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)
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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