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# 25.4 Chapter Summary

In this chapter, we learned about Minimum Spanning Trees and the Cut Property:
• MST: the lightest set of edges in a graph possible such that all the vertices are connected and acyclic.
• The Cut Property: given any cut, the minimum weight crossing edge is in the MST.
• Cut: an assignment of a graph’s nodes to two non-empty sets
• Crossing Edge: an edge which connects a node from one set to a node from the other set.
We also learned about how to find MSTs of a graph with two algorithms:
• Prim's Algorithm: Construct MST through a mechanism similar to Dijkstra's Algorithm, with the only difference of inserting vertices into the fringe not based on distance to goal vertex but distance to the MST under construction.
• Runtime:
$O((|V| + |E| )log(|V|))$
• Kruskal's Algorithm: Construct MST by first sorting edges from lightest to heaviest, then add edges sequentially if no cycles are formed until there are V - 1 edges.
• Runtime:
• $O(|E| log |E|)$
(unsorted edges)
• $O(|E| log* |V|)$
(sorted edges)