Graph Algorithms

1
Graph Algorithms

• Graph theory plays an important role in computer
  science
• Used to model many problems
• Example ) model a network topology as a graph

2
Weighted Adjacency Matrix (1)
G (V, E, w)
1
1
2
3
1
3
1
5
2
A
2
3
3
3
Weighted Adjacency Matrix (2)

• The space required to store a graph with n
  vertices is O(n2)
• Useful for dense graphs
• Dense graphs
• example complete graph
• Sparse graphs
• example ring

4
Graph algorithms I will cover

• Discuss algorithms for dense graphs
• Minimum spanning tree Prims algorithm
• Single-source shortest paths Dijkstras
  algorithm
• All-pairs shortest paths
• Dijkstras algorithm
• Floyds algorithm (if time permits)

5
Minimum Spanning Tree
a

• Spanning tree of G a subgraph of G that is a
  tree containing all vertices of G
• Minimum spanning tree (MST) of G
• G a weighted undirected graph
• A spanning tree with the minimum weight
• Applications of MST
• the minimum length of cable necessary to connect
  computers

1
3
5
b
c
1
2
d
The minimum weight 4
6
Prims algorithm (serial)

• A greedy algorithm
• select a starting vertex (root)
• choose the next vertex and edge that are
  guaranteed to be a MST
• continue until all vertices have been selected

7
G original graph
A the weighted adjacency matrix for G
a
b
c
d
a
a
1
3
b
5
A
b
c
c
1
d
2
d
8
Select the root
Dv the least weight from any vertex to v in
MST during its construction
a
b
c
d
a
a
1
3
b
A
5
b
c
c
update
root
d
1
2
d
D
9
Step 1 select the next vertex and edge
a
b
c
d
a
a
1
3
b
A
5
b
c
c
d
1
2
d
D
Pick a minimum weighted edge in D, which is not
in a MST ?minDa,Dc,Ddmin1,5,11
10
a
b
c
d
a
a
1
3
b
A
5
b
c
c
d
1
2
d
2
D
minDa,Ad,a min1, 81
minDc,Ad,c min5,22
11
Step 2 select the next vertex and edge
a
b
c
d
a
a
1
3
b
A
5
b
c
c
d
1
2
d
D
Pick a minimum weighted edge in D, which is not
in a MST ? minDa,Dc1
12
a
b
c
d
a
a
1
3
b
A
5
b
c
c
d
1
2
d
D
minDc,Aa,c min2,32
13
Step 3 select the next vertex and edge
a
b
c
d
a
a
1
3
b
A
5
b
c
c
d
1
2
d
D
Pick a minimum weighted edge in D, which is not
in a MST ? minDc2
14
Final MST
a
b
c
d
a
a
1
3
b
A
5
b
c
c
d
1
2
d
D
The minimum weight 4.
15
Time complexity of Prims algorithm (serial)

• The number of steps n-1
• Finding a minimum weighted edge in the array D in
  each step O(n)
• The total complexity ?(n2)

16
Parallel Formulation of Prims algorithm
a
b
c
d

• Partition the adjacency matrix A and the distance
  array D among p processors
• Each processor handles n/p vertices.
• Example) n 4, p 2, n/p4/2 2

a
a
b
b
c
c
d
d
processor 0
processor 1
17
Select the root
a
b
c
d
a
a
a
b
b
1
3
c
c
5
b
c
d
d
1
2
d
Processor 0 broadcasts the root to all other
processors
processor 0
processor 1
18
Update the array D
a
b
c
d
a
a
a
b
b
1
3
c
c
5
b
update
c
d
d
root
1
2
d
processor 0
processor 1
19
Step 1 select the next vertex and edge

• Find the minimum in D, which is not in MST by
  using All-to-one reduction
• Store it (the next vertex) in processor 0
• Processor 0 broadcasts the next vertex to all
  other processors by using one-to-all broadcast

20
processor 0
processor 1
a
b
c
d
a
min11
min5,11
1
3
one-to-all reduction
5
Processor 0 picks a vertex d
b
c
Processor 0 broadcasts d
1
2
d
all-to-one broadcast
Pick a minimum weighted edge in D, which is not
in a MST
21
Finding a minimum all-to-one reduction
1
4,7,1,2,3
3
2
4
3,5,2,8,6
8,7,3,9,5
9,6,8,4,7
22
Finding a minimum all-to-one reduction
1
min4,7,1,2,31
3
2
4
min3,5,2,8,62
min8,7,3,9,53
min9,6,8,4,74
23
Finding a minimum all-to-one reduction
1
min4,7,1,2,31
2
4
3
3
2
4
min3,5,2,8,62
min8,7,3,9,53
min9,6,8,4,74
1
Again, on find min1,2,3,4 1
24
processor 0
processor 1
a
b
c
d
a
a
a
b
b
1
3
c
c
5
b
d
d
c
1
update
2
2
d
minDa,Ad,a min1, 81
minDc,Ad,c min5,22
25
Step 2 select the next vertex and edge
processor 0
processor 1
a
b
c
d
a
min11
min22
1
3
5
one-to-all reduction
b
Processor 0 picks a vertex a
c
Processor 0 broadcasts a
1
2
d
all-to-one broadcast
26
processor 0
processor 1
a
b
c
d
a
a
a
1
b
b
3
c
c
5
b
c
d
d
1
2
d
minDc,Aa,c min2,32
27
Step 3 select the next vertex and edge
processor 0
processor 1
a
b
c
d
a
1
min ?
min22
3
5
b
one-to-all reduction
c
Processor 0 picks a vertex c
1
Processor 0 broadcasts c
2
d
all-to-one broadcast
28
processor 0
processor 1
Final MST
a
b
c
d
a
a
a
1
b
b
3
c
c
5
b
c
d
d
1
2
d
29
Time complexity of Prims algorithm (parallel)

• The number of steps ?(n)
• Finding a minimum weighted edge in the array D in
  each step
• The total complexity

computation
communication
30
Speedup and Efficiency

• Speedup
• Efficiency

31
Cost-optimal

• For a cost-optical parallel formulation,
• the cost of solving MST in parallel grows at he
  same rate as does the cost of the serial
  algorithm.