1. Shortest Path Problems

2. Introduction
Many problems can be modeled using graphs with weights assigned to their edges:
Airline flight times
Telephone communication costs
Computer networks response times

3. Where’s my motivation? Fastest way to get to school by car
Finding the cheapest flight home
How much wood could a woodchuck chuck if a woodchuck could chuck wood?

4. Optimal driving time UCLA The Red Garter the beach (dude) In N’ Out25
The Red Garter 19 5 9
the beach (dude) 21 16 31
In N’ Out 36
My cardboard box on Sunset

5. Tokyo Subway Map

6. Applications
- Maps (Map Quest, Google Maps) - Routing Systems

7. Dijkstra's algorithm Steps: 1) Choose/given a starting point.
A solution to the single-source shortest path problem in graph theory. 
Works on both directed and undirected graphs.
Steps:
1) Choose/given a starting point.
2) Find the shortest path to the next vertex –
label the vertex.
3) Label the remaining vertices from the shortest
vertex.
4) Systematically check each vertex to be sure
that they are labelled correctly.

8. Using the previous example, we will find the shortest path from a to c.25 r 19 9 5 b 16 21 31 i 36 c

9. (a) = 0 25
(r) = 19 9 5
(b) = 16 21 31
(i) = 36
(c) =

10. (a) = 0 25
(r) = 19 9 5
(b) = 19 16 21 31
(i) = 36
(c) =

11. (a) = 0 25
(r) = 19 9 5
(b) = 19 16 21 31
(i) = 35 36
(c) =

12. (a) = 0 25
(r) = 24 19 9 5
(b) = 19 16 21 31
(i) = 35 36
(c) =

13. (a) = 0 25
(r) = 24 19 9 5
(b) = 19 16 21 31
(i) = 35 36
(c) = 50

14. (a) = 0 25
(r) = 24 19 9 5
(b) = 19 16 21 31
(i) = 9 36
(c) = 50

15. (a) = 0 25
(r) = 24 19 9 5
(b) = 19 16 21 31
(i) = 9 36
(c) = 50

16. (a) = 0 25
(r) = 24 19 9 5
(b) = 19 16 21 31
(i) = 9 36
(c) = 45

17. (a) = 0 25
(r) = 24 19 9 5
(b) = 19 16 21 31
(i) = 9 36
(c) = 45

18. Example ∞ Distance(source) = 0 Distance (all vertices but source) = ∞∞
Distance(source) = 0
Distance (all vertices but source) = ∞ A 2 B 4 1 3 10 C 2 D 2 E ∞ ∞ ∞ 5 8 4 6 F 1 G ∞ ∞
Pick vertex in List with minimum distance.

19. Example: Initialization∞
Distance(source) = 0
Distance (all vertices but source) = ∞ A 2 B 4 1 3 10 C 2 D 2 E ∞ ∞ 1 5 8 4 6 F 1 G ∞ ∞

20. Example 2 Distance(source) = 0 Distance (all vertices but source) = ∞2
Distance(source) = 0
Distance (all vertices but source) = ∞ A 2 B 4 1 3 10 C 2 D 2 E ∞ ∞ 1 5 8 4 6 F 1 G ∞ ∞

21. Example 2 Distance(source) = 0 Distance (all vertices but source) = ∞2
Distance(source) = 0
Distance (all vertices but source) = ∞ A 2 B 4 1 3 10 C 2 D 2 E ∞ 3 1 5 8 4 6 F 1 G ∞ ∞

22. Example 2 Distance(source) = 0 Distance (all vertices but source) = ∞2
Distance(source) = 0
Distance (all vertices but source) = ∞ A 2 B 4 1 3 10 C 2 D 2 E 3 3 1 5 8 4 6 F 1 G ∞ ∞

23. Example 2 Distance(source) = 0 Distance (all vertices but source) = ∞2
Distance(source) = 0
Distance (all vertices but source) = ∞ A 2 B 4 1 3 10 C 2 D 2 E 3 3 1 5 8 4 6 F 1 G 9 ∞

24. Example 2 Distance(source) = 0 Distance (all vertices but source) = ∞2
Distance(source) = 0
Distance (all vertices but source) = ∞ A 2 B 4 1 3 10 C 2 D 2 E 3 3 1 5 8 4 6 F 1 G 9 5

25. Example 2 Distance(source) = 0 Distance (all vertices but source) = ∞2
Distance(source) = 0
Distance (all vertices but source) = ∞ A 2 B 4 1 3 10 C 2 D 2 E 3 3 1 5 8 4 6 F 1 G 6 5