This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Solutions to Midterm 1 Peng Du University of California, San Diego 1 (10 points) 1.1 Part (a) The DFS forest is showed in Fig. 1 where pre and post numbers are written in the form [pre,post] beside each node. a b c f d e h g i [1,2] [3,6] [9,18] [7,8] [4,5] [14,15] [12,13] [11,16] [10,17] Fig.1. The DFS forest. 1.2 Part (b) The shortest path tree is showed in Fig. 2 where the distances are written next to each node. 2 (10 points) 2.1 Part (a) Polynomial time. We only need to output all nodes with zero in-degrees, which takes O ( | V | + | E | ) time. 2.2 Part (b) Exponential time. We have proved in homework 2-4 that graphs can have expo- nentially many cycles. It takes the computer at least exponential time to output all of them. 2 a b c e d f h g i 6 7 8 5 8 1 3 7 Fig.2. The shortest path tree. 2.3 Part (c) Exponential time. There are n ! many topological orderings for a graph with n nodes and no edges. Since n ! > 2 n when n is large enough, we need at least exponential time to output all of them.exponential time to output all of them....
View Full Document
This note was uploaded on 01/21/2010 for the course CSE 661930 taught by Professor Freund,yoav during the Fall '09 term at UCSD.
- Fall '09