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PROFESSOR: Since our shortest paths tree algorithms like Dijkstras are not going to work for minimum spanning trees, we need something else. So what Im going to use is something called the cut property. Its a really beautiful, simple idea that is a little surprising that it works, as well see in a moment. But it will allow us to compute minimum spanning trees in a bunch of different ways all resting on the foundation that is known as the cut property. So the basic idea here, Im going to first introduce a term which is a cut, which is an assignment of a graphs nodes to one of two sets. And as an example here, we have a graph. And Ive decided some vertices are gray and some are white and just totally arbitrary which ones I pick. Second term I want to define is a crossing edge. So a crossing edge is any edge which connects vertices from the gray side to the white side. Now, heres the surprising fact that the cut property tells us. If you make up any cut-- anything you want at all
