Discrete Mathematics with Graph Theory (3rd Edition) 341

Discrete Mathematics with Graph Theory (3rd Edition) 341 -...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Section 12.3 339 Compo Size Vertices Compo Size Vertices 1 3 A,J,H 1 3 A,J,H aMHJ 2 1 B aMKM 2 1 B --+ 3 1 C --+ 3 1 C 7 5 G,M,F,D,E 7 7 G,M,F,D,E,K,L 9 1 I 9 1 I 11 2 K,L Compo Size Vertices Compo Size Vertices addCK 1 3 A,J,H aMBC 1 3 A,J,H --+ 2 1 B --+ 7 9 G, M, F, D, E, K, L, C, B 7 8 G, M, F, D, E, K, L, C 9 1 I 9 1 I Compo Size Vertices addJL --+ 7 12 G, M, F, D, E, K, L, C, B, A, J, H 9 1 I Compo Size Vertices add fA --+ 7 13 G, M, F, D, E, K, L, C, B, A, J, H, I (b) [BB] Each time a vertex is relabeled, the component to which it belongs contains at least twice as many vertices as before. Thus, if the label on a vertex changes t times, 2t ~ n; so t ~ log2 n as required. (c) [BB] Since the initial sorting of edges requires 0 (N log N) comparisons, we have only to show that the relabeling process described in (a) and (b) can also be accomplished within this bound. By (b), the total number of vertex relabelings is O(n log n). Ifn ~ N, we are done. Ifn = N + 1, then n log n ~ (2N) log N 2 = 4N log N and again we are done, except when
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern