Question

I am comparing two algorithms, Prim’s and Kruskal’s.

I understand the basic concept of time complexity and when the two work best (sparse/dense graphs)

I found this on the Internet, but I am struggling to convert it to English.

dense graph:  Prim = O(N2)
              Kruskal = O(N2*log(N))

sparse graph: Prim = O(N2)
              Kruskal = O(N log(N))

It’s a bit of a long shot, but could anyone explain what is going on here?

Answer 1

Prim is O(N^2), where N is the number of vertices.

Kruskal is O(E log E), where E is the number of edges. The “E log E” comes from a good algorithm sorting the edges. You can then process it in linear E time.

In a dense graph, E ~ N^2. So Kruskal would be O( N^2 log N^2 ), which is simply O( N^2 log N ).