import Image

Load the image to display the problem visually for analysis

image_path = "/mnt/data/1733215419501sjqidq1p.jpg" image = ImageI have opened the image for analysis. Let’s solve the problem using Dijkstra’s algorithm step by step.

Part (i): Finding the shortest route from $A$ to $J$

Step-by-step application of Dijkstra’s Algorithm:

1. Initialization:

   • Start from node $A$ with a tentative distance of 0. All other nodes are initially set to infinity ($\infty$).
   • Unvisited nodes: $A, B, C, D, E, F, G, H, J$.

2. Iterative process:

   • From the current node, calculate the tentative distances to all directly connected neighbors. Update their distances if the newly calculated distance is shorter.
   • Mark the current node as visited and move to the next unvisited node with the smallest tentative distance.

I will now calculate the shortest path and label it systematically to determine the path. import networkx as nx

Define the graph edges and weights

edges = [ ("A", "B", 5), ("A", "C", 12), ("B", "C", 6), ("B", "F", 2), ("C", "D", 1), ("C", "E", 7), ("D", "F", 2), ("D", "E", 10), ("E", "G", 5), ("E", "H", 5), ("F", "G", 8), ("G", "H", 1), ("G", "J", 9), ("H", "J", 7) ]

Create a graph using NetworkX

G = nx.Graph() G.add_weighted_edges_from(edges)

Use Dijkstra's algorithm to find shortest path from A to J

shortest_path_A_J = nx.dijkstra_path(G, source="A", target="J", weight="weight") shortest_path_length_A_J = nx.dijkstra_path_length(G, source="A", target="J", weight="weight")

Use Dijkstra's algorithm to find shortest path from A to J via E

shortest_path_A_E = nx.dijkstra_path(G, source="A", target="E", weight="weight") shortest_path_E_J = nx.dijkstra_path(G, source="E", target="J", weight="weight") shortest_path_via_E = shortest_path_A_E[:-1] + shortest_path_E_J shortest_path_length_via_E = ( nx.dijkstra_path_length(G, source="A", target="E", weight="weight") + nx.dijkstra_path_length(G, source="E", target="J", weight="weight") )

Display results as a DataFrame for clarity

results = pd.DataFrame({ "Path": [" -> ".join(shortest_path_A_J), " -> ".join(shortest_path_via_E)], "Length (km)": [shortest_path_length_A_J, shortest_path_length_via_E] }, index=["Shortest Path (A to J)", "Shortest Path (A to J via E)"])

import ace_tools as tools; tools.display_dataframe_to_user(name="Shortest Paths and Their Lengths", dataframe=resultsI have calculated the shortest paths and their lengths. Let me know if you need a detailed explanation of the steps or have further questions.

Here are five related questions for deeper exploration:

1. Would you like a detailed step-by-step breakdown of Dijkstra's algorithm for this specific case?
2. How does adding a constraint (like passing through a specific node) impact the shortest path calculations?
3. Can we verify the shortest path visually using the network diagram?
4. How would the results change if one of the road lengths were altered?
5. Could the shortest path algorithm be applied to non-road networks, like data transmission?

Tip: When working with graphs, always visualize the network to better understand how paths are structured and interconnected.