Allpairs shortest paths: Floyds Algorithm

1
All-pairs shortest paths Floyds Algorithm

• G (V, E, w), V v1, v2, , vn
• For a subnet v1, v2, , vk, k n,
• the length of the shortest path from vi
  to vj whose intermediate vertices belong to the
  set v1, v2, , vk
• Matrix

2
The matrix D (0)
G original graph
A the weighted adjacency matrix for G
1
2
3
4
1
1
1
3
2
5
AD(0)
2
3
3
1
4
2
4
3
The matrix D (0)
the length of the path from 1 to 3 is 3

• Consider all paths going through the vertex 1 in
  D(0)
• Example)

1
2
3
4
1
1
1
3
2
5
D(0)
2
3
3
the length of the path from 2 to 1 is 1
1
4
2
4
4
The matrix D (1)

• Consider all paths going through the vertex 1in
  D(0)
• Example)

1
2
3
4
1
1
1
3
4
2
5
D(0)
2
3
3
1
4
2
4
5
The matrix D(1)

• Consider all paths going through the vertex 1 in
  D(0)

1
2
3
4
1
1
1
3
4
2
5
D(1)
2
4
3
3
1
4
2
4
6
The matrix D(2)

• Consider all paths going through the vertex 2 in
  D(1)
• Example)

1
2
3
4
1
1
1
3
2
5
D(1)
2
3
3
1
4
2
4
7
The matrix D(2)

• Consider all paths going through the vertex 2 in
  D(1)

1
2
3
4
1
2
1
1
3
2
5
D(2)
2
3
3
2
1
4
2
4
8
The matrix D(3)

• Consider all paths going through the vertex 3 in
  D(2)

1
2
3
4
1
1
1
3
2
5
D(3)
2
3
3
1
4
2
4
9
The matrix D(4)

• Consider all paths going through the vertex 4 in
  D(3)

1
2
3
4
1
1
1
3
3
2
5
D(4)
2
3
3
3
1
4
2
4
10
Parallel Formulation
1
2
3
4

• D(k) is partitioned into p parts
• Each processor computes a partition of D(k)

1
2
3
4
p 4
11
2-D Block Mapping

• Matrix D(k) is divided into p blocks of size
  
• Each block is assigned to one of the p processors

n 8, p 16
D(k)
12
2-D Block Mapping

• Example)

1
2
3
4
P1,2
P1,1
1
1
1
3
2
5
D(0)
2
3
3
1
4
2
4
P2.1
P2,2
13
The matrix D(1)

• Consider all paths going through the vertex 1 in
  D(0)

1
2
3
4
P1,2
P1,1
1
1
1
3
2
5
D(1)
2
3
3
1
4
2
4
P2.1
P2,2
one-to-all broadcast
14
The matrix D(2)

• Consider all paths going through the vertex 2 in
  D(1)

1
2
3
4
P1,2
P1,1
1
1
1
3
2
5
D(2)
2
3
3
1
4
2
4
P2.1
P2,2
one-to-all broadcast
15
The matrix D(3)

• Consider all paths going through the vertex 3 in
  D(2)

1
3
2
4
P1,2
P1,1
1
1
1
3
2
5
D(3)
2
3
3
1
4
2
4
P2.1
P2,2
one-to-all broadcast
16
The matrix D(4)

• Consider all paths going through the vertex 4 in
  D(3)

1
3
2
4
P1,2
P1,1
1
1
1
3
2
5
D(4)
2
3
3
1
4
2
4
P2.1
P2,2
one-to-all broadcast
17
Performance

• Parallel computer with a cross-bisection
  bandwidth of ?(p)
• Example) bisection width of hypercube is p/2
• On the kth iteration,

k (5) column
Each processor sends elements of the
5th row or column
D(5)
k (5) row
18
Communication time

• On the kth iteration,
• (phase 1) each row performs one-to-all broadcast
  at the same time
• (phase 2) each column performs one-to-all
  broadcast at the same time

k (5) column
This broadcast requires
D(5)
k (5) row
19
Time complexity of Floyds algorithm (parallel)

• On the kth iteration,
• Each processor is assigned n2/p elements of D(k)
• The time to compute corresponding D(k) values is
  ?(n2/p).
• The broadcast requires
• The parallel run time is

computation
communication
20
Speedup and Efficiency

• Speedup
• Efficiency

21
Cost-optimal

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