 and 8. Following subgraph shows vertices and their key values, only the vertices with finite key values are shown. The vertices included in MST are shown in green color.

Step 2: Pick the vertex with minimum key value and which is not already included in the MST (not in mstSET). The vertex 1 is picked and added to mstSet. So mstSet now becomes {0, 1}. Update the key values of adjacent vertices of 1. The key value of vertex 2 becomes 8.

Step 3: Pick the vertex with minimum key value and which is not already included in the MST (not in mstSET). We can either pick vertex 7 or vertex 2, let vertex 7 is picked. So mstSet now becomes {0, 1, 7}. Update the key values of adjacent vertices of 7. The key value of vertex 6 and 8 becomes finite (1 and 7 respectively).

Step 4: Pick the vertex with minimum key value and not already included in MST (not in mstSET). Vertex 6 is picked. So mstSet now becomes {0, 1, 7, 6}. Update the key values of adjacent vertices of 6. The key value of vertex 5 and 8 are updated.

Step 5: Repeat the above steps until mstSet includes all vertices of given graph. Finally, we get the following graph.

Coding implementation of Prim’s algorithm

Use a boolean array mstSet[] to represent the set of vertices included in MST. If a value mstSet[v] is true, then vertex v is included in MST, otherwise not. Array key[] is used to store key values of all vertices. Another array parent[] to store indexes of parent nodes in MST. The parent array is the output array, which is used to show the constructed MST.

C#

// A C# program for Prim's Minimum

// Spanning Tree (MST) algorithm.

// The program is for adjacency

// matrix representation of the graph

 

using System;

class MST {

 

    // Number of vertices in the graph

    static int V = 5;

 

    // A utility function to find

    // the vertex with minimum key

    // value, from the set of vertices

    // not yet included in MST

    static int minKey(int[] key, bool[] mstSet)

    {

 

        // Initialize min value

        int min = int.MaxValue, min_index = -1;

 

        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[]

    static void printMST(int[] parent, int[, ] graph)

    {

        Console.WriteLine("Edge \tWeight");

        for (int i = 1; i < V; i++)

            Console.WriteLine(parent[i] + " - " + i + "\t"

                              + graph[i, parent[i]]);

    }

 

    // Function to construct and

    // print MST for a graph represented

    // using adjacency matrix representation

    static void primMST(int[, ] graph)

    {

 

        // Array to store constructed MST

        int[] parent = new int[V];

 

        // Key values used to pick

        // minimum weight edge in cut

        int[] key = new int[V];

 

        // To represent set of vertices

        // included in MST

        bool[] mstSet = new bool[V];

 

        // Initialize all keys

        // as INFINITE

        for (int i = 0; i < V; i++) {

            key[i] = int.MaxValue;

            mstSet[i] = false;

        }

 

        // Always include first 1st vertex in MST.

        // Make key 0 so that this vertex is

        // picked as first vertex

        // First node is always root of MST

        key[0] = 0;

        parent[0] = -1;

 

        // 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] != 0 && mstSet[v] == false

                    && graph[u, v] < key[v]) {

                    parent[v] = u;

                    key[v] = graph[u, v];

                }

        }

 

        // print the constructed MST

        printMST(parent, graph);

    }

 

    // Driver's Code

    public static void Main()

    {

 

        /* Let us create the following graph

        2 3

        (0)--(1)--(2)

        | / \ |

        6| 8/ \5 |7

        | / \ |

        (3)-------(4)

            9 */

 

        int[, ] graph = new int[, ] { { 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);

    }

}

 

// This code is contributed by anuj_67.

Output

Edge     Weight

0 - 1     2 

1 - 2     3 

0 - 3     6 

1 - 4     5 

• Time Complexity: O(V²), If the input graph is represented using an adjacency list, then the time complexity of Prim’s algorithm can be reduced to O(E log V) with the help of a binary heap.  In this implementation, we are always considering the spanning tree to start from the root of the graph
• Auxiliary Space: O(V)