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3. Spanning Trees: Prim’s Algorithm
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.
Sections
This section introduces Prim's algorithm for constructing a minimum cost spanning tree in a weighted graph.
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.
Minimum Spanning Tree
A spanning tree of a weighted graph that has the minimum sum of edge weights.
Greedy Algorithm
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.
Minimum Separator Lemma
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.
Practice Exercises
Total Questions
2
Estimated Time
4 min
Passing Score
70%
Instructions
- Read each question carefully
- You can use hints if you need help
- Complete all questions before submitting
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