Unexpected Behavior in Prim's Algorithm Implementation - Minimum Spanning Tree Not Returning Correct Edges
After trying multiple solutions online, I still can't figure this out. I've been trying to implement Prim's Algorithm in Python to find the minimum spanning tree of a weighted undirected graph using the `networkx` library, but I keep getting unexpected results. My implementation seems to return edges that do not form a valid minimum spanning tree. Here's the code snippet I'm using: ```python import networkx as nx # Create a weighted undirected graph G = nx.Graph() G.add_weighted_edges_from([(1, 2, 3), (1, 3, 1), (2, 3, 1), (2, 4, 6), (3, 4, 5)]) # Prim's algorithm to find the minimum spanning tree mst = nx.minimum_spanning_tree(G) # Print the edges of the minimum spanning tree print(mst.edges(data=True)) ``` When I run this code, the output is `[(1, 3, {'weight': 1}), (1, 2, {'weight': 3}), (2, 4, {'weight': 6})]`, which includes the edge `(2, 4)` that I believe should not be part of the minimum spanning tree since it has a higher weight compared to the other edges. I thought that Prim's algorithm should always select the lowest weight edge connecting the tree to a new vertex, but it seems to be adding edges that violate this condition. I've also tried manually implementing the algorithm without using `networkx` to see if the issue lies within that library. The manual implementation is as follows: ```python import heapq def prim_algorithm(graph, start): mst_edges = [] visited = set([start]) edges = [(weight, start, to) for to, weight in graph[start].items()] heapq.heapify(edges) while edges: weight, frm, to = heapq.heappop(edges) if to not in visited: visited.add(to) mst_edges.append((frm, to, weight)) for to_next, weight in graph[to].items(): if to_next not in visited: heapq.heappush(edges, (weight, to, to_next)) return mst_edges # Representation of the graph as an adjacency list graph = {1: {2: 3, 3: 1}, 2: {1: 3, 3: 1, 4: 6}, 3: {1: 1, 2: 1, 4: 5}, 4: {2: 6, 3: 5}} result = prim_algorithm(graph, 1) print(result) ``` However, I still end up with edges that include the higher weight ones. What am I missing in terms of the implementation? Is there a specific edge case I should consider? Any help would be greatly appreciated! Is there a simpler solution I'm overlooking?