https://file.aiquickdraw.com/mathbot/file/17248618322102tdcre5s.jpg

1. Use Dijkstra’s algorithm to find the shortest path spanning tree for the following weighted directed graph with vertices A, B, C, D, and E given. Consider the starting vertex as E.



Let's solve this new graph using Dijkstra's algorithm with **E** as the starting vertex.

### Graph Representation:

Based on the provided image, the directed weighted edges are as follows:

- E → D (weight 3)
- E → A (weight 5)
- D → A (weight 2)
- D → C (weight 6)
- A → B (weight 3)
- A → C (weight 4)
- C → B (weight 2)
- B → D (weight 4)

### Applying Dijkstra's Algorithm:

We will initialize the distances and compute the shortest paths from vertex **E**. The algorithm will iteratively update the tentative distances to the vertices until the shortest paths are found.

Let me calculate the shortest path tree from vertex **E**.### Result of Dijkstra's Algorithm (starting from vertex **E**):

The shortest path tree provides the following distances to each vertex:

- **E**: 0 (starting point)
- **D**: 3 (via E → D)
- **A**: 5 (via E → A)
- **C**: 9 (via E → D → C)
- **B**: 8 (via E → A → B)

### Previous Nodes:
This table shows the previous node for each vertex, which helps in reconstructing the shortest paths:

- **E**: None (starting point)
- **D**: E (previous node is E)
- **A**: E (previous node is E)
- **C**: D (previous node is D)
- **B**: A (previous node is A)

### Shortest Paths:
- E → D (3)
- E → A (5)
- E → D → C (9)
- E → A → B (8)

### Do you have any questions or would you like more details?

Here are 5 follow-up questions for further exploration:
1. How does the shortest path change if you start from vertex **D** instead of **E**?
2. Can you verify the shortest path from **E** to **C** by hand calculation?
3. What modifications to the graph would result in different shortest paths?
4. How does Dijkstra’s algorithm handle graphs with cycles but without negative edges?
5. Could you explain how this algorithm would scale with larger graphs?

**Tip:** When dealing with more complex graphs or larger datasets, using an adjacency list for graph representation (like in Dijkstra's algorithm) optimizes both space and time efficiency.

Learn how to apply Dijkstra’s algorithm to find the shortest path spanning tree in a weighted directed graph, starting from vertex E. This problem involves graph theory concepts and practical application of algorithms for shortest paths.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation