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Prim's algorithm is a greedy method used to construct a minimum cost spanning tree in a weighted undirected graph. It builds the spanning tree by continuously selecting the edge with the smallest weight connecting the current tree to a vertex not yet included in the tree. The algorithm's correctness is established through the minimum separator lemma, ensuring that the smallest edge connecting two distinct subsets of the graph is always included in the minimum spanning tree.
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Sections
References
ch30 part a.pdfClass Notes
Memorization
What we have learnt
- Prim's algorithm constructs...
- The correctness of the algo...
- Prim's algorithm can be ini...
Final Test
Revision Tests
What we have learnt
- Prim's algorithm constructs a minimum spanning tree by adding the minimum weight edge at each step.
- The correctness of the algorithm is guaranteed by the minimum separator lemma, which states that the smallest edge connecting two subsets must be in every minimum spanning tree.
- Prim's algorithm can be initialized from any vertex to systematically build the minimum spanning tree.
Key Concepts
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Term: Minimum Spanning Tree
Definition: A spanning tree of a weighted graph that has the minimum sum of edge weights.
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Term: Greedy Algorithm
Definition: An algorithm that makes a series of choices, each of which looks best at the moment, aiming for a locally optimal solution hoping it leads to a globally optimal solution.
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Term: Minimum Separator Lemma
Definition: A theoretical property stating that, in a connected graph, the smallest edge that connects two distinct partitions of the vertex set must be included in every minimum spanning tree.