Section 12.3
335
connected to
A
via two paths of edges and, as we have noted, Dijkstra's algorithm provides just one
path. Also note that if a vertex
X
is encountered on the shortest path from
A
to
B,
as given by the
algorithm, then the shortest path from
A
to
X
given by the algorithm agrees with the first part of the
shortest path from
A
to
B.
Exercises
12.3
1. (a) [BB] We want five edges (since there are six vertices). Choose
BC,
then
AD, FE,
and
DE.
We
would like next to choose
AE,
but this would complete a circuit with
AD
and
DE,
so we choose
AC
and obtain the spanning tree shown, of weight 13.
(c)
(a)
Bl0[)A:
(b)
~3~2A
35
2~
~
A
2
~3
S
BET
C
E
C
E
1
2
2
6
3
C
F
D
D
(b) As in part (a), we need five edges. Choose
AD,
then
BC,
then
EF,
then
AE.
Now
DE
is no
good, so
AC
will finish the job. The weight of a minimum spanning tree is 122 as shown.
(c) We want seven edges (since there are eight vertices). First choose
SC,
then
CE,
then
AD,
then
BC
and
DT.
Now
BE
cannot be chosen since it would complete a circuit with previously chosen
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 Summer '10
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 Graph Theory, Shortest path problem, edges, Dijkstra

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