1. Lecture 23
CSE 331
Oct 24, 2011

2. Warning
Will not always do COMPLETE proofs
More work for YOU

3. Last lecture
Convert optimal schedule O to Ô such that Ô has no inversions
(a) Exists an inversion (i,j) such that i is scheduled right before j (di > dj)
Repeat O(n2) times
(a.5) Swap i and j to get O’
(b) O’ has one less inversion than O
(c) Max lateness(O’) ≤ Max lateness(O)

4. Max lateness(O’) ≤ Max lateness(O)
di > dj
Same lateness
Same lateness O i j t1 t3 O’ j i t2
Lateness of j in O’ ≤ Lateness of j in O
Lateness of i in O’ ≤ Lateness of j in O
Lateness of i in O’ =
t3 - di
< t3 - dj
= Lateness of j in O

6. Rest of today
Shortest Path Problem

7. Reading Assignment
Sec 2.5 of [KT]

8. 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

9. Naïve Algorithm
Ω(n!) time

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

11. 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)

12. 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