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Section 12.5
351
7
8
9
11.
(a)
[BB]
ti
(b)
7
11
5
3
1
2
(c)
(d)
14
13
~:
2
5
9
6W'-'<]
3
4
10
4
3
~
12. (a) The argument that we used to answer Exercise 3 works here as well. Use the strong form of
mathematical induction on
k,
a vertex label, the result being clearly true for
k
=
1. Suppose
k
> 1 and the result is true for all
i
in the interval 1
~
i
<
k;
thus, there is a path from vertex
1 to the vertex labeled
i
which uses only edges required by the algorithm. We must prove the
result for
k:
We must prove that there is a path from 1 to the vertex with label
k
which uses
only edges required by the algorithm. Vertex
k
was so labeled because the algorithm found this
vertex unlabeled but adjacent to a previously labeled vertex
i.
At the time a vertex is labeled, the
algorithm chooses the smallest unused label, so
i
<
k.
By the induction hypothesis, there is a
path from 1 to
i
using edges required by the algorithm; also, the algorithm uses the edge
ik
to
label vertex
k.
Thus, there is a path from 1 to

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