Dijkstra's shortest path algorithm is run on the graph, starting at vertex A. When a vertex is
dequeued, 0 or more adjacent vertices' distances are updated.
For each iteration of the while loop in Dijkstra's algorithm, find the vertex dequeued and the adjacent
vertices updated. Enter updated vertices as A, B, C or "none" if no adjacent vertices are updated.
5
E
C
Iteration
Vertex dequeued
Adjacent vertices updated
1
Ex: C
Ex: A, B, C or none
1
2
6
2
B
A
9
3
3
4
8
5
D

Question

Accepted Answer

∻Explanation∻ ⚹Dijkstra's algorithm is used to find the shortest path from a starting vertex to all other vertices in a weighted graph.⚹ ∻Step-by-step Instruction∻ ‖ ▪ ⋮ Define the main functions required here: return a "Hello World" string. ‖ ‖ ▪ ⋮ Choose to return a string because Hello World contains multiple characters. ‖ ‖ ▪ ⋮ Define concrete use cases and testing cases for the function: when user calls function_name(), it should return "Hello World" ‖ To solve the problem of identifying the vertex dequeued and the adjacent vertices updated for each iteration of Dijkstra's algorithm, follow these steps: 1. **Initialize the Algorithm:** - Start at vertex A. - Set the initial distances to all vertices as infinity, except for the starting vertex A, which is set to 0. - Use a priority queue to keep track of the vertices to be processed. 2. **Process Each Vertex:** - Dequeue the vertex with the smallest distance from the priority queue. - For each adjacent vertex, calculate the potential new distance. - If the new distance is smaller than the current known distance, update the distance and enqueue the adjacent vertex. 3. **Record the Iterations:** - For each iteration, record the vertex dequeued and the adjacent vertices whose distances were updated. Here is a Python implementation of Dijkstra's algorithm with comments explaining each step: ```python import heapq def dijkstra(graph, start): # Initialize distances with infinity distances = {vertex: float('infinity') for vertex in graph} # Distance to the start vertex is 0 distances[start] = 0 # Priority queue to process vertices priority_queue = [(0, start)] # To keep track of the order of dequeued vertices and updated vertices iterations = [] while priority_queue: # Dequeue the vertex with the smallest distance current_distance, current_vertex = heapq.heappop(priority_queue) # Record the dequeued vertex updated_vertices = [] # Process each adjacent vertex for neighbor, weight in graph[current_vertex].items(): distance = current_distance + weight # If a shorter path is found if distance < distances[neighbor]: distances[neighbor] = distance heapq.heappush(priority_queue, (distance, neighbor)) # Record the updated vertex updated_vertices.append(neighbor) # Record the iteration details iterations.append((current_vertex, updated_vertices if updated_vertices else "none")) return iterations # Example graph represented as an adjacency list graph = { 'A': {'B': 9, 'E': 2}, 'B': {'C': 6}, 'C': {'D': 3}, 'D': {'E': 8}, 'E': {'C': 7} } # Run Dijkstra's algorithm starting from vertex 'A' iterations = dijkstra(graph, 'A') for i, (vertex, updated) in enumerate(iterations, 1): print(f"Iteration {i}: Vertex dequeued: {vertex}, Adjacent vertices updated: {updated}") ``` ∻Time Complexity∻ ⚹The time complexity of Dijkstra's algorithm using a priority queue (min-heap) is $O((V + E) \log V)$, where $V$ is the number of vertices and $E$ is the number of edges.⚹ ∻Space Complexity∻ ⚹The space complexity is $O(V + E)$ due to the storage of the graph, distances, and priority queue.⚹

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