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Solve using the Prims algorithm. Show all steps, minimum spanning tree, and final cost. Start from vertex D D 4 B 1 2 10 E 6 с 3 A

Solve using the Prims algorithm. Show all steps, minimum spanning tree, and final cost. Start from vertex D D 4 B 1 2 10 E 6 с 3 A

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**Title: Solving Minimum Spanning Tree Using Prim's Algorithm** **Objective:** Solve using Prim’s algorithm. Show all steps, the minimum spanning tree, and the final cost starting from vertex D. ### Diagram Explanation: The graph depicted includes five vertices labeled A, B, C, D, and E. The edges are associated with the following weights: - B to C: 10 - C to A: 3 - C to E: 6 - D to B: 4 - D to E: 1 - D to C: 2 ### Steps to Solve Using Prim's Algorithm: 1. **Initialization:** - Start with the initial vertex D. - The set of selected edges is initially empty. 2. **Step-by-Step Edge Selection:** - From D, choose the edge with the smallest weight. - Select edge D to E (weight 1). - Move to E. From the available edges (E to C: 6, back to D is ignored), choose the smallest. - Select edge D to C (weight 2). - Move to C. From C, choose the next smallest edge. - Select edge C to A (weight 3). - Finally, from A or D, select the smallest available edge to connect to a new vertex. - Select edge D to B (weight 4). 3. **Resulting Minimum Spanning Tree:** - The selected edges form the minimum spanning tree: D–E, D–C, C–A, D–B. 4. **Final Cost Calculation:** - Sum of weights in the minimum spanning tree: 1 (D-E) + 2 (D-C) + 3 (C-A) + 4 (D-B) = 10. **Conclusion:** The Minimum Spanning Tree (MST) for the given graph starting from vertex D, using Prim’s algorithm, includes edges D–E, D–C, C–A, and D–B with a total cost of 10. This ensures all vertices are connected with the minimum possible total edge weight.
Transcribed Image Text:**Title: Solving Minimum Spanning Tree Using Prim's Algorithm** **Objective:** Solve using Prim’s algorithm. Show all steps, the minimum spanning tree, and the final cost starting from vertex D. ### Diagram Explanation: The graph depicted includes five vertices labeled A, B, C, D, and E. The edges are associated with the following weights: - B to C: 10 - C to A: 3 - C to E: 6 - D to B: 4 - D to E: 1 - D to C: 2 ### Steps to Solve Using Prim's Algorithm: 1. **Initialization:** - Start with the initial vertex D. - The set of selected edges is initially empty. 2. **Step-by-Step Edge Selection:** - From D, choose the edge with the smallest weight. - Select edge D to E (weight 1). - Move to E. From the available edges (E to C: 6, back to D is ignored), choose the smallest. - Select edge D to C (weight 2). - Move to C. From C, choose the next smallest edge. - Select edge C to A (weight 3). - Finally, from A or D, select the smallest available edge to connect to a new vertex. - Select edge D to B (weight 4). 3. **Resulting Minimum Spanning Tree:** - The selected edges form the minimum spanning tree: D–E, D–C, C–A, D–B. 4. **Final Cost Calculation:** - Sum of weights in the minimum spanning tree: 1 (D-E) + 2 (D-C) + 3 (C-A) + 4 (D-B) = 10. **Conclusion:** The Minimum Spanning Tree (MST) for the given graph starting from vertex D, using Prim’s algorithm, includes edges D–E, D–C, C–A, and D–B with a total cost of 10. This ensures all vertices are connected with the minimum possible total edge weight.
Expert Solution
Step 1: Introduction:

  1. Spanning: A spanning tree must contain all the vertices of the original graph. In other words, it "spans" the graph.

  2. Tree Structure: It is a tree, meaning it is acyclic (no closed loops or cycles) and connected (there is a path between any two vertices).

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