What should you do when several candidate edges have the same minimum weight?
Conditions
- Multiple edges connect the visited set to unvisited nodes.
- These edges have identical minimum weights.
Reasoning, step by step
- Identify all edges crossing from the visited set to unvisited nodes.
- Find the minimum weight among these edges.
- Select any edge that has this minimum weight.
- Add the selected edge and its unvisited endpoint to the MST.
Example
The narrator says, "Three edges all have a weight of three. Pick one of these. I've chosen to add AC."
Common misconceptions
- Believing that a specific tie-breaking rule (like alphabetical order) is mandatory.
- Thinking that ties invalidate the algorithm.
- Assuming that different tie-breaking choices always result in different total weights.
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Related questions
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.
Edge B-E is skipped because both of its endpoints (B and E) are already in the visited set (MST). Prim's algorithm only considers edges that connect a visited vertex to an unvisited vertex to avoid creating cycles.
Conditions: Both endpoints of the edge are already marked as visited.; The algorithm strictly follows the rule of adding only crossing 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.
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