This table represents a distance matrix for five points $A, B, C, D, E$. The task is to find the shortest path from point $A$ to point $E$.

Approach:

We can use Dijkstra's algorithm to find the shortest path from $A$ to $E$. Here's the breakdown:

1. Graph Representation:

   • Each point ($A, B, C, D, E$) is a node.
   • The distances between the points are the edge weights.

2. Steps:

   • Initialize the distances from $A$ to all other points as infinity ($\infty$) except $A$ itself, which is 0.
   • Process each node by selecting the one with the smallest tentative distance.
   • Update the distances to neighboring nodes if a shorter path is found.

Let’s solve this step by step. heapq

Graph representation from the table

graph = { 'A': {'B': 2, 'C': 3}, 'B': {'A': 2, 'C': 5, 'D': 1}, 'C': {'A': 3, 'B': 5, 'D': 2, 'E': 2}, 'D': {'B': 1, 'C': 2, 'E': 4}, 'E': {'C': 2, 'D': 4} }

def dijkstra(graph, start, end): # Priority queue to store (distance, node) queue = [(0, start)] distances = {node: float('inf') for node in graph} distances[start] = 0 visited = set()

while queue:
    # Pop the node with the smallest distance
    current_distance, current_node = heapq.heappop(queue)

    if current_node in visited:
        continue
    visited.add(current_node)

    # Check neighbors
    for neighbor, weight in graph[current_node].items():
        distance = current_distance + weight
        if distance < distances[neighbor]:
            distances[neighbor] = distance
            heapq.heappush(queue, (distance, neighbor))

return distances[end]

Find the shortest path from A to E

shortest_path_length = dijkstra(graph, 'A', 'E') shortest_path_lengthThe shortest path from point $A$ to point $E$ has a total length of 5.

Let me know if you want:

1. A detailed step-by-step explanation of the algorithm used.
2. The shortest path (not just the length but the sequence of points).
3. A visual representation of the graph.
4. A comparison to other shortest-path algorithms.
5. Help in applying this algorithm to a different dataset.

Tip: Dijkstra's algorithm is efficient for non-negative weights, but for graphs with negative weights, consider using the Bellman-Ford algorithm.