Shortest Paths
The Algorithms
IE170: Algorithms in Systems Engineering:
Lecture 20
Jeff Linderoth
Department of Industrial and Systems Engineering
Lehigh University
March 19, 2007
Jeff Linderoth
IE170:Lecture 20
Shortest Paths
The Algorithms
Taking Stock
Last Time
Minimum Spanning Trees
Strongly Connected Components
This Time
Shortest Paths
Jeff Linderoth
IE170:Lecture 20
Shortest Paths
The Algorithms
Shortest Path Properties
Shortest Paths—Definitions
For the next few lectures, we will have a
directed
graph
G
= (
V, E
)
, and a weight function
w
:
E
→
R

E

.
The
weight
of a path
P
=
{
v
0
, v
1
, . . . v
k
}
is simply the weight
of the edges taken on the sequence of nodes:
w
(
P
) =
k
i
=1
w
v
i

1
,v
i
.
We are interested in finding the shortestpath weights from
u
to
v
, which we will denote
δ
(
u, v
)
.
We use the convention that
δ
(
u, v
) =
∞
if there is no path
from
u
to
v
in
G
Jeff Linderoth
IE170:Lecture 20
Shortest Paths
The Algorithms
Shortest Path Properties
Example
The example (hopefully) makes it clear that shortest paths are
organized as a tree
Many algorithms work like a generalization of BFS to
weighted graphs.
Jeff Linderoth
IE170:Lecture 20
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Shortest Paths
The Algorithms
Shortest Path Properties
Shortest Path Variants
SingleSource
: Find the shortest path from
s
∈
V
to
every
vertex
v
∈
V
SingleDestination
: Find the shortest path from every vertex
v
∈
V
to a
given
destination vertex
t
∈
V
SinglePair
: Find the shortest path from given
s
∈
V
to given
t
∈
V
. There is now way known that is better (in the worst
case) that solving the singlesource version.
AllPairs
: Find the shortest path from
every
u
∈
V
to
every
vertex
v
∈
V
Jeff Linderoth
IE170:Lecture 20
Shortest Paths
The Algorithms
Shortest Path Properties
Negative Weight Edges
In Minimum Spanning Tree, negative weight edges posed no
significant challenge to the algorithms. However, for shortest
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 Spring '07
 Ralphs
 Graph Theory, Systems Engineering, Shortest path problem, shortest path, Jeff Linderoth

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