PRELIM 2
ORIE 321/521
March 13, 2008
Closed book exam. Justify all work.
1. In the context of the BellmanFord shortest path algorithm, suppose
G
= (
V, E
) is a digraph with
vertices
V
=
{
1
,
· · ·
, n
}
and arc cost data
c
i,j
,
(
i, j
)
∈
E
, not necessarily nonnegative.
(a) (10 points) Explain why any directed path from node 1 to node
n
with more than
n

2 inter
mediate nodes must contain a cycle.
(b) (10 points) Show that if
G
contains no cycles of negative total length then there is a shortest
path from node 1 to node
n
with
n

2 or fewer intermediate nodes.
2. The following allocation model was considered in the homework: Certain sensitive instruments may
fail during the course of a planned scientific expedition, with the probability that the
i
th instrument
will fail
j
times during the expedition given by
p
i,j
. Each time the
i
th instrument fails, it must be
replaced by a copy shipped to the expedition at cost
c
i
. Also given for each instrument is its weight
w
i
, and we consider the problem of determining the composition of a
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 Spring '10
 TROTTER
 Graph Theory, Recursion, Shortest path problem, recursive equations

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