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Good morning, everyone. Glad you are all here bright and early. I'm counting the days till the TA's outnumber the students. They'll show up. We return to a familiar story. This is part two, the Empire Strikes Back. So last time, our adversary, the graph, came to us with a problem. We have a source, and we had a directed graph, and we had weights on the edges, and they were all nonnegative. And there was happiness. And we triumphed over the Empire by designing Dijkstra's algorithm, and very efficiently finding single source shortest paths, shortest path weight from s to every other vertex. Today, however, the Death Star has a new trick up its sleeve, and we have negative weights, potentially. And we're going to have to somehow deal with, in particular, negative weight cycles. And we saw that when we have a negative weight cycle, we can just keep going around, and around, and around, and go back in time farther, and farther, and farther. And we can get t...

hi everybody um glad to have you all back for our next to last installment of theory of computation today um we are going to embark on the very last big topic for the semester and that is in some ways going to be following on what we started a couple of lectures back when we looked at probabilistic turing machines and probabilistic computation and its associated class bpp now what we're going to discuss is in some sense a probabilistic version of np and that's going to be a complexity class called ip which stands for interactive proof systems um and so we're going to present that model and look at a couple of examples uh i would just like to say at the beginning that the this model is a very important one it's really uh has been the starting point for a great deal of research in complexity theory so we're just really going to be touching on it but there's a lot more that people have pursued with this model and it's also a connection in to the c...