Minimum Spanning Trees: Master the Algorithms for Optimal Networks

Question 1

In Kruskal's Algorithm, what data structure is primarily used to keep track of connected components?

Accepted Answer

Union-Find Data Structure

Answer

Adjacency List

Answer

Hash Table

Answer

Queue

Question 2

In a connected graph, how many unique MSTs can exist?

Accepted Answer

V - 1

Answer

1

Answer

E - V + 1

Answer

V - E + 1

Question 3

How can the concept of a minimum spanning tree be extended to more complex graph structures, such as directed graphs or graphs with negative weights?

Accepted Answer

Directed graphs can be converted to undirected graphs, and negative weights can be handled using modified versions of Kruskal's or Prim's algorithms.

Answer

MSTs are exclusively applicable to undirected graphs with positive weights.

Answer

In directed graphs, the weight of an MST is always negative.

Answer

MSTs cannot be applied to graphs with negative weights.

Question 4

Which of the following is not an application of MSTs?

Accepted Answer

Shortest path finding

Answer

Network design

Answer

Cluster analysis

Answer

Image segmentation

Question 5

Consider a graph with V vertices and E edges. What is the maximum possible number of edges in a Minimum Spanning Forest?

Accepted Answer

V - 1

Answer

V

Answer

E

Answer

V + 1

Question 6

Which characteristic uniquely defines Minimum Spanning Trees (MSTs)?

Accepted Answer

They connect all vertices in a weighted graph with the least total edge weight.

Answer

They are guaranteed to be unique for any given graph.

Answer

They can have cycles.

Answer

They cannot contain negative weight edges.

Question 7

In Kruskal's algorithm, what is the primary role of the disjoint-set data structure?

Accepted Answer

To efficiently manage the connected components of the graph.

Answer

To organize edges in ascending order based on weight.

Answer

To store the vertices included in the MST.

Answer

To calculate the total weight of the MST.

Question 8

In the context of MSTs, what is the purpose of a disjoint-set data structure?

Accepted Answer

To efficiently determine if two vertices belong to the same connected component

Answer

To store the MST as a set of edges

Answer

To calculate the total weight of the MST

Question 9

In a weighted graph, an edge (u, v) is considered a safe edge for MST if:

Accepted Answer

It belongs to some MST of the graph

Answer

Its weight is the smallest among all edges in the graph

Answer

Its weight is less than the sum of weights of any two other edges

Answer

It connects two MSTs of the graph

Question 10

Consider a connected weighted graph G. Which of the following statements is true about the MST of G?

Accepted Answer

The sum of weights of any subset of edges in the MST is less than or equal to the sum of weights of the same subset of edges in any other spanning tree of G

Answer

The MST always has the maximum number of edges among all spanning trees of G

Question 11

Which of the following is an advantage of using Minimum Spanning Trees in network design?

Accepted Answer

It minimizes the total cost of the network while ensuring connectivity

Answer

It guarantees that the network is always acyclic

Answer

It helps in finding the shortest path between any two points in the network

Question 12

In Prim's algorithm, which vertex is always chosen initially?

Accepted Answer

The vertex with the smallest weight

Answer

A random vertex

Answer

The vertex with the most edges

Answer

The vertex with the largest weight

Question 13

When all graph vertices have the same degree, which heuristic is commonly used to improve the efficiency of Prim's algorithm?

Accepted Answer

Degree heuristic

Answer

Random heuristic

Answer

Depth-first heuristic

Answer

Weight heuristic

Question 14

Which term measures the performance of an MST by comparing its total weight to the optimal MST?

Accepted Answer

Approximation ratio

Answer

Memory usage

Answer

Number of vertices

Answer

Time complexity

Question 15

To implement Prim's algorithm efficiently, which data structure is commonly utilized?

Accepted Answer

Min-priority queue

Answer

Stack

Answer

Queue

Answer

Hash Table

Question 16

In a graph with 'n' vertices, what is the maximum number of edges in its Minimum Spanning Tree?

Accepted Answer

n-1

Answer

n

Answer

n^2

Answer

n+1

Question 17

Which application does NOT utilize Minimum Spanning Trees?

Accepted Answer

Finding the shortest path between two cities

Answer

Designing communication networks

Answer

Building a power grid

Answer

Connecting computers in a LAN

Question 18

In a weighted graph with cycles, how many distinct Minimum Spanning Trees can it have?

Accepted Answer

One or more

Answer

Exactly one

Answer

It depends on the number of vertices.

Answer

None

Question 19

What is the primary objective of identifying a Minimum Spanning Tree in a graph?

Accepted Answer

To connect all vertices with the minimum possible total edge weight.

Answer

To determine the longest path in the graph.

Answer

To find the shortest path between two given vertices.

Question 20

What is the defining characteristic of a minimum spanning tree (MST) in a weighted, undirected graph?

Accepted Answer

It connects all vertices with the minimum possible total edge weight.

Answer

It contains all edges with the smallest weights.

Answer

It always forms a single path between any two vertices.

Question 21

Which of the following algorithms is NOT designed to find a minimum spanning tree?

Accepted Answer

Dijkstra's Algorithm

Answer

Prim's Algorithm

Answer

Borůvka's Algorithm

Answer

Kruskal's Algorithm

Question 22

In Kruskal's algorithm, how are edges selected to build the Minimum Spanning Tree (MST)?

Accepted Answer

Edges are sorted by weight in ascending order, and the next edge with the smallest weight that doesn't create a cycle is added.

Answer

Edges are selected randomly.

Answer

Edges are sorted by weight in descending order, and the next edge with the highest weight that doesn't create a cycle is added.

Question 23

Prim's algorithm builds an MST by repeatedly:

Accepted Answer

Adding the edge with the lowest weight that connects a vertex already in the MST to a vertex outside of it.

Answer

Removing the edge with the highest weight that doesn't disconnect the graph.

Answer

Finding the shortest path between any two vertices in the graph.

Question 24

A graph has 7 vertices. What is the maximum number of edges a minimum spanning tree of this graph can have?

Accepted Answer

6

Answer

The number of edges in the MST depends on the edge weights.

Answer

7

Answer

12

Question 25

What data structure is commonly used to efficiently detect cycles during the implementation of Kruskal's algorithm?

Accepted Answer

Disjoint-set data structure (Union-Find)

Answer

Stack

Answer

Queue

Answer

Binary Search Tree

Question 26

In which of these situations would finding a minimum spanning tree be MOST applicable?

Accepted Answer

Designing a network cable layout to minimize the total cable length needed to connect all devices.

Answer

Sorting a list of numbers in ascending order.

Answer

Finding the shortest route between two cities on a map.

Question 27

If the weight of a single edge in a graph is decreased, what impact does it have on the minimum spanning tree?

Accepted Answer

The MST may or may not change.

Answer

The MST will never change.

Answer

The MST will always change.

Question 28

If all edges in a graph have distinct weights, what can we conclude about the minimum spanning tree?

Accepted Answer

The minimum spanning tree is unique.

Answer

There might be multiple minimum spanning trees.

Answer

The graph must be a complete graph.

Answer

A minimum spanning tree cannot be determined.

Question 29

Which of the following algorithms is NOT a greedy algorithm for finding a Minimum Spanning Tree?

Accepted Answer

Depth-First Search (DFS)

Answer

Borůvka's Algorithm

Answer

Kruskal's Algorithm

Answer

Prim's Algorithm

Question 30

What data structure is typically used to implement Kruskal's Algorithm?

Accepted Answer

Disjoint Set Union (DSU)

Answer

Heap

Answer

Queue

Answer

Stack

Question 31

In a weighted graph with n vertices, how many edges does a Minimum Spanning Tree (MST) contain?

Accepted Answer

n-1

Answer

n+1

Answer

n

Answer

2n

Question 32

Which data structure is used in Prim's Algorithm to efficiently identify the next closest vertex to the growing MST?

Accepted Answer

Min-Heap

Answer

Max-Heap

Answer

Queue

Answer

Stack

Question 33

What is the time complexity of Kruskal's Algorithm when using a disjoint set data structure with path compression?

Accepted Answer

O(E log V)

Answer

O(V log V)

Answer

O(V^2)

Answer

O(E log E)

Question 34

Consider a graph with negative edge weights. What can be said about using standard algorithms like Kruskal's and Prim's to find the Minimum Spanning Tree (MST) in such a graph?

Accepted Answer

They may not produce a valid MST.

Answer

MST algorithms will run in polynomial time even with negative weights.

Answer

The MST will always be unique, regardless of negative weights.

Answer

Negative weights are always handled correctly by MST algorithms.

Question 35

A connected, undirected graph with n vertices will always have how many edges in its Minimum Spanning Tree?

Accepted Answer

n-1

Answer

n+1

Answer

2n

Answer

n

Question 36

Which of these is NOT a common application of Minimum Spanning Trees (MSTs)?

Accepted Answer

Finding the shortest path between two specific nodes.

Answer

Connecting cities with minimal road construction.

Answer

Planning efficient cable layouts.

Answer

Optimizing communication networks.

Question 37

In Kruskal's Algorithm, what is the significance of the 'cut' property during the correctness proof?

Accepted Answer

The lightest edge crossing any cut must be included in the MST.

Answer

The heaviest edge crossing any cut must be excluded from the MST.

Answer

Edges crossing a cut can be arbitrarily added or removed without affecting the MST.

Answer

All edges crossing a cut must have the same weight.

Question 38

What is the fundamental difference between Kruskal's and Prim's MST algorithms?

Accepted Answer

Kruskal's adds edges in ascending weight order, while Prim's grows the MST from a starting vertex.

Answer

Kruskal's is deterministic, while Prim's is non-deterministic.

Answer

Kruskal's uses a disjoint set data structure, while Prim's uses a heap.

Answer

Prim's is more efficient for dense graphs, while Kruskal's is better for sparse graphs.

Question 39

Which algorithm is NOT commonly used for finding Minimum Spanning Trees?

Accepted Answer

Depth First Search (DFS)

Answer

Borůvka's algorithm

Answer

Kruskal's algorithm

Answer

Prim's algorithm

Question 40

In Kruskal's algorithm, what data structure is used to maintain the connected components of the graph?

Accepted Answer

Disjoint Set Union (DSU)

Answer

Queue

Answer

Min-Heap

Answer

Stack

Question 41

What fundamental property is guaranteed by a Minimum Spanning Tree (MST)?

Accepted Answer

The MST has the minimum total edge weight among all spanning trees of the graph.

Answer

The MST has the minimum number of edges needed to connect all vertices.

Answer

The MST has a balanced height.

Answer

The MST includes the maximum number of vertices in the graph.

Question 42

In which real-world scenario is finding a Minimum Spanning Tree (MST) particularly beneficial?

Accepted Answer

Designing a network infrastructure that minimizes cabling costs while connecting all nodes.

Answer

Detecting cycles within a graph to identify redundancies.

Answer

Sorting a list of numbers in ascending order.

Answer

Determining the shortest path between two specific points in a network.

Question 43

What is a key advantage of Prim's algorithm over Kruskal's algorithm for finding a Minimum Spanning Tree (MST)?

Accepted Answer

Prim's algorithm constructs the MST incrementally, adding one edge at a time.

Answer

Prim's algorithm has a faster time complexity than Kruskal's in all scenarios.

Answer

Prim's algorithm is generally easier to implement than Kruskal's.

Answer

Prim's algorithm is guaranteed to find the absolute optimal MST in all cases.

Question 44

What is the time complexity of Kruskal's algorithm for finding a Minimum Spanning Tree (MST) in a graph with E edges and V vertices?

Accepted Answer

O(E log V)

Answer

O(V log E)

Answer

O(V^2)

Answer

O(E)

Question 45

In which type of graph is it impossible to guarantee a unique Minimum Spanning Tree (MST)?

Accepted Answer

A graph with negative-weight edges.

Answer

A connected graph.

Answer

A complete graph.

Answer

A tree.

Question 46

What is the primary purpose of using a priority queue in Prim's algorithm for finding a Minimum Spanning Tree (MST)?

Accepted Answer

The priority queue efficiently selects the edge with the smallest weight from the set of candidate edges to add to the growing MST.

Answer

The priority queue detects cycles in the graph to ensure that the MST remains acyclic.

Answer

The priority queue calculates the total weight of the MST as it is constructed.

Answer

The priority queue stores all the vertices in the graph for efficient access.

Question 47

Which of the following is NOT a typical application of Minimum Spanning Trees (MSTs)?

Accepted Answer

Finding the Hamiltonian path in a graph, which visits each vertex exactly once.

Answer

Segmenting images into distinct regions based on similarities.

Answer

Optimizing network design by minimizing connection costs.

Answer

Clustering data points into groups based on proximity.

Question 48

What is the significance of the edges in a Minimum Spanning Tree (MST) forming a forest?

Accepted Answer

The MST is composed of a collection of subtrees that collectively span all the vertices in the graph.

Answer

The MST is guaranteed to be cycle-free due to its forest structure.

Answer

The MST is a single connected component of the original graph, ensuring connectivity between all vertices.

Question 49

Which algorithm is NOT used for finding a Minimum Spanning Tree (MST)?

Accepted Answer

Dijkstra's Algorithm, which is designed to find the shortest path between two specific vertices in a graph.

Answer

Prim's Algorithm.

Answer

Kruskal's Algorithm.

Answer

Borůvka's Algorithm.

Question 50

In a graph with 10 vertices, what is the maximum number of edges that can be present in its Minimum Spanning Tree?

Accepted Answer

9

Answer

45

Answer

19

Answer

10

Question 51

Which of the following data structures is commonly used in Prim's algorithm to efficiently store the edges considered for inclusion in the MST?

Accepted Answer

Min-Priority Queue

Answer

Queue

Answer

Hash Table

Answer

Stack

Question 52

A graph is represented by an adjacency matrix. Which of these methods is more suitable for finding the MST using Kruskal's algorithm?

Accepted Answer

Sorting the edges by weight

Answer

Using breadth-first search

Answer

Using depth-first search

Question 53

Which property of Minimum Spanning Trees ensures that they are acyclic?

Accepted Answer

Cut Property

Answer

Connectivity Property

Answer

Path Property

Answer

Cycle Property

Question 54

Consider a weighted graph where all edge weights are distinct. How many different Minimum Spanning Trees (MSTs) are possible?

Accepted Answer

1

Answer

An infinite number

Answer

2

Answer

Multiple, but the number depends on the graph

Question 55

You are given a graph with negative edge weights. Which of the following statements is TRUE about finding its MST?

Accepted Answer

Standard MST algorithms like Kruskal's and Prim's may not produce a correct MST.

Answer

The MST will always be unique, even with negative weights.

Answer

Negative weights have no impact on MST algorithms.

Question 56

In a real-world application, which of the following scenarios would **MOST** likely benefit from finding a Minimum Spanning Tree?

Accepted Answer

Connecting multiple cities with a network of roads using the least amount of cable

Answer

Finding the shortest path between two points in a maze

Answer

Scheduling tasks in a project management system

Question 57

What is the time complexity of Kruskal's algorithm for finding the MST of a graph with 'V' vertices and 'E' edges, assuming efficient data structures are used?

Accepted Answer

O(E log E)

Answer

O(E + V)

Answer

O(V log V)

Answer

O(V^2)

Question 58

Suppose you have an undirected weighted graph with 'V' vertices and 'E' edges.  What is the maximum number of edges you need to examine when using Prim's algorithm to find the MST?

Accepted Answer

E

Answer

V

Answer

V-1

Answer

E-V