Prims Algorithm Graph Theory

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

A group of edges that connects two set of vertices in a graph is called cut in graph theory. So, at every step of Prim’s algorithm, we find a cut (of two sets, one contains the vertices already included in MST and other contains 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.

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

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

How to implement the above algorithm?
We 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.

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 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.

Pick the vertex with minimum key value and not already included in 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.

Pick the vertex with minimum key value and not already included in 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).

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.

We repeat the above steps until mstSet includes all vertices of given graph. Finally, we get the following graph.

Python program for Prim's Minimum Spanning Tree (MST) algorithm

The program is for adjacency matrix representation of the graph

Code:

import sys
class Graph():

def __init__(self, vertices):
    self.V = vertices
    self.graph = [[0 for column in range(vertices)]
                for row in range(vertices)]

# A utility function to print the constructed MST stored in parent[]
def printMST(self, parent):
    print "Edge \tWeight"
    for i in range(1, self.V):
        print parent[i], "-", i, "\t", self.graph[i][ parent[i] ]

# A utility function to find the vertex with
# minimum distance value, from the set of vertices
# not yet included in shortest path tree
def minKey(self, key, mstSet):

    # Initilaize min value
    min = sys.maxint

    for v in range(self.V):
        if key[v] < min and mstSet[v] == False:
            min = key[v]
            min_index = v

    return min_index

# Function to construct and print MST for a graph
# represented using adjacency matrix representation
def primMST(self):

    # Key values used to pick minimum weight edge in cut
    key = [sys.maxint] * self.V
    parent = [None] * self.V # Array to store constructed MST
    # Make key 0 so that this vertex is picked as first vertex
    key[0] = 0
    mstSet = [False] * self.V

    parent[0] = -1 # First node is always the root of

    for cout in range(self.V):

        # Pick the minimum distance vertex from
        # the set of vertices not yet processed.
        # u is always equal to src in first iteration
        u = self.minKey(key, mstSet)

        # Put the minimum distance vertex in
        # the shortest path tree
        mstSet[u] = 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 shotest path tree
        for v in range(self.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 self.graph[u][v] > 0 and mstSet[v] == False and key[v] > self.graph[u][v]:
                    key[v] = self.graph[u][v]
                    parent[v] = u

    self.printMST(parent)

g = Graph(5)
g.graph = [ [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]]

g.primMST();

Time Complexity of the above program is O(V^2). If the input graph is represented using adjacency list, then the time complexity of Prim’s algorithm can be reduced to O(E log V) with the help of binary heap.