Section 12.4
347
9. (a) [BB] Here are the distances
dt
and the
t
0
1
2
3
4
5
6
7
corresponding values of
Pt.
dt
0
5
7
4
5
7
6
6
Pt
1 0 0 0 0
4
4
3
(b) Here are the distances
dt
and the
t
0
1
2
3
4
5
6
7
8
corresponding values of
Pt.
dt
0
1
2
3
4
4
5
5
5
Pt
1 0 0 0 1
3
0
2
1
10. We use induction on t.
If
t
=
1, there is a path to
VI
if and only if
VOVI
is an arc. At the first iteration
of the loop in Step 2, with
t
=
1, the algorithm checks the value of
do
+
w(vo,
VI)
=
w(vo,
VI) and,
if this is finite, assigns this number to d
l
and sets PI
=
0, as we claimed. Now assume
t
> 1 and
the algorithm has worked as claimed for each
j
< t. Consider a path from
Vo
to to
Vt.
The last arc
on this is of the form
VjVt
with
j
<
t
(since the vertices are labeled canonically). Moreover, since
Vj
lies on a path from
Vo,
the induction hypothesis implies that the algorithm has identified the length
dj
of a shortest path to
Vj.
A shortest path to
Vt
has length the least value of
dj
+
w(Vj,Vt)
for all
j =
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 Summer '10
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 Graph Theory, Mathematical Induction, Recursion, Inductive Reasoning, Mathematical logic, Structural induction

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