1. Lecture 26
CSE 331
Nov 1, 2010

2. Blog posts/Group Scribe leader
Please sign up if you have not
There are many lectures even in November with <3 students
If I have a pick a blogger/leader I will only pick THREE/lecture
Will lose out on 5%-10% of your grade

3. Shortest Path problem 15 5 s u v 100 Input: Directed graph G=(V,E)
Edge lengths, le for e in E
“start” vertex s in V 15 5 s u v
Output: All shortest paths from s to v in V 5 s u

4. Naïve Algorithm
Ω(n!) time

5. Dijkstra’s shortest path algorithm
E. W. Dijkstra ( )

6. Dijkstra’s shortest path algorithm1
d’(v) = min e=(u,v) in E, u in R d(u)+le 1 2 4 3 y 4 3 u
d(s) = 0
d(u) = 1 s x 2 4
d(v) = 2
d(x) = 2
d(y) = 3
d(z) = 4 v z 5 4 2 s v
Input: Directed G=(V,E), le ≥ 0, s in V u
R = {s}, d(s) =0
Shortest paths x
While there is a v not in R with (u,v) in E, u in R z y
Pick w that minimizes d’(w)
Add w to R
d(w) = d’(w)

7. Couple of remarks
The Dijkstra’s algo does not explicitly compute the shortest paths
Can maintain “shortest path tree” separately
Dijkstra’s algorithm does not work with negative weights
Left as an exercise

8. Rest of today’s agenda Prove correctness of Dijkstra’s algorithm
Efficient implementation of Dijkstra’s algorithm