Differences
This shows you the differences between two versions of the page.
| Both sides previous revisionPrevious revisionNext revision | Previous revisionNext revisionBoth sides next revision | ||
| courses:cs211:winter2012:journals:paul:home [2012/04/04 17:35] – [Chapter 7] nguyenp | courses:cs211:winter2012:journals:paul:home [2012/04/06 01:54] – [Chapter 7] nguyenp | ||
|---|---|---|---|
| Line 756: | Line 756: | ||
| * Section 7.2 - Maximum Flows and Minimmum Cuts in a Network | * Section 7.2 - Maximum Flows and Minimmum Cuts in a Network | ||
| * Cut: s-t cut is a partition (A,B) where s is in A and t is in B | * Cut: s-t cut is a partition (A,B) where s is in A and t is in B | ||
| + | * Problem: Find an s-t cut of minimum capacity | ||
| + | * The capacity of a cut is the sum of all the possible stuff that can go through the edge at once | ||
| + | * Flow Value Lemma: Let f be any flow, and let (A, B) be any s-t cut. Then, the value of the flow is = f_out(A) – f_in(A). | ||
| + | * Weak Duality: let f be any flow and let (A, B) be any s-t cut. Then the value of the flow is at most the cut’s capacity. | ||
| + | * Corollary: Let f be any flow, and let (A, B) be any cut. If v(f) = cap(A, B), then f is a max flow and (A, B) is a min cut | ||
| + | * Augmenting path theorem: Flow f is a max flow iff there are no augmenting paths | ||
| + | * Max-flow min-cut theorem: The value of the max flow is equal to the value of the min cut. | ||
| + | * More explanation on page 350 | ||
| + | * Boom and we are done. Just use the same stuff as in the past section and we have all we need. | ||
| + | * Section 7.5 - A First Application: | ||
| + | * Given an undirected bipartite graph | ||
| + | * We must match so each node appears at most once on each edge | ||
| + | * How do we find one with the max number of nodes? This is our problem. | ||
| + | * Here's how we do it. We got a bunch of red and blue nodes. Add nodes s and t, which will be the start and end nodes. Have the start node s have a unit edge (edge of size 1) connect it to all the red nodes. Have all the blue nodes have a unit edge connect them to the end node t. Still have same edges in the middle. | ||
| + | * Now, use same algorithm as in the first section. BOOOM!!! Right? Yep. It's pretty intuitive, these later sections are more like practical uses rather than actual algorithms. I guess this is kind of good since pretty much all computer science problems go back to graph theory. | ||
| + | * Extensions: The structore of bipartite graphs with no perfect matching | ||
| + | * Dicusses how to tell if it is impossible to find a perfect matching in a bipartite graph. | ||
| + | * Can easily find an algorithm based on the following theorem: | ||
| + | * Assume that the bipartite graph G=(V,E) has two sides X and Y such that |X| and |Y|. Then the graph G either has a perfect matching or these is a subset A such that |\G(A)|< | ||
| + | * Section 7.7 - Extensions to Max-Flow Problem | ||
| * | * | ||
