Minimum Spanning Trees: Algorithms and Applications

Question 1

Which of the following is NOT a property of a Minimum Spanning Tree (MST)?

Accepted Answer

Is a forest

Answer

Contains all the vertices of the original graph

Answer

Has the minimum possible total weight

Answer

Is a connected graph

Question 2

In Kruskal's algorithm, what is the purpose of the union-find data structure?

Accepted Answer

To determine if two vertices are in the same connected component

Answer

To find the edge with the minimum weight

Answer

To store the edges of the MST

Answer

To calculate the total weight of the MST

Question 3

Which of the following is a valid criterion for selecting the next edge to add to the MST in Prim's algorithm?

Accepted Answer

Smallest weight among all edges not already in the MST

Answer

Largest weight among all edges not already in the MST

Answer

Randomly chosen from all edges not already in the MST

Answer

Edge with the smallest degree

Question 4

Which of the following applications of MSTs is used in network design?

Accepted Answer

Minimum cost spanning tree

Answer

Clustering

Answer

Primality testing

Answer

Finding connected components

Question 5

In Kruskal's algorithm, which edge is added to the MST first?

Accepted Answer

Edge with the minimum weight

Answer

Edge with the maximum weight

Answer

Edge with the smallest number of vertices

Answer

Randomly chosen edge

Question 6

How do cycles in an undirected graph affect the uniqueness and computational complexity of finding its minimum spanning tree (MST)?

Accepted Answer

Cycles make the MST non-unique and potentially increase the complexity of its computation.

Answer

Cycles guarantee a unique MST and simplify its computation.

Answer

Cycles have no impact on the uniqueness or computational complexity of the MST.

Question 7

What is the primary objective of finding a minimum spanning tree (MST) in a weighted graph?

Accepted Answer

To find a subset of edges that connects all nodes with minimal total weight

Answer

To determine the shortest path between every pair of nodes

Answer

To divide the graph into clusters with the fewest possible connections within each cluster

Answer

To maximize the number of edges in the graph while keeping the total weight minimal

Question 8

In Kruskal's algorithm, which data structure is employed to efficiently track connected components?

Accepted Answer

Union-find data structure

Answer

Adjacency matrix

Answer

Depth-first search tree

Answer

Breadth-first search tree

Question 9

In Kruskal's algorithm, which edge should be considered initially?

Accepted Answer

The edge with the smallest weight

Answer

The edge with the largest weight

Answer

An arbitrary edge

Answer

An edge that does not introduce a cycle

Question 10

Which of the following is NOT an advantage of Prim's algorithm over Kruskal's algorithm?

Accepted Answer

Ability to find multiple MSTs

Answer

Ease of implementation

Answer

Faster execution for sparse graphs

Answer

Elimination of the need for a union-find data structure

Question 11

Which of the following is not a property of a minimum spanning tree (MST)?

Accepted Answer

May not be unique for a given graph

Answer

Contains all vertices of the graph

Answer

Has the minimum possible total weight

Answer

Always forms a connected graph

Question 12

Which MST algorithm operates by iteratively merging components based on edge weights?

Accepted Answer

Kruskal's algorithm

Answer

Prim's algorithm

Answer

Dijkstra's algorithm

Answer

Bellman-Ford algorithm

Question 13

What is the time complexity of Kruskal's algorithm for finding MSTs in a graph with V vertices and E edges?

Accepted Answer

O(E log V)

Answer

O(V^2)

Answer

O(V log V)

Answer

O(E^2)

Question 14

Which data structure is typically used to represent disjoint sets in Kruskal's algorithm?

Accepted Answer

Union-find data structure

Answer

Array

Answer

Linked list

Answer

Heap

Question 15

In Prim's algorithm, which criterion determines the next edge to be added to the growing MST?

Accepted Answer

Minimum weight

Answer

Maximum weight

Answer

Smallest source vertex

Answer

Largest destination vertex

Question 16

What is the key difference between a spanning tree and a minimum spanning tree?

Accepted Answer

A minimum spanning tree is a spanning tree with the minimum possible total weight.

Answer

A spanning tree is a minimum spanning tree with all edges having positive weights.

Answer

A spanning tree has all vertices connected, while a minimum spanning tree does not.

Answer

There is no difference between a spanning tree and a minimum spanning tree.

Question 17

In Prim's algorithm, which data structure is typically used to track the vertices not yet included in the MST efficiently?

Accepted Answer

Min-heap

Answer

Queue

Answer

Stack

Answer

Adjacency list

Question 18

Which data structure is commonly used to represent weighted edges in both Kruskal's and Prim's algorithms?

Accepted Answer

Priority queue

Answer

Adjacency list

Answer

Adjacency matrix

Answer

Spanning tree

Question 19

Which of the following is a fundamental property of a minimum spanning tree (MST)?

Accepted Answer

It minimizes the total weight of edges included in the tree.

Answer

It connects all vertices in a graph with the minimum possible number of edges.

Answer

It maximizes the sum of weights of edges in the tree.

Answer

It forms a unique path between every pair of vertices.

Question 20

Which data structure is most efficient for implementing the disjoint-set union (DSU) operation in Kruskal's algorithm?

Accepted Answer

Union-Find

Answer

Array

Answer

Linked List

Answer

Hash Table

Question 21

For a weighted undirected graph G with |V| vertices and |E| edges, what is the minimum number of edges required to form its spanning tree?

Accepted Answer

|V| - 1

Answer

|V|

Answer

|E|

Answer

|E| - |V| + 1

Question 22

Which step is NOT included in Prim's algorithm?

Accepted Answer

Updating edge weights.

Answer

Selecting a starting vertex.

Answer

Adding the lowest-weight edge to the MST.

Answer

Stopping when all vertices are added to the MST.

Question 23

What is the purpose of using a priority queue in Prim's algorithm?

Accepted Answer

To maintain a list of candidate edges sorted by weight.

Answer

To keep track of visited vertices.

Answer

To compute the minimum spanning weight.

Answer

To perform disjoint-set union operations.

Question 24

In a dense graph with a large number of edges, which algorithm is more efficient for finding a minimum spanning tree?

Accepted Answer

Kruskal's algorithm

Answer

Prim's algorithm

Answer

Both algorithms are equally efficient

Answer

Neither algorithm is efficient for dense graphs

Question 25

Which algorithm is used to find MSTs by merging subtrees based on edge weights?

Accepted Answer

Kruskal's algorithm

Answer

Dijkstra's algorithm

Answer

Bellman-Ford algorithm

Answer

Floyd-Warshall algorithm

Question 26

What data structure is commonly used to efficiently maintain and select the minimum weight edges in Kruskal's algorithm?

Accepted Answer

Disjoint-set union (DSU)

Answer

Array

Answer

List

Answer

Queue

Question 27

In Prim's algorithm, which data structure is typically used to manage the set of candidate edges not yet included in the MST?

Accepted Answer

Priority queue

Answer

Stack

Answer

Hash table

Answer

Linked list

Question 28

Finding the MST of a weighted graph where weights represent distances is useful for solving which problem?

Accepted Answer

Finding the minimum cost network to connect cities

Answer

Finding the shortest path between two vertices

Answer

Finding the maximum flow in a network

Answer

Detecting cycles in a graph

Question 29

What is a Minimum Spanning Forest (MSF)?

Accepted Answer

A collection of MSTs for a graph with multiple connected components

Answer

A graph with the minimum number of vertices

Answer

A forest with the maximum number of edges

Answer

A tree with the minimum total weight

Question 30

Which algorithm can be used to find MSTs in graphs with negative weight edges?

Accepted Answer

Borůvka's algorithm

Answer

Kruskal's algorithm

Answer

Prim's algorithm

Answer

Dijkstra's algorithm

Question 31

Which of the following is NOT a property of a minimum spanning tree (MST)?

Accepted Answer

It maximizes the total weight of the edges.

Answer

It minimizes the total weight of the edges.

Answer

It can be found using Kruskal's or Prim's algorithm.

Answer

It connects all vertices in the graph.

Question 32

Which of the following algorithms is specifically designed for finding an MST in a weighted, undirected graph?

Accepted Answer

Kruskal's algorithm or Prim's algorithm

Answer

Bellman-Ford algorithm

Answer

Floyd-Warshall algorithm

Answer

Dijkstra's algorithm

Question 33

Provide a real-world application where MSTs can be used to solve practical problems.

Accepted Answer

Optimizing the layout of a network of roads or pipelines

Answer

Clustering data points

Answer

Finding the shortest path between two nodes

Answer

Scheduling tasks

Question 34

Which of the following statements accurately describes Prim's algorithm?

Accepted Answer

It iteratively selects the lowest-weight edge that connects a vertex in the growing tree to a vertex not yet in the tree.

Answer

It sorts the edges in decreasing order of weight and adds them to the MST.

Question 35

In the context of Kruskal's algorithm, which data structure plays a vital role in efficiently managing disjoint sets of vertices?

Accepted Answer

Union-Find data structure

Answer

Adjacency list

Answer

Hash table

Answer

Binary search tree

Question 36

Identify a potential drawback of Prim's algorithm when compared to Kruskal's algorithm.

Accepted Answer

It may be less efficient for dense graphs with a large number of edges.

Answer

It is more difficult to implement.

Answer

It cannot handle graphs with negative edge weights.

Answer

It requires significantly more memory.

Question 37

In Prim's algorithm, what is the primary purpose of utilizing a disjoint-set data structure?

Accepted Answer

To efficiently determine whether two vertices belong to the same connected component in the growing MST.

Answer

To calculate the weights of the edges in the MST.

Answer

To store the edges of the MST.

Question 38

What is the primary objective of finding a minimum spanning tree (MST) in a graph?

Accepted Answer

To identify a subgraph connecting all vertices with the lowest possible total edge weight.

Answer

To determine a subgraph with the fewest number of edges.

Answer

To find a subgraph with the maximum total edge weight.

Answer

To discover a subgraph containing the largest number of cycles.

Question 39

Which algorithm utilizes a priority queue to efficiently construct a minimum spanning tree?

Accepted Answer

Prim's algorithm

Answer

Floyd-Warshall algorithm

Answer

Kruskal's algorithm

Answer

Dijkstra's algorithm

Question 40

Which of the following scenarios exemplifies a real-world application of minimum spanning trees?

Accepted Answer

Designing a network infrastructure to connect multiple locations with minimal cabling cost.

Answer

Computing the shortest path between two points on a map.

Answer

Finding the maximum clique in a social network graph.

Answer

Detecting cycles in a railway network to avoid potential conflicts.

Question 41

What distinguishes a minimum spanning forest from a minimum spanning tree?

Accepted Answer

A minimum spanning forest is a collection of MSTs when the input graph is not connected.

Answer

A minimum spanning forest contains cycles, while a minimum spanning tree does not.

Answer

A minimum spanning tree has more vertices than a minimum spanning forest.

Answer

A minimum spanning forest has more edges than a minimum spanning tree.

Question 42

In Prim's algorithm, which data structure plays a crucial role in efficiently managing the selection of edges for the growing MST?

Accepted Answer

Priority queue

Answer

Disjoint-set forest

Answer

Hash table

Answer

Graph adjacency list

Question 43

Given a connected graph G with n vertices, what is the maximum number of edges that can be present in any MST of G?

Accepted Answer

n - 1

Answer

n

Answer

n + 1

Answer

2n - 1

Question 44

In Prim's algorithm, which data structure is used to efficiently store and retrieve the edges with the minimum weight connecting the current MST to the remaining vertices?

Accepted Answer

Min-priority queue

Answer

Hash table

Answer

Linked list

Answer

Stack

Question 45

For a graph with 5 vertices, what is the maximum number of edges possible in its minimum spanning tree?

Accepted Answer

4

Answer

5

Answer

10

Answer

6

Question 46

Which real-world application would benefit most from using a minimum spanning tree algorithm?

Accepted Answer

Designing a network of communication cables to connect multiple cities at the lowest cost.

Answer

Finding the shortest route between two locations on a map.

Answer

Scheduling tasks in a project management system to minimize completion time.

Question 47

If a graph has multiple minimum spanning trees (MSTs), will they all have the same total edge weight?

Accepted Answer

True

Answer

Only if the graph is complete.

Answer

False

Question 48

In a weighted graph, is the edge with the smallest weight always part of the minimum spanning tree?

Accepted Answer

False

Answer

Only if the graph is connected.

Answer

True

Answer

Only if the graph is undirected.

Question 49

For finding the MST in a sparse graph (with fewer edges), which algorithm is generally preferred due to its lower time complexity?

Accepted Answer

Kruskal's Algorithm

Answer

Both are equally efficient for sparse graphs.

Answer

Prim's Algorithm

Answer

Neither algorithm is suitable for sparse graphs.

Question 50

Which of the following is NOT a characteristic of a minimum spanning tree?

Accepted Answer

It contains cycles.

Answer

It is a subgraph of the original graph.

Answer

It connects all vertices in the graph.

Answer

It has the minimum total weight.

Question 51

Which algorithm for finding minimum spanning trees utilizes a priority queue?

Accepted Answer

Prim's algorithm

Answer

Floyd-Warshall algorithm

Answer

Kruskal's algorithm

Answer

Dijkstra's algorithm

Question 52

Given a graph with n vertices and m edges, what is the time complexity of Kruskal's algorithm for finding the minimum spanning tree?

Accepted Answer

O(m log n)

Answer

O(n log n)

Answer

O(m^2)

Answer

O(n^2)

Question 53

Which of the following is a valid application of minimum spanning trees?

Accepted Answer

Network design

Answer

Sorting

Answer

Searching

Question 54

Consider a graph with n vertices and m edges, where m > n. Which of these statements is true about its minimum spanning tree?

Accepted Answer

It has n-1 edges

Answer

It has m-n+1 edges

Answer

It has m edges

Answer

It has n+1 edges

Question 55

In a graph with n vertices and m edges, where m < n, what can be concluded about its minimum spanning tree?

Accepted Answer

It has a minimum spanning tree that is a forest

Answer

It has a unique minimum spanning tree

Answer

It has a minimum spanning tree that is a cycle

Answer

It does not have a minimum spanning tree