Why is edge B-E skipped even though it has weight 3?
Conditions
- Both endpoints of the edge are already marked as visited.
- The algorithm strictly follows the rule of adding only crossing edges.
Reasoning, step by step
- Check the status of both endpoints of edge B-E.
- Observe that both B and E are in the 'visited' list.
- Conclude that adding B-E would create a cycle within the current tree.
- Discard B-E and look for the next smallest edge connecting to an unvisited node.
Example
The narrator explains, "Notice at this point, the edge BE with a weight of 3 is the smallest edge. But both vertices are already in the MST, so we do not pick it."
Common misconceptions
- Thinking that the smallest available edge is always chosen, regardless of endpoint status.
- Believing that skipping internal edges is optional.
- Confusing Prim's algorithm with Kruskal's, which sorts all edges globally.
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Related questions
When multiple candidate edges share the same minimum weight, you can pick any one of them. The choice does not affect the validity of the greedy step, though it may lead to a different MST if edge weights are not distinct.
Conditions: Multiple edges connect the visited set to unvisited nodes.; These edges have identical minimum weights.
The time complexity of Prim's algorithm depends on the data structures used. Using an adjacency matrix with searching gives , while using a binary heap with an adjacency list gives , where is the number of vertices and is the number of edges.
Conditions: The graph is represented using specific data structures.; denotes the number of vertices.; denotes the number of edges.
No, the minimum spanning tree is not always unique. It is unique only if all edge weights in the graph are distinct.
Conditions: The graph is weighted.; Uniqueness is guaranteed only when all edge weights are pairwise distinct.
To start Prim's algorithm, create an empty list called 'visited' to track touched nodes, then pick an arbitrary starting node and add it to the visited list.
Conditions: The graph is weighted and connected.; You have a mechanism to track visited vertices (e.g., a list or set).
To compute the total weight of the MST, sum the weights of all edges that were selected and added to the tree during the execution of Prim's algorithm.
Conditions: The MST edge set has been fully determined.; You have access to the weights of the selected edges.
Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.