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)