请用AOPS的视频风格制作这个题目的讲解视频---**Question Stem:**
The one-way routes connecting towns A, M, C, X, Y, and Z are shown in the figure below (not drawn to scale). The distances in kilometers along each route are marked. Traveling along these routes, what is the shortest distance from A to Z in kilometers?
如图所示, A、M、C、X、Y、Z几座小镇之间通过单行道相连。图中标明了每条单行道的长度, 单位为千米。请问沿单行道从 A 行驶到 Z 的最短距离为多少 千米? (图形未按比例绘制)
**Chart/Diagram Description:**
Type: Directed graph (network diagram).
Main Elements:
- Nodes: Six nodes represented by circles labeled A, M, C, X, Y, and Z.
- Edges: Directed lines (arrows) connecting the nodes, representing one-way routes. Each edge is labeled with a numerical weight representing distance in kilometers.
- Edges and Weights:
- A to X: 5
- A to M: 8
- X to M: 2
- X to Y: 10
- M to Y: 6
- M to C: 14
- Y to C: 5
- Y to Z: 17
- C to Z: 10
- M to Z: 25
- Layout: Nodes are arranged generally from left to right, starting with A on the left, X and Y in the upper/middle section, M in the lower/middle section, C to the right of M and Y, and Z on the far right.
**Options:**
(A) 28
(B) 29
(C) 30
(D) 31
(E) 32
视频信息
答案文本
视频字幕
Welcome to this shortest path problem! We have six towns connected by one-way routes with different distances. Our goal is to find the shortest path from town A to town Z. This is a classic application of Dijkstra's algorithm, which systematically explores all possible paths to guarantee we find the optimal solution.
Now let's apply Dijkstra's algorithm step by step. First, we initialize all distances. We set the distance to our starting node A as zero, and all other nodes as infinity. Node A is highlighted in yellow as our current node. From A, we can reach X with distance 5 and M with distance 8. These are highlighted in red as we update our distance table.
Next, we select node X with the smallest unvisited distance of 5. From X, we can update M's distance from 8 to 7, since the path A to X to M gives us 5 plus 2 equals 7. We also set Y's distance to 15. Now M has the smallest unvisited distance of 7, so we process it next. From M, we update Y's distance to 13, set C's distance to 21, and Z's distance to 32.
Now we select Y with distance 13. From Y, we can improve C's distance to 18 and Z's distance to 30. Next, we process C with distance 18. From C to Z costs 10, giving us a total of 28, which is better than 30. Finally, we reach Z with the optimal distance of 28 kilometers. The shortest path is A to X to M to Y to C to Z, highlighted in red.
To summarize our solution: Using Dijkstra's algorithm, we systematically found the shortest path from A to Z. The optimal route is A to X to M to Y to C to Z, with distances of 5, 2, 6, 5, and 10 kilometers respectively. Adding these up: 5 plus 2 plus 6 plus 5 plus 10 equals 28 kilometers. Therefore, the answer is A, 28 kilometers.