Minimum Spanning Tree Algorithm

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Questions and Answers

What is the main concept of the Greedy method?

Select the best option at each stage (correct)

Use dynamic programming to solve the problem

Divide the problem into smaller sub-problems

Use backtracking to find the optimal solution

The Divide and Conquer technique can be applied to all types of problems.

False

What is a feasible solution in the context of the Greedy method?

A subset that satisfies given constraints

In the Greedy method, a problem is solved by determining a subset to satisfy some _______________________.

constraints Signup and view all the answers

What is the goal of the Greedy method?

To find an optimal solution that maximizes or minimizes an objective function Signup and view all the answers

Match the algorithm with its description:

Kruskal's Algorithm = A minimum-spanning-tree algorithm that uses a greedy approach Prim's Algorithm = A minimum-spanning-tree algorithm that uses a divide-and-conquer approach Greedy Method = A general method that solves problems by making the locally optimal choice Heapify = A process of building a heap data structure Signup and view all the answers

The Greedy method always finds an optimal solution.

False Signup and view all the answers

What is the main difference between the Divide and Conquer technique and the Greedy method?

Divide and Conquer breaks down the problem into smaller sub-problems, while the Greedy method makes the locally optimal choice at each stage. Signup and view all the answers

What is the main purpose of Kruskal's algorithm?

To construct a minimum spanning tree Signup and view all the answers

In Kruskal's algorithm, edges are arranged in decreasing order of their weights.

False Signup and view all the answers

What is the role of the Heapify function in Kruskal's algorithm?

To construct a heap out of the edge costs. Signup and view all the answers

In Kruskal's algorithm, a minimum spanning tree T is constructed by selecting edges that do not form a ______ with the edges already included in T.

cycle Signup and view all the answers

Match the following algorithms with their purposes:

Kruskal's Algorithm = To construct a minimum spanning tree Prim's Algorithm = To find the shortest path in a graph Greedy Method = To make the locally optimal choice at each step Heapify = To sort the edges of a graph in increasing order Signup and view all the answers

Prim's Algorithm is used to construct a minimum spanning tree.

True Signup and view all the answers

What is the primary goal of Kruskal's Algorithm?

To find the minimum spanning tree of a graph Signup and view all the answers

The Greedy Method is used to find the shortest path in a graph.

False Signup and view all the answers

What is the value of near[k] when k = 7 in the first iteration?

4 Signup and view all the answers

The minimum spanning tree is a subset of the edges in the original graph that connect all the vertices together with a minimum total ______________.

cost Signup and view all the answers

What is the value of j in the second iteration?

2 Signup and view all the answers

Prim's Algorithm is used to find the minimum spanning tree of a graph.

True Signup and view all the answers

What is the value of mincost after the first iteration?

69 Signup and view all the answers

Match the following algorithms with their respective applications:

Kruskal's Algorithm = Minimum Spanning Tree Prim's Algorithm = Minimum Spanning Tree Heapify = Sorting Dijkstra's Algorithm = Shortest Path Signup and view all the answers

Study Notes

Greedy Method

A problem-solving approach that suggests devising an algorithm that works in stages, considering one input at a time
At each stage, a decision is made regarding whether a particular input is in an optimal solution
The greedy method is used to find a feasible solution that either maximizes or minimizes a given objective function

Minimum Spanning Tree

A minimum spanning tree is a subset of edges that connect all nodes in a graph with the minimum total edge cost
Kruskal's algorithm is a greedy algorithm used to construct a minimum spanning tree
The algorithm works by arranging all edges in increasing order of weight and selecting the edge with minimum weight that does not form a cycle with the edges already included in the tree

Example Problem

Find the maximum value for the objective function Z = 3x + 4y, subjected to certain constraints
Use the greedy method to find the optimal solution by determining the subset of inputs that satisfies the constraints and maximizes the objective function

Steps to Construct a Minimum Spanning Tree

Arrange all edges in increasing order of weight
Select an edge with minimum weight as the first edge of the spanning tree
Select the next edge with minimum weight that does not form a cycle with the edges already included in the tree
Continue step 3 until the tree contains (n-1) edges, where n is the number of vertices of the graph

Key Concepts

Feasible solution: a subset that satisfies the constraints of the problem
Optimal solution: a feasible solution that either maximizes or minimizes the objective function

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Description

This quiz is about finding the minimum spanning tree using a greedy algorithm. It involves selecting the edge with the minimum cost and updating the near array.