Prim's Algorithm

Preview

1. What is Prim’s Algorithm used for in computer science?
2. How does Prim’s Algorithm differ from Kruskal’s Algorithm?
3. What is the time complexity of Prim’s Algorithm for finding the minimum spanning tree?
4. How does Prim’s Algorithm work at each iteration to select the next edge to add to the tree?
5. What modifications could be made to Prim’s Algorithm to make it more efficient or suitable for different types of graphs?

Explain

Prim’s Algorithm is a commonly used algorithm in computer science that is used to find the minimum spanning tree (MST) of a given graph. The MST is the subset of edges that connects all the vertices of a graph without forming a cycle, while minimizing the total weight of the tree.

Prim’s Algorithm starts by selecting an arbitrary vertex, adding it to the MST and marking it as visited. Then, it examines all the edges connected to this vertex and chooses the one with the minimum weight that leads to an unvisited vertex. It then adds this vertex to the MST, marks it as visited, and repeats the process until all the vertices have been included in the MST.

Example:

Let’s consider the following weighted undirected graph with 6 vertices and 7 edges:

We will use Prim’s Algorithm to find the minimum spanning tree of this graph:

Step 1: Select an arbitrary vertex (let’s choose vertex A) and add it to the MST.

Step 2: Consider all the edges that are connected to vertex A and choose the one with the minimum weight that leads to an unvisited vertex. This edge is AB with weight 2. Add vertex B to the MST and mark it as visited.

Step 3: Consider all the edges that are connected to vertices A and B, and choose the one with the minimum weight that leads to an unvisited vertex. This edge is CD with weight 3. Add vertex C to the MST and mark it as visited.

Step 4: Consider all the edges that are connected to vertices A, B, and C, and choose the one with the minimum weight that leads to an unvisited vertex. This edge is BE with weight 4. Add vertex E to the MST and mark it as visited.

Step 5: Consider all the edges that are connected to vertices A, B, C, and E, and choose the one with the minimum weight that leads to an unvisited vertex. This edge is AD with weight 5. Add vertex D to the MST and mark it as visited.

Step 6: Consider all the edges that are connected to vertices A, B, C, E, and D, and choose the one with the minimum weight that leads to an unvisited vertex. This edge is CE with weight 1. Add vertex E to the MST and mark it as visited.

After Step 6, all the vertices have been included in the MST, and we have found the minimum spanning tree of the graph:

Keypoint

1. Prim’s algorithm is a greedy algorithm used to find the minimum spanning tree of an undirected graph.
2. The algorithm starts with a given vertex and adds the nearest neighbour to the tree set.
3. The algorithm chooses the minimum edge weight while adding vertices to the tree set.
4. The algorithm terminates when all vertices have been added to the tree set.
5. The algorithm requires a priority queue to store the minimum weight edge at each step.
6. The algorithm has a time complexity of O(E log V).
7. Prim’s Algorithm can be used in both dense and sparse graphs.

Review

1. What is the main objective of Prim’s Algorithm?

Ans: The main objective of Prim’s Algorithm is to find the Minimum Spanning Tree (MST) for a given connected and weighted undirected graph.

2. How does Prim’s Algorithm work?

Ans: Prim’s Algorithm works by starting with an arbitrary vertex and adding the edge with minimum weight to the existing MST in each step until all vertices are included.

3. What is the time complexity of Prim’s Algorithm?

Ans: The time complexity of Prim’s Algorithm is O(E log V), where E is the number of edges and V is the number of vertices in the graph.

4. Can Prim’s Algorithm be used for directed graphs?

Ans: No, Prim’s Algorithm is specifically designed for undirected graphs. For directed graphs, other algorithms like Dijkstra’s Algorithm or Kruskal’s Algorithm can be used.

5. What is the difference between Prim’s Algorithm and Kruskal’s Algorithm?

Ans: Prim’s Algorithm starts with an arbitrary vertex and gradually builds the MST by adding the edge with the minimum weight at each step. Kruskal’s Algorithm, on the other hand, starts with the edges of the graph arranged in increasing order of their weight, and adds them to the MST one by one as long as they do not create a cycle.