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Section 10.4
20. [BB]
It
is indeed necessary to continue.
An
identical set of
d(
i,
j)
for
k
=
r
and
k
=
r
+
1 simply indicates that the shortest path from each
Vi
to each
Vj
passing through
V1,
.•. ,
Vr
has the same length as the shortest
path through
V1,
•.. ,
Vr +
1. In the graph shown to the right, for instance,
the values of
d(
i,
j)
do not change until
k
=
3 since the shortest path
between pairs of vertices cannot be reduced until vertex
V3
is used.
285
21.
It
is sufficient to prove that for each
r
=
1,2,
... ,
n,
after the outer loop has completed the
k
=
r
iteration, the number
d(i,j)
is the length of a shortest path from
Vi
to
Vj
through the intermediate
vertices
V1, V2,
...
,Vr .
We use mathematical induction on
r.
Suppose
r
=
1.
If
there is a path from
Vi
to
Vj
requiring at most the intermediate vertex
Vb
the shortest path is
ViVj
or
ViV1Vj.
With
k
=
1, the
algorithm compares the lengths of these paths and makes
d(
i,
j)
the smaller of these numbers. Thus
the result is true. Assume the result is true for
k
=
r.
Thus, when the algorithm has completed the
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 Summer '10
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 Graph Theory

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