Lecture 23

CSE 331Oct 24, 2011

Reminder

2 points for Piazza participation

3 points for mini-project

Online Office Hr @10:00-11:00pm

Shortest Path Problem

http://xkcd.com/85/

Another more important application

Is BGP a known acronym for you?

Routing uses shortest path algorithm

Shortest Path problem

Input: Directed graph G=(V,E)

Edge lengths, le for e in E

“start” vertex s in V

Output: All shortest paths from s to all nodes in V

100

155

s

u

w

5

s

u

155

s

u

w

Dijkstra’s shortest path algorithmE. W. Dijkstra (1930-2002)

Dijkstra’s shortest path algorithm

Input: Directed G=(V,E), le ≥ 0, s in V

R = {s}, d(s) =0

While there is a x not in R with (u,x) in E, u in R

d’(w) = min e=(u,w) in E, u in R d(u)+le

Pick w that minimizes d’(w) Add w to Rd(w) = d’(w)

ss

ww

uu

zz

xx

yy

1

2

4

3

3

1

2

1

2

d(s) = 0

1

4

2 ss

uu

d(u) = 1

4

2

ww

d(w) = 2

5

xx

d(x) = 2

3

4

yy

d(y) = 3

zz

d(z) = 4

Shortest paths

Shortest paths

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

Rest of Today’s agenda

Prove the correctness of Dijkstra’s Algorithm

Runtime analysis of Dijkstra’s Algorithm

Reading AssignmentSec 4.4 of [KT]

Building a fiber network

Lay down fibers to connect n locations

All n locations should be connected

Laying down a fiber costs money

What is the cheapest way to lay down the fibers?