Is the minimum spanning tree always unique?
Conditions
- The graph is weighted.
- Uniqueness is guaranteed only when all edge weights are pairwise distinct.
Reasoning, step by step
- Check if all edge weights in the graph are different.
- If yes, the MST is unique.
- If no (there are ties), there may be multiple valid MSTs.
- Verify that any chosen MST has the same minimum total weight.
Example
The narrator states, "Note that if edge weights are distinct, this minimum spanning tree will be unique."
Common misconceptions
- Assuming that every graph has exactly one MST.
- Believing that different MSTs can have different total weights.
- Thinking that tie-breaking rules determine a unique MST regardless of weight distribution.
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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.
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.
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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