1. Lecture 22
CSE 331
Oct 15, 2014

2. Graded Quiz 1
Will hand them out at the END of the lecture

3. Online OH

4. Two definitions for schedules
Idle time
Max “gap” between two consecutively scheduled tasks
Idle time =0
Idle time =1
Inversion
(i,j) is an inversion if i is scheduled before j but di > dj i j
f=1
For every i in 1..n do
Schedule job i from si=f to fi=f+ti
f=f+ti i j
0 idle time and 0 inversions for greedy schedule

5. Proof structure
Any two schedules with 0 idle time and 0 inversions have the same max lateness
Greedy schedule has 0 idle time and 0 inversions
There is an optimal schedule with 0 idle time and 0 inversions

6. Today’s agenda
“Exchange” argument to convert an optimal solution into a 0 inversion one

7. Rest of Today
Shortest Path Problem

8. Reading Assignment
Sec 2.5 of [KT]

9. Shortest Path problem s 100 Input: Directed graph G=(V,E) w15 5 s u w
100
Input: Directed graph G=(V,E)
Edge lengths, le for e in E
“start” vertex s in V 15 5 s u w 5 s u
Output: All shortest paths from s to all nodes in V

10. Naïve Algorithm
Ω(n!) time

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

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

13. 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