Discrete Mathematics with Graph Theory (3rd Edition) 341

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

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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
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