Shortest paths in graphs with negative edge weights
•
Let
G
= (
V, E
) be a directed graph with edge weights
c
uv
for each edge (
u, v
)
∈
E
.
•
Let

V

=
n
and

E

=
m
.
•
Cost
c
(
p
) of a path
p
= (
v
0
, v
1
, . . . , v
k
) is
∑
k
i
=1
c
v
i

1
,v
i
.
•
Shortest path from
s
to
t
is a path with minimum cost among all paths from
s
to
t
.
•
Problem: find a shortest path from
s
to all vertices.
•
Need the following fact:
– FACT 1:
If
G
has no negative cycles then there is a shortest path from
s
to
t
with at most
n

1 edges.
1
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BellmanFord Algorithm
•
Works in the presence of negative weight edges but no negative cycles.
•
Let
opt
(
i, v
) be the minimum cost of a path from
v
to
t
using at most
i
edges.
•
To compute
opt
(
n

1
, s
).
•
Let
P
be an optimal path with cost
opt
(
i, v
).
•
There are two mutually exclusive cases:
–
CASE 1:
P
uses at most
i

1 edges:
*
opt
(
i, v
) =
opt
(
i

1
, v
).
–
CASE 2:
P
uses
i
edges:
*
Let the first edge of
P
be (
v, w
).
*
opt
(
i, v
) =
c
vw
+
opt
(
i

1
, w
).
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 Spring '10
 N/A
 Graph Theory, negative cycle, negative cycles, t. • O

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