1. Management Science 461 Lecture 2b – Shortest Paths September 16, 2008

2. 2 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)

3. 3 Physical Abstraction Euler and the Königsberg Bridges - 1736

4. 4 Shortest Path Problems Find the shortest (distance, time, cost) path between two nodes (e.g. 1  5) 1 2 3 4 5 8 2 5 4 3 5 9 Node = Vertex Arc = Edge = Link Route = Path Tour = Circuit = Cycle

5. 5 Algebraic formulation 1 2 3 4 5 8 2 5 4 3 5 9

6. 6 Flow and Integrality Constraints subject to x 12 + x 13 = 1 (node 1) x 12 + x 32 + x 42 = x 23 + x 24 + x 25 (node 2) x 13 + x 23 + x 43 = x 32 + x 34 (node 3) x 34 + x 24 = x 43 + x 42 + x 45 (node 4) x 25 + x 45 = 1(node 5) x ij = 0 or 1 for all i, j 1 2 3 4 5

7. 7 Rearranging, we get… subject to - x 12 - x 13 = -1 (node 1) x 12 + x 32 + x 42 - x 23 - x 24 - x 25 = 0 (node 2) x 13 + x 23 + x 43 - x 32 - x 34 = 0 (node 3) x 34 + x 24 - x 43 - x 42 - x 45 = 0 (node 4) x 25 + x 45 = 1(node 5) x ij = 0 or 1 for all i, j 1 2 3 4 5 Flow in = flow out  flow in – flow out = 0 Do these -1, 0, 1 values look familiar?

8. 8 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

9. 9 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)

10. 10 Dijkstra’s Algorithm 8 2 5 4 3 5 9 1 2 3 4 5

11. 11 Step 0 – Place Labels 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist:  Pred: 2 Dist:  Pred: 5 Dist:  Pred: 4 Dist:  Pred: 3

12. 12 Step 1 – Choose Minimum Label 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist:  Pred: 2 Dist:  Pred: 5 Dist:  Pred: 4 Dist:  Pred: 3

13. 13 Step 2 – Update Adjacent Labels 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist: 8 Pred: 1 Dist:  Pred: 5 Dist:  Pred: 4 Dist: 2 Pred: 1

14. 14 Step 2b – Mark Node As Optimal 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist: 8 Pred: 1 Dist:  Pred: 5 Dist:  Pred: 4 Dist: 2 Pred: 1

15. 15 Step 3 – Choose Next Min Dist 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist: 8 Pred: 1 Dist:  Pred: 5 Dist:  Pred: 4 Dist: 2 Pred: 1

16. 16 Step 4 – Update Adjacent Labels 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist: 7 Pred: 3 Dist:  Pred: 5 Dist: 6 Pred: 3 Dist: 2 Pred: 1

17. 17 Step 4b – Mark Node As Optimal 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist: 7 Pred: 3 Dist:  Pred: 5 Dist: 6 Pred: 3 Dist: 2 Pred: 1

18. 18 Step 5 – Choose Next Min Dist 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist: 7 Pred: 3 Dist:  Pred: 5 Dist: 6 Pred: 3 Dist: 2 Pred: 1

19. 19 Step 6 – Update Adjacent Labels 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist: 7 Pred: 3 Dist: 15 Pred: 4 Dist: 6 Pred: 3 Dist: 2 Pred: 1

20. 20 Step 6b – Mark Node As Optimal 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist: 7 Pred: 3 Dist: 15 Pred: 5 Dist: 6 Pred: 3 Dist: 2 Pred: 1

21. 21 Step 7 – Choose Next Min Dist 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist: 7 Pred: 3 Dist: 15 Pred: 5 Dist: 6 Pred: 3 Dist: 2 Pred: 1

22. 22 Step 8 – Update Adjacent Labels 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist: 7 Pred: 3 Dist: 12 Pred: 2 Dist: 6 Pred: 3 Dist: 2 Pred: 1

23. 23 Step 8b – Mark Node As Optimal 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist: 7 Pred: 3 Dist: 12 Pred: 2 Dist: 6 Pred: 3 Dist: 2 Pred: 1

24. 24 Step 9 – Choose Next Min Dist 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist: 7 Pred: 3 Dist: 12 Pred: 2 Dist: 6 Pred: 3 Dist: 2 Pred: 1

25. 25 Step 10 – Update Adjacent Labels 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist: 7 Pred: 3 Dist: 12 Pred: 2 Dist: 6 Pred: 3 Dist: 2 Pred: 1

26. 26 Step 10b – Mark Node As Optimal 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist: 7 Pred: 3 Dist: 12 Pred: 2 Dist: 6 Pred: 3 Dist: 2 Pred: 1

27. 27 Final Solution – Min Dist Tree 8 2 5 4 3 5 9 1 2 3 4 5 Dist: 0 Pred: 1 Dist: 7 Pred: 3 Dist: 12 Pred: 2 Dist: 6 Pred: 3 Dist: 2 Pred: 1