Management
Science 461
Lecture 2b – Shortest Paths
September 16, 2008
Graph Theory
Branch of mathematics
 “Graph” has a very specific definition:

 Collection
of objects
 Collection of relationships between objects

Both collections can have associated data
(costs, weights)
2
Physical Abstraction

Euler and the Königsberg Bridges - 1736
3
Shortest Path Problems

Find the shortest (distance, time, cost)
path between two nodes (e.g. 15)
2
5
8
5
5
1
Arc = Edge = Link
3
2
Route = Path
9
3
4
Node = Vertex
Tour = Circuit = Cycle
4
4
Algebraic formulation
1 if the edge connecting node i to j is selected
xij  
0 otherwise
2
5
8
5
5
1
3
2
9
3
4
minimize
8 x12  2 x13  5 x32  5 x23
 3x24  3x42  4 x34
 4 x43  5 x25  9 x45
4
5
Flow and Integrality Constraints
subject to
x12 + x13 = 1
x12 + x32 + x42 = x23 + x24 + x25
x13 + x23 + x43 = x32 + x34
x34 + x24 = x43 + x42 + x45
x25 + x45 = 1
2
xij = 0 or 1 for all i, j
(node 1)
(node 2)
(node 3)
(node 4)
(node 5)
5
1
3
4
6
Rearranging, we get…
Flow in = flow out
subject to

flow in – flow out = 0
- x12 - x13 = -1
x12 + x32 + x42 - x23 - x24 - x25 = 0
x13 + x23 + x43 - x32 - x34 = 0
x34 + x24 - x43 - x42 - x45 = 0
x25 + x45 = 1
2
xij = 0 or 1 for all i, j
(node 1)
(node 2)
(node 3)
(node 4)
(node 5)
5
Do these -1, 0, 1 values look
familiar?
1
3
4
7
Shortest Path Problems

Solve using Excel’s Solver

However, a network with several hundred nodes
would be out of reach, even for Premium Solver
 e.g.

Edmonton
Efficient algorithms need to be used – like
Dijkstra’s algorithm
8
Dijkstra’s algorithm

“Label-setting” algorithm
 At
each node, keep track of minimum
distance from origin to the node, and how we
got there

Very fast – polynomial time algorithm
 Time
to solve increases polynomially with
problem size (not exponential)
9
Dijkstra’s Algorithm
2
5
8
5
5
1
3
9
2
3
4
4
10
Step 0 – Place Labels
2
Dist: 
Pred: 2
5
8
Dist: 0
Pred: 1
Dist: 
Pred: 5
5
5
1
3
9
2
Dist: 
Pred: 3
3
4
4
Dist: 
Pred: 4
11
Step 1 – Choose Minimum
Label
2
Dist: 
Pred: 2
5
8
Dist: 0
Pred: 1
Dist: 
Pred: 5
5
5
1
3
9
2
Dist: 
Pred: 3
3
4
4
Dist: 
Pred: 4
12
Step 2 – Update Adjacent
Labels
2
Dist: 8
Pred: 1
5
8
Dist: 0
Pred: 1
Dist: 
Pred: 5
5
5
1
3
9
2
Dist: 2
Pred: 1
3
4
4
Dist: 
Pred: 4
13
Step 2b – Mark Node As
Optimal
2
Dist: 8
Pred: 1
5
8
Dist: 0
Pred: 1
Dist: 
Pred: 5
5
5
1
3
9
2
Dist: 2
Pred: 1
3
4
4
Dist: 
Pred: 4
14
Step 3 – Choose Next Min Dist
2
Dist: 8
Pred: 1
5
8
Dist: 0
Pred: 1
Dist: 
Pred: 5
5
5
1
3
9
2
Dist: 2
Pred: 1
3
4
4
Dist: 
Pred: 4
15
Step 4 – Update Adjacent
Labels
2
Dist: 7
Pred: 3
5
8
Dist: 0
Pred: 1
Dist: 
Pred: 5
5
5
1
3
9
2
Dist: 2
Pred: 1
3
4
4
Dist: 6
Pred: 3
16
Step 4b – Mark Node As
Optimal
2
Dist: 7
Pred: 3
5
8
Dist: 0
Pred: 1
Dist: 
Pred: 5
5
5
1
3
9
2
Dist: 2
Pred: 1
3
4
4
Dist: 6
Pred: 3
17
Step 5 – Choose Next Min Dist
2
Dist: 7
Pred: 3
5
8
Dist: 0
Pred: 1
Dist: 
Pred: 5
5
5
1
3
9
2
Dist: 2
Pred: 1
3
4
4
Dist: 6
Pred: 3
18
Step 6 – Update Adjacent
Labels
2
Dist: 7
Pred: 3
5
8
Dist: 0
Pred: 1
Dist: 15
Pred: 4
5
5
1
3
9
2
Dist: 2
Pred: 1
3
4
4
Dist: 6
Pred: 3
19
Step 6b – Mark Node As
Optimal
2
Dist: 7
Pred: 3
5
8
Dist: 0
Pred: 1
Dist: 15
Pred: 5
5
5
1
3
9
2
Dist: 2
Pred: 1
3
4
4
Dist: 6
Pred: 3
20
Step 7 – Choose Next Min Dist
2
Dist: 7
Pred: 3
5
8
Dist: 0
Pred: 1
Dist: 15
Pred: 5
5
5
1
3
9
2
Dist: 2
Pred: 1
3
4
4
Dist: 6
Pred: 3
21
Step 8 – Update Adjacent
Labels
2
Dist: 7
Pred: 3
5
8
Dist: 0
Pred: 1
Dist: 12
Pred: 2
5
5
1
3
9
2
Dist: 2
Pred: 1
3
4
4
Dist: 6
Pred: 3
22
Step 8b – Mark Node As
Optimal
2
Dist: 7
Pred: 3
5
8
Dist: 0
Pred: 1
Dist: 12
Pred: 2
5
5
1
3
9
2
Dist: 2
Pred: 1
3
4
4
Dist: 6
Pred: 3
23
Step 9 – Choose Next Min Dist
2
Dist: 7
Pred: 3
5
8
Dist: 0
Pred: 1
Dist: 12
Pred: 2
5
5
1
3
9
2
Dist: 2
Pred: 1
3
4
4
Dist: 6
Pred: 3
24
Step 10 – Update Adjacent
Labels
2
Dist: 7
Pred: 3
5
8
Dist: 0
Pred: 1
Dist: 12
Pred: 2
5
5
1
3
9
2
Dist: 2
Pred: 1
3
4
4
Dist: 6
Pred: 3
25
Step 10b – Mark Node As
Optimal
2
Dist: 7
Pred: 3
5
8
Dist: 0
Pred: 1
Dist: 12
Pred: 2
5
5
1
3
9
2
Dist: 2
Pred: 1
3
4
4
Dist: 6
Pred: 3
26
Final Solution – Min Dist Tree
2
Dist: 7
Pred: 3
5
8
Dist: 0
Pred: 1
Dist: 12
Pred: 2
5
5
1
3
9
2
Dist: 2
Pred: 1
3
4
4
Dist: 6
Pred: 3
27