What are the two main strategies for minimum cost spanning trees?

Prim's strategy and Kruskal's Algorithm

How does Kruskal's Algorithm avoid creating cycles?

By skipping edges that connect vertices within the same component.

Adding the smallest edge connecting different components.

Kruskal's Algorithm assembles the tree bottom-up, while Prim's Algorithm grows the tree gradually.

They may lead to multiple possible spanning trees with different edge sets.

Kruskal's Algorithm and Prim's Algorithm both find the same minimum cost spanning tree but follow distinct processes, with Kruskal's focusing on sorting edges and connecting components without cycles, while Prim's grows the tree incrementally.
In scenarios where edge weights are repeated, multiple spanning trees with the same minimum cost could emerge due to different edge sets resulting from varying vertex ordering, showcasing the impact of seemingly minor details on the final outcome.

What are the two main strategies for minimum cost spanning trees?
Prim's strategy and Kruskal's Algorithm
How does Kruskal's Algorithm avoid creating cycles?
By skipping edges that connect vertices within the same component.
What is the key principle behind Kruskal's Algorithm?
Adding the smallest edge connecting different components.
How does Kruskal's Algorithm differ from Prim's Algorithm in tree construction?
Kruskal's Algorithm assembles the tree bottom-up, while Prim's Algorithm grows the tree gradually.
How do repeated edge weights impact the construction of spanning trees?
They may lead to multiple possible spanning trees with different edge sets.

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Minimum cost spanning trees are discussed, with two main strategies: Prim's strategy and Kruskal's Algorithm.
Weighted undirected graphs are used, with a weight function for each edge, assuming graph connectivity.
Kruskal's Algorithm starts with disconnected vertices forming components, merging them with the smallest connecting edge.
Example of Kruskal's Algorithm implementation involves sorting edges in ascending order and connecting components.
The algorithm avoids creating cycles by skipping edges that connect vertices within the same component.
Kruskal's Algorithm and Prim's Algorithm find the same minimum cost spanning tree but in different sequences.
Kruskal's Algorithm involves sorting edges by weight, tracking added edges, and connecting components without cycles.
The algorithm ensures the smallest edge connecting different components is added, following the minimum separator lemma.
Kruskal's Algorithm assembles the tree bottom-up, while Prim's Algorithm grows the tree gradually.
Repeated edge weights may lead to multiple possible spanning trees, with different choices affecting the outcome.

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Edge weights repeating can lead to different spanning trees with the same minimum cost but different edge sets, based on vertex ordering.