3.5 Prims and Kruskals Algorithms - Greedy Method

Summary

This script explains the concept of a minimum cost spanning tree (MCST) in graph theory. It defines a spanning tree and illustrates how to calculate the number of possible spanning trees in a graph. The focus then shifts to finding the MCST using greedy algorithms like Prim's and Kruskal's, which efficiently select edges to minimize the total cost without forming cycles. The script also addresses scenarios with missing edge weights and non-connected graphs, emphasizing the importance of connected components in spanning tree formation.

Takeaways

• 🌳 A spanning tree is a subgraph of a connected graph that includes all the vertices and exactly (n-1) edges where n is the number of vertices.
• 🔄 The number of different spanning trees possible for a graph can be calculated using combinations, specifically 'n choose (n-1)' where n is the number of edges.
• 🔢 The formula to calculate the number of different spanning trees is (number of edges) choose (number of vertices - 1) minus the number of cycles.
• 🏗️ Prim's algorithm is a greedy method to find the minimum cost spanning tree by initially selecting the smallest edge and then always choosing the smallest edge connected to the already selected vertices.
• 🛠️ Kruskal's algorithm is another greedy method that selects the smallest edge and continues to do so without forming cycles until (n-1) edges are chosen.
• ⏱️ Kruskal's algorithm has a time complexity of O(n^2), but this can be improved to O(n log n) using a min-heap data structure.
• 🔄 Both Prim's and Kruskal's algorithms can be applied to non-connected graphs, but they will only find spanning trees for the individual connected components.
• 💡 The minimum cost of a missing edge in a graph can be deduced by understanding the cycle it would form if included in a spanning tree.
• 📉 In a weighted graph, different spanning trees can have varying costs, and the goal is to find the one with the minimum cost.
• 🚫 A graph must be connected to have a spanning tree; disconnected graphs do not support the concept of a single spanning tree.

Q & A

What is a spanning tree?

A spanning tree is a subgraph of a graph that includes all the vertices of the graph but only a subset of the edges, specifically the number of edges is equal to the number of vertices minus one, and it contains no cycles.

How many different spanning trees can be formed from a graph with six vertices and six edges?

The number of different spanning trees that can be formed is given by the combination formula \( \binom{6}{5} \), which equals 6.

What is the formula to calculate the number of different spanning trees possible from a given graph?

The formula to calculate the number of different spanning trees is to take the number of edges available, subtract the number of cycles formed by those edges, and then subtract one to account for the number of vertices.

How does the cost of a spanning tree in a weighted graph differ from one spanning tree to another?

The cost of a spanning tree in a weighted graph can vary because different sets of edges can be chosen to form the tree, each with a different total weight.

What is the minimum cost spanning tree?

The minimum cost spanning tree is the spanning tree that has the lowest possible total edge weight among all possible spanning trees for a given weighted graph.

What are the two greedy algorithms for finding the minimum cost spanning tree?

The two greedy algorithms for finding the minimum cost spanning tree are Prim's algorithm and Kruskal's algorithm.

How does Prim's algorithm work to find the minimum cost spanning tree?

Prim's algorithm starts with an arbitrary vertex and grows the minimum cost spanning tree by always adding the smallest edge that connects a new vertex to the tree.

How does Kruskal's algorithm work to find the minimum cost spanning tree?

Kruskal's algorithm sorts all the edges in the graph by weight and adds them to the minimum cost spanning tree in ascending order, ensuring no cycles are formed.

What is the time complexity of Kruskal's algorithm without using a data structure like a min-heap?

The time complexity of Kruskal's algorithm without using a min-heap is \( \Theta(n^2) \), where n is the number of vertices.

How can Kruskal's algorithm be optimized using a min-heap?

Using a min-heap, Kruskal's algorithm can achieve a time complexity of \( \Theta(n \log n) \) by efficiently finding the next minimum edge to add to the tree.

Can Prim's or Kruskal's algorithm find a minimum cost spanning tree for a non-connected graph?

No, Prim's or Kruskal's algorithm cannot find a minimum cost spanning tree for a non-connected graph because a spanning tree requires all vertices to be connected.

What is the minimum possible weight for an edge that is not included in the minimum cost spanning tree?

The minimum possible weight for an edge not included in the minimum cost spanning tree is the weight that would have been chosen if it were available before the edge that actually formed a cycle.