1. Discrete Mathematics and Its Applications (5th Edition)
Shortest Paths
Text
Discrete Mathematics and Its Applications (5th Edition)
Kenneth H. Rosen
Chapter 8.6
Based on slides from Chuck Allison, Michael T. Goodrich, and Roberto Tamassia
By Longin Jan Latecki

2. Weighted Graphs
Graphs that have a number assigned to each edge are called weighted graphs. SF LA
DEN
CHI
ATL
MIA
BOS NY

3. Weighted Graphs MILES SF LA DEN CHI ATL MIA BOS NY 860 2534 191 1855
722
908
957
760
606
834
349
2451
1090
595

4. Weighted Graphs FARES SF LA DEN CHI ATL MIA BOS NY $129 $79 $39 $99
$59
$69
$89
$79
$99
$89
$39
$129
$69

5. Weighted Graphs FLIGHT TIMES SF LA DEN CHI ATL MIA BOS NY 4:05 2:10
0:50
2:55
1:50
2:10
2:20
1:55
1:40
2:00
3:50
2:45
1:15
1:30

6. Weighted Graphs
A weighted graph is a graph in which each edge (u, v) has a weight w(u, v). Each weight is a real number.
Weights can represent distance, cost, time, capacity, etc.
The length of a path in a weighted graph is the sum of the weights on the edges.
Dijkstra’s Algorithm finds the shortest path between two vertices.

14. Dijkstra's Algorithm

15. Dijkstra Animation
Demo

16. processed:
fromNode:
distance: # # #
index:
IndexOfMin 1 5 7 2 3 4 6 20 40 15 35 10 50 75

17. processed:
fromNode:
distance: # # #
index: 1 5 7 2 3 4 6 20 40 15 35 10 50 75
Unprocessed node adjacent to 2 is 4.
# > = 55
so replace # with 55

18. processed:
fromNode:
distance: # #
index:
IndexOfMin 1 5 7 2 3 4 6 20 40 15 35 10 50 75

19. processed:
fromNode:
distance: # #
index:
IndexOfMin 1 5 7 2 3 4 6 20 40 15 35 10 50 75
Unprocessed node adjacent to 5 is 3.
35 > = 30
so replace 35 with 30

20. processed:
fromNode:
distance: # #
index:
IndexOfMin 1 5 7 2 3 4 6 20 40 15 35 10 50 75
Unprocessed node adjacent to 5 is 6.
# > = 70
so replace # with 70

21. processed:
fromNode:
distance: #
index:
IndexOfMin 1 5 7 2 3 4 6 20 40 15 35 10 50 75
Unprocessed node adjacent to 5 is 7.
# > = 95
so replace # with 95

22. processed:
fromNode:
distance:
index:
IndexOfMin 1 5 7 2 3 4 6 20 40 15 35 10 50 75

23. processed:
fromNode:
distance:
index:
IndexOfMin 1 5 7 2 3 4 6 20 40 15 35 10 50 75
Unprocessed node adjacent to 3 is 4.
55 < = 65
no change in array

24. processed:
fromNode:
distance:
index:
IndexOfMin 1 5 7 2 3 4 6 20 40 15 35 10 50 75

25. processed:
fromNode:
distance:
index:
IndexOfMin 1 5 7 2 3 4 6 20 40 15 35 10 50 75
Unprocessed node adjacent to 4 is 6.
70 > = 65
so replace 70 with 65

26. processed:
fromNode:
distance:
index:
IndexOfMin 1 5 7 2 3 4 6 20 40 15 35 10 50 75

27. processed:
fromNode:
distance:
index:
IndexOfMin 1 5 7 2 3 4 6 20 40 15 35 10 50 75

28. processed:
fromNode:
distance:
index:
IndexOfMin 1 5 7 2 3 4 6 20 40 15 35 10 50 75
Unprocessed node adjacent to 6 is 7.
95 > = 80
so replace 95 with 80

29. processed:
fromNode:
distance:
index:
IndexOfMin 1 5 7 2 3 4 6 20 40 15 35 10 50 75

30. processed:
fromNode:
distance:
index:
IndexOfMin 1 5 7 2 3 4 6 20 40 15 35 10 50 75
All nodes have been processed
Algorithm finishes.

31. Theorems
Dijkstra’s algorithm finds the length of a shortest path between two vertices in a connected simple undirected weighted graph.
Dijkstra’s algorithm uses O(n2) operations (additions and comparisons) to find the length of the shortest path between two vertices in a connected simple undirected weighted graph.

32. Problem: shortest path from a to zd 5 5 4 7 3 4 1 2 a z 3 4 c 5 e 5 g a b c d e f g z ∞ x
4(a)
3(a)