https://r2.erweima.ai/mathbot/image/1727207602775im947kqm.jpg

The image outlines a problem that applies **Prim's Algorithm** to find the minimum cost of wiring between houses A, B, C, and D. Here’s a breakdown of the problem:

1. **Description**: 
   - A delivery person needs to wire multiple houses in a neighborhood with the **minimum possible wiring cost**.
   
2. **Vertices and Edges**: The houses (A, B, C, D) represent the vertices of a graph, and the wiring costs between them represent the edges.
   - A-D: 1
   - A-C: 4
   - A-B: 1
   - B-D: 5
   - B-C: 2
   - C-D: 1

3. **Graph Visualization**: 
   - A is connected to B, C, and D with weights 1, 4, and 1, respectively.
   - B is connected to D and C with weights 5 and 2.
   - C is connected to D with a weight of 1.
   
4. **Prim’s Algorithm** (Minimum Spanning Tree - MST):
   - Prim’s algorithm starts from a node and grows the MST by adding the smallest edge that connects a new vertex to the growing tree.
   
   - **Steps**:
     1. Start from A.
     2. Add the smallest edge: A-B (cost 1).
     3. Add the next smallest edge: A-D (cost 1).
     4. Add the next smallest edge: C-D (cost 1).

5. **Total Cost of Wiring**: 
   - From the calculations, the total cost is **4**. This is confirmed in the image with the route notation `(1 + 1 + 1 = 4)`.

### Do you need further explanation on Prim's Algorithm, or how the minimum cost is calculated?

Here are 5 related questions to expand on this problem:
1. How does Prim’s Algorithm differ from Kruskal’s Algorithm?
2. Can Prim’s Algorithm be used with directed graphs?
3. What is the time complexity of Prim’s Algorithm?
4. Why is the edge with the lowest weight chosen at each step?
5. Can this approach be applied to a fully connected graph?

**Tip**: When using Prim's Algorithm, always ensure that you keep track of visited nodes and only add edges that connect to unvisited nodes to avoid cycles.

A delivery person needs to wire multiple houses (A, B, C, D) in a neighborhood with the minimum possible wiring cost using Prim's Algorithm.

This solution explains how to find the minimum cost to wire four houses (A, B, C, D) using Prim's Algorithm. The process focuses on selecting the smallest edge to form a Minimum Spanning Tree. The total wiring cost is calculated to be 4 units.

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