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Shortest Paths

Description: This quiz covers the concept of shortest paths in graph theory, including Dijkstra's algorithm, Floyd-Warshall algorithm, and Bellman-Ford algorithm.
Number of Questions: 15
Created by:
Tags: graph theory shortest paths dijkstra's algorithm floyd-warshall algorithm bellman-ford algorithm
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What is the primary goal of shortest path algorithms?

  1. Finding the shortest path between two nodes in a graph

  2. Determining the longest path between two nodes in a graph

  3. Calculating the total weight of all paths in a graph

  4. Identifying cycles in a graph

Show answer
Correct Option: A
Explanation:

Shortest path algorithms aim to find the path with the minimum total weight or distance between two specified nodes in a graph.

Which algorithm is commonly used for finding the shortest path between a single source node and all other nodes in a graph?

  1. Dijkstra's Algorithm

  2. Floyd-Warshall Algorithm

  3. Bellman-Ford Algorithm

  4. Prim's Algorithm

Show answer
Correct Option: A
Explanation:

Dijkstra's Algorithm efficiently finds the shortest paths from a single source node to all other nodes in a weighted graph.

What is the time complexity of Dijkstra's Algorithm for finding the shortest paths in a graph with V vertices and E edges?

  1. O(V^2)

  2. O(E log V)

  3. O(V log V)

  4. O(E^2)

Show answer
Correct Option: B
Explanation:

Dijkstra's Algorithm has a time complexity of O(E log V), where V is the number of vertices and E is the number of edges in the graph.

Which algorithm is suitable for finding the shortest paths between all pairs of nodes in a graph?

  1. Dijkstra's Algorithm

  2. Floyd-Warshall Algorithm

  3. Bellman-Ford Algorithm

  4. Kruskal's Algorithm

Show answer
Correct Option: B
Explanation:

Floyd-Warshall Algorithm efficiently computes the shortest paths between all pairs of nodes in a weighted graph.

What is the time complexity of the Floyd-Warshall Algorithm for finding the shortest paths between all pairs of nodes in a graph with V vertices?

  1. O(V^2)

  2. O(V^3)

  3. O(V log V)

  4. O(E log V)

Show answer
Correct Option: B
Explanation:

Floyd-Warshall Algorithm has a time complexity of O(V^3), where V is the number of vertices in the graph.

Which algorithm can handle negative edge weights in a graph while finding the shortest paths?

  1. Dijkstra's Algorithm

  2. Floyd-Warshall Algorithm

  3. Bellman-Ford Algorithm

  4. Prim's Algorithm

Show answer
Correct Option: C
Explanation:

Bellman-Ford Algorithm is capable of finding the shortest paths in graphs with negative edge weights, even if there are negative cycles.

What is the time complexity of the Bellman-Ford Algorithm for finding the shortest paths in a graph with V vertices and E edges?

  1. O(V^2)

  2. O(E log V)

  3. O(V log V)

  4. O(V^3)

Show answer
Correct Option:
Explanation:

Bellman-Ford Algorithm has a time complexity of O(V * E), where V is the number of vertices and E is the number of edges in the graph.

In Dijkstra's Algorithm, which data structure is typically used to efficiently maintain the set of unvisited nodes?

  1. Queue

  2. Stack

  3. Binary Search Tree

  4. Hash Table

Show answer
Correct Option: A
Explanation:

Dijkstra's Algorithm commonly uses a queue (priority queue) to efficiently manage the set of unvisited nodes, prioritizing nodes based on their distance from the source node.

In Floyd-Warshall Algorithm, what is the significance of the intermediate nodes in the dynamic programming approach?

  1. They represent the shortest paths between all pairs of nodes

  2. They are used to compute the shortest paths between all pairs of nodes

  3. They are ignored in the computation of shortest paths

  4. They are used to store the distances between all pairs of nodes

Show answer
Correct Option: B
Explanation:

In Floyd-Warshall Algorithm, intermediate nodes are used to compute the shortest paths between all pairs of nodes by considering all possible paths that pass through those intermediate nodes.

In Bellman-Ford Algorithm, what is the purpose of the relaxation operation?

  1. To update the distance estimates of nodes

  2. To identify negative cycles in the graph

  3. To terminate the algorithm when the shortest paths are found

  4. To check if there are any unvisited nodes

Show answer
Correct Option: A
Explanation:

The relaxation operation in Bellman-Ford Algorithm updates the distance estimates of nodes by considering all possible paths from the source node to each node, ensuring that the shortest paths are found.

Which of the following is a necessary condition for the existence of a negative cycle in a graph?

  1. The graph must have at least one negative edge

  2. The graph must be directed

  3. The graph must be connected

  4. The graph must have an odd number of vertices

Show answer
Correct Option: A
Explanation:

The presence of at least one negative edge in a graph is a necessary condition for the existence of a negative cycle.

In the context of shortest paths, what is the significance of a negative cycle?

  1. It indicates that there is no shortest path between two nodes

  2. It implies that the shortest path between two nodes is not unique

  3. It means that the shortest path between two nodes has a negative weight

  4. It suggests that the graph contains a loop with a negative total weight

Show answer
Correct Option: D
Explanation:

The presence of a negative cycle in a graph indicates that there is a loop with a negative total weight, which can lead to infinitely decreasing path weights.

Which of the following algorithms can detect negative cycles in a graph?

  1. Dijkstra's Algorithm

  2. Floyd-Warshall Algorithm

  3. Bellman-Ford Algorithm

  4. Prim's Algorithm

Show answer
Correct Option: C
Explanation:

Bellman-Ford Algorithm is capable of detecting negative cycles in a graph during its execution.

In the context of shortest paths, what is the significance of a directed acyclic graph (DAG)?

  1. DAGs always have unique shortest paths between any two nodes

  2. DAGs never contain negative cycles

  3. DAGs can be efficiently traversed using depth-first search

  4. DAGs have the same shortest path properties as undirected graphs

Show answer
Correct Option: A
Explanation:

In a directed acyclic graph (DAG), there is always a unique shortest path between any two nodes, and these paths can be efficiently found using topological sorting.

Which of the following is a disadvantage of using Dijkstra's Algorithm for finding shortest paths?

  1. It can handle negative edge weights

  2. It is not suitable for finding shortest paths between all pairs of nodes

  3. It is more efficient than Floyd-Warshall Algorithm

  4. It is not guaranteed to find the shortest path if there are negative cycles

Show answer
Correct Option: B
Explanation:

Dijkstra's Algorithm is not suitable for finding shortest paths between all pairs of nodes in a graph, as it only finds the shortest paths from a single source node to all other nodes.

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