Homework 9

Background: Ch. 23.1, 23.4, 23.5, 27.2, 27.3

Due November 13th:

• 9.1: CLR Exercise 23.1-3 (pg. 468).
• 9.2: Extra credit. CLR Exercise 23.1-6 (pg. 468).
• 9.3: CLR Exercise 23.4-1 (pg. 487).
• 9.4: CLR Exercise 23.5-1 (pg. 494).
• 9.5: CLR Exercise 23.5-5 (pg. 494).
• 9.6: Extra credit. In class, we came up with an alternate algorithm for strongly connected components. The algorithm went something like this: Start with a graph. Find a cycle in the graph. If there is no cycle, we're done. If there is, mark all nodes in the cycle as being in the same strongly connected component. Now collapse all those nodes into a single node and repeat. Describe an efficient implementation of this procedure and give its running time.
• 9.7: Give a O(|V|+|E|) (linear time) algorithm for maximal (not maximum) bipartite matching.
• 9.8: Give an example bipartite graph with 2k left nodes and 2k right nodes with a maximal matching of size k and a maximum matching of size 2k.
• 9.9: Extra credit. In the previous question, the ratio of the size of the maximum matching to the maximal matching is 2:1. Prove that this is the largest possible ratio. We can conclude from this that a maximal matching is an approximation (to within a factor of 2) of the maximum matching.
• 9.10: Book gives a algorithm for network flow. When applied to bipartite matching, is this better or worse than the O(|V||E|) algorithm we discussed?