CSC 213

1
CSC 213 Large Scale Programming

• Lecture 34
• Weighted Graphs Shortest Paths

2
Todays Goal

• Look at graphs that have weight issues
• How these weights exist
• What the weights mean
• Problems that can arise with weighted graphs
• (Sensitive topic for some Graphs, be careful!)

3
Weighted Graphs

• Edges have numerical weight
• Weights could be distance, cost, travel time, c.
• In this example, weights are miles between the
  two airports

849
PVD
ORD
1843
142
SFO
802
LGA
1205
1743
337
1387
HNL
2555
1099
1233
LAX
1120
DFW
MIA
4
Shortest Path Problem

• Find minimum weight path between u v
• Paths weight is sum of the edge weights
• Here is shortest path between PVD HNL

849
PVD
ORD
1843
142
SFO
802
LGA
1205
1743
337
1387
HNL
2555
1099
1233
LAX
1120
DFW
MIA
5
Shortest Path Assumptions

• Subpath of shortest path is also shortest path
• Shortest path from a vertex to all others is tree
• Shortest paths from Providence

849
PVD
ORD
1843
142
SFO
802
LGA
1205
1743
337
1387
HNL
2555
1099
1233
LAX
1120
DFW
MIA
6
Dijkstras Algorithm

• Finds shortest paths from vertex s to all other
  vertices
• Relies upon several basic assumptions
• Graph is connected
• Edge weights are nonnegative (0 or larger)

7
Dijkstras Algorithm

• Beginning from s, grow cloud of vertices
• Record distance from s through cloud to each
  vertex v not in the cloud
• At each step
• Find vertex, w, with smallest distance not in
  cloud
• Add w to the cloud
• See if this creates smaller path to any vertices
  adjacent to w

8
Edge Relaxation

• Consider Edge e with endvertices u z
• u added to cloud most recently
• z is not in the cloud
• Relaxation of e updates distance to z

d(u) 50
d(z) 60
10
e
z
u
s
d(z) 75

9
Example
0
A
4
8
2
4
2
8
7
1
C
B
D
3
9
?
?
2
5
E
F
0
0
A
A
4
4
8
8
2
2
3
2
8
3
2
7
7
1
7
1
C
B
D
C
B
D
3
9
3
9
5
11
5
8
2
5
2
5
E
F
E
F
10
Example (cont.)
0
A
4
8
2
3
2
7
7
1
C
B
D
3
9
5
8
2
5
E
F
0
A
4
8
2
3
2
7
7
1
C
B
D
3
9
5
8
2
5
E
F
11
Why Dijkstras Algorithm Works

• Dijkstra was smart. Smarter even than me
• Algorthm is greedy. Vertices added in order of
  increasing distance
• Suppose didnt find shortestdistance to F, but
  foundfind shortest distance to D
• Edge from D to F relaxed when D added to cloud
• As long as d(F)gtd(D), Fs distance correct

0
A
4
8
2
3
2
7
7
1
C
B
D
3
9
5
8
2
5
E
F
12
Why No Negative-Weight Edges?

• Suppose Edge has negative weight
• After adding an incident vertex to cloud, may
  discover smaller distance to opposite vertex
• But distance set when vertex added to cloud!

0
A
4
8
6
4
5
7
7
1
C
B
D
0
-8
5
9
Cs true distance is 1, but added to cloud with
distance of 5!
2
5
E
F
13
Not All Is Lost

• Shortest paths algorithms that work with negative
  weights exist
• Bellman-Ford algorithm takes time O(nm)
• DAG-based algorithms takes time O(n m)
• Merlin algorithm takes time O(n m)

14
For Next Lecture

• I will not be here on Friday
• Need to go to programming contest/conference
• Minimum spanning trees lecture now on Monday
• Use extra time productively
• Work on programming assignment or lab
• Also remember portfolios due soon!