Graphs/Spanning Tree and Shortest Paths MCQ Quiz - Objective Question with Answer for Graphs/Spanning Tree and Shortest Paths - Download Free PDF

The correct answer is Greedy algorithm.

Key Points

• Dijkstra's algorithm is a graph search algorithm that solves the single-source shortest path problem for a graph with non-negative edge weights.
• It finds the shortest path from the source node to all other nodes in the graph.
• Dijkstra's algorithm follows the Greedy algorithm paradigm, where it makes a series of choices, each of which looks best at the moment, to find the shortest path.
• It maintains a set of nodes whose shortest distance from the source is already known and repeatedly selects the node with the smallest known distance.

Additional Information

• Branch and bound is a general algorithm for finding optimal solutions of various optimization problems, particularly in discrete and combinatorial optimization. It is not specifically used in Dijkstra's algorithm.
• Backtracking is a general algorithm for finding solutions by trying partial solutions and then abandoning them if they are not suitable. It is used in problems like the N-Queens problem, but not in Dijkstra's algorithm.
• Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is used in algorithms like the Floyd-Warshall algorithm, but Dijkstra's algorithm does not use this paradigm.

The correct answer is 7

Concept:

Minimum spanning tree: Minimum spanning tree is a subset of the edges of a connected edge weighted undirected graph that connects all the vertices together, without any cycles and with minimum possible edge weight.

Explanation:

Graph has 4 vertices with weights 1, 2, 3, 4, 5, and 6.

Use the cut and cycle property of minimum spanning tree.

Here, for a spanning tree, 3 edges are must. Edge weights 1 and 2 must be included because 2 edges can’t form a cycle. So, according to Kruskal algorithm minimum 2 edge weights are in increasing order will be included.

Now, 4 edge weights are remaining (3, 4, 5, 6).

Assign edge weight 3 such as it forms a cycle so that it can’t be included for minimum spanning tree.

Now from remaining edge weights choose the minimum one.

Graph:

Minimum spanning tree:

So, maximum weight of minimum spanning tree = 7

The correct answer is More than one of the above.

Key Points

• Graph traversal algorithms are techniques used to visit all the nodes in a graph systematically.
  • Common graph traversal algorithms include Depth-First Search (DFS) and Breadth-First Search (BFS).
• Greedy, Divide and Conquer, and Dynamic Programming are not graph traversal algorithms.
  • Greedy algorithms make the optimal choice at each step to find the overall optimal way to solve the problem.
  • Divide and Conquer algorithms break the problem into smaller sub-problems, solve each sub-problem recursively, and combine their solutions to solve the original problem.
  • Dynamic Programming is used to solve problems by breaking them down into simpler sub-problems and storing the results of these sub-problems to avoid redundant work.

Additional Information

• Depth-First Search (DFS) and Breadth-First Search (BFS) are the primary graph traversal algorithms used in various applications, including pathfinding and topological sorting.
• While Greedy, Divide and Conquer, and Dynamic Programming are important algorithm design paradigms, they are not specifically used for graph traversal.
• Understanding the appropriate use cases for each algorithmic approach is crucial for solving complex computational problems efficiently.

The correct answer is A - II, B - I, C - IV, D - III

Key Points

• Dijkstra's Algorithm:
  • It is used to find the shortest path in a weighted graph with non-negative edge weights. Thus, A matches with II.
• Floyd-Warshall Algorithm:
  • This algorithm finds the shortest path between all pairs of vertices in a graph with positive or negative edge weights. Thus, B matches with I.
• Bellman-Ford Algorithm:
  • This algorithm is used to determine the strongest connected components in a directed graph. Thus, C matches with IV.
• Prim's Algorithm:
  • This algorithm sorts elements by repeatedly moving them post neighboring elements that are smaller. Thus, D matches with III.

Therefore, the correct option is A - II, B - I, C - IV, D - III. So, the correct option is 2).

Key Points

• The number of edges in MST with n nodes is (n-1).
• The weight of MST of a graph is always unique. However, there may be different ways to get this weight (if there are edges with the same weights).
• The weight of MST is the sum of the weights of edges in MST.
• The maximum path length between two vertices is (n-1) for MST with n vertices.
• There exists only one path from one vertex to another in MST.
• Removal of any edge from MST disconnects the graph.
• For a graph having edges with distinct weights, MST is unique.

Option 1: T has n-1 edges 

Option 2: T has n vertices

True, For every minimum spanning tree has the n vertices with n-1 edges that are connected.

Option 4: R has n/2 edges

False.

Hence the correct answer is T has n/2 edges.

Answer:3 to 3

Concept:

A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges(V – 1 ) of a connected, edge-weighted undirected graph G(V, E) that connects all the vertices together, without any cycles and with the minimum possible total edge weight.

Explanation:

The minimum weight in the graph is 0.1 choosing this we get.

Suppose we get two trees T₁ and T_2.

To connect to those two trees we got 3 possible edges of weight 0.9.

Hence we can choose any one of those 3 edges.

No of Minimum weight spanning tree is 3.

• Breadth-first search (BFS) and Depth First Search (DFS) is an algorithm for traversing or searching tree or graph data structures.
• Depth First Search (DFS) uses Stack data structure. DFS uses backtracking technique. Remember backtracking can proceed by Stack.
• Breadth First Search (BFS) algorithm traverses a graph in a breadthwise manner and uses a queue to remember to get the next vertex to start a search, when a dead end occurs in any iteration.

• Breadth-first search (BFS) and Depth First Search (DFS) is an algorithm for traversing or searching tree or graph data structures.
• Breadth First Search (BFS) algorithm traverses a graph in a breadthwise manner and uses a queue to remember to get the next vertex to start a search, when a dead end occurs in any iteration.
• Depth First Search (DFS) uses Stack data structure. DFS uses backtracking technique. Remember backtracking can proceed by Stack.

At a distance four from the root, tree will be like:

In this tree, maximum possible value of n at a distance 4 from the root is 31.

Alternate solution:

A tree with height h has maximum number of nodes = 2^h+1 – 1

Here it is given that t is the nth vertex at a distance 4. We have to find the maximum value of n.

Here height = 4

Maximum value of n = 2⁴⁺¹ – 1 = 31

Prim's algorithm: spanning Tree

Dijkstra's algorithm: single source shortest path

Bellman ford algorithm: single source shortest path

Floyd warshalls algorithm: all pair shortest path

Concept:

A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges(V – 1 ) of a connected, edge-weighted undirected graph G(V, E) that connects all the vertices together, without any cycles and with the minimum possible total edge weight.

Explanation:

Edge set for the given graph = {2, 3, 4, 5, 6, 7, 8}

For 5 vertices, we need 4 edges in MST,

So, edge set for MST = {2, 3, 4, 6}

Minimum spanning tree

Minimum cost 2 + 3 + 4 + 6 = 15

Concept:

A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges(V – 1 ) of a connected, edge-weighted undirected graph G(V, E) that connects all the vertices together, without any cycles and with the minimum possible total edge weight.

Example:

Graph G(V, E)

G(V, V – 1) → minimum spanning tree

If edge weights are distinct then there exist unique MST.

Hence Statement I is correct.

When edge weights are distinct and positive in that case shortest path between any two vertices of the graph is not always unique. Shortest path can be different even if the edge weights are unique.

Example:

Distance between A → C (3) and A → B → C (1 + 2 = 3), both paths has same distance.

Hence statement II is incorrect.

I. If e is the lightest edge of some cycle in G, then every MST of G includes e          (INCORRECT)

Let G = (V, E) is a graph with 4 vertices and 5 edges. 

Here, cycle contains the edge weights (3, 4, 5) but edge weight 3 is not included in the MST.

So, according to the cut-property of MST if is the lightest edge of some cycle in G, then every MST of G may or may not include e.

II. If e is the heaviest edge of some cycle in G, then every MST of G excludes e          (CORRECT)

According to the cycle property of MST, same example as above. Here, cycle contains the edge weights (1, 2, 3)  and 3 is the heaviest so edge weight 3 is excluded in the MST.

If there is heaviest edge in cycle, then we will exclude it from the MST because there are other low-cost edges are available to include in MST (because edge weights are distinct here).

Concept:

This graph doesn’t have any cycle, hence there must exist a topological ordering.

R has an indegree from P and S. Q has indegree from P, R and S. Hence,

1. P and S must appear R in the topological ordering.
2. P, S, R must appear before Q in the topological ordering.
3. Sequence of P and S has no indegree can be interchanged.

Therefore, the valid topological orders are PSRQ and SPRQ.

Explanation:

For topological ordering, the starting vertex must have 0 indegree. Hence, any one from P or S can be chosen.

If we choose P, then we need to choose the next vertex with indegree only from P. Hence, S is chosen.

After P and S, we choose a vertex with indegree only from P and S. Hence. R then Q.

Same could be performed with S instead of P.

Concept:

A spanning tree is a subset of Graph G, which has all the vertices covered with the minimum possible number of edges. Spanning tree doesn’t have cycles and it cannot be disconnected.

Formula:

Number of spanning trees possible with n nodes = nn-2

Calculation:

Here, the number of vertices is 7.