SingleSource Shortest Paths SSSP: Dijkstras Algorithm

1
Single-Source Shortest Paths (SSSP) Dijkstras
Algorithm

• Find the shortest paths from a vertex s to all
  other vertices
• Solve SSSP for graphs with non-negative weights
• Greedy method incrementally find the shortest
  paths from s to the other vertices by choosing a
  closest vertex

2
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
3
Select the source
Lv the least weight from s to v
a
b
c
d
a
a
1
3
b
A
5
b
c
c
Source
update
d
1
2
d
L
4
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
L
Pick a minimum weighted edge from the source b ?
minLa,Lc,Ld1
5
a
b
c
d
a
a
1
3
b
A
5
b
c
c
d
1
2
update
d
3
L
minLa,LdAd,a min1, 81
minLc,LdAd,c min5,33
6
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
L
Pick a minimum weighted edge from the source b ?
minLa,Lc1
7
a
b
c
d
a
a
1
3
b
A
5
b
c
c
update
d
1
2
d
L
minLc,LaAa,c min3,43
8
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
L
Pick a minimum weighted edge from the source b ?
minLc3
9
Final Shortest Paths
a
b
c
d
a
a
1
3
b
A
5
b
c
c
d
1
2
d
L
10
Time complexity of Dijkstras algorithm (serial)

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

11
Parallel Formulation of Dijkstras algorithm
a
b
c
d

• Partition the adjacency matrix A and the distance
  array L into 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
12
Select the source
a
b
c
d
a
a
a
b
b
1
3
c
c
5
b
update
c
d
d
source
1
2
d
processor 0
processor 1
13
Step 1 select the next vertex and edge

• Find the minimum in L by using all-to-one
  reduction
• Store it (the next vertex) in processor 0
• Processor 0 broadcast the next vertex to all
  other processors by using one-to-all broadcast

14
processor 0
processor 1
a
b
c
d
a
min11
min5,11
1
Ld1
3
all-to-one reduction
5
Processor 0 picks a vertex d
b
c
Processor 0 broadcasts d
1
2
d
Ld1
one-to-all broadcast
15
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
3
d
minLa,LdAd,a min1, 81
minLc,LdAd,c min5,33
16
Step 2 select the next vertex and edge
processor 0
processor 1
a
b
c
d
a
min11
min33
1
3
Lc3
5
all-to-one reduction
b
Processor 0 picks a vertex a
c
Processor 0 broadcasts a
1
2
d
La1
one-to-all broadcast
17
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
minLc,LaAa,c min3,43
18
Step 3 select the next vertex and edge
processor 0
processor 1
a
b
c
d
a
1
min ?
min33
3
Lc3
5
all-to-one reduction
b
c
Processor 0 picks a vertex c
Processor 0 broadcasts c
1
2
d
Lc3
one-to-all broadcast
19
processor 0
processor 1
Final shortest paths
a
b
c
d
a
a
a
1
b
b
3
c
c
5
b
c
d
d
1
2
d
20
Time complexity of Dijkstras algorithm
(parallel)

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

computation
communication
21
Speedup and Efficiency

• Speedup
• Efficiency

22
Cost-optimal

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

23
All-Pairs Shortest Paths (ASP)

• Find the shortest paths between all pairs of
  vertices
• Dijkstras Algorithm
• solve ASP for graphs with non-negative weights
• Floyds algorithm
• solve ASP for graphs with non-negative weights
• solve ASP for graphs with negative weights
  provided there is no negative-weight cycles

24
Dijkstras all-pairs shortest path algorithm

• Serial algorithm
• Parallel Formulations
• Source-partitioned
• Source-parallel

25
Dijkstras Algorithm (serial)

• for each vertex, use Dijkstras single-source
  shortest path algorithm
• takes O(n3) time

a
a
a
a
1
1
1
1
3
3
3
3
5
5
5
5
b
b
b
b
c
c
c
c
1
1
1
1
2
2
2
2
d
d
d
d
26
Source-Partitioned Formulation (parallel)

• Assign each vertex to a processor
• Each processor executes the single-source
  shortest path algorithm

a
a
a
a
1
1
1
1
3
3
3
3
5
5
5
5
b
b
b
b
c
c
c
c
1
1
1
1
2
2
2
2
d
d
d
d
p0
p1
p2
p3
27
Source-Partitioned Speedup and Efficiency

• The parallel run time
• Speedup
• Efficiency if p ?(n)

28
Source-Parallel Formulation (parallel)

• Assign each vertex to a set of processors
• p/n processors, p gt n
• The set of p/n processors executes the
  single-source shortest path algorithm
• example) p 8, n 4, each group has p/n 2
  processors, where n is the number of vertices

p0
p1
p2
p3
p4
p5
p6
p7
group 1
group 2
group 3
group 4
29
Dijkstras single-source shortest paths
performed by two processors p2, p3.
source
a
a
a
a
1
1
1
1
source
3
3
3
3
5
5
5
5
b
b
b
b
c
c
c
c
source
1
1
1
1
2
2
2
2
d
d
d
d
source
group 1
group 2
group 3
group 4
p0
p1
p2
p3
p4
p5
p6
p7
30
Source-Parallel Time complexity

• The parallel run time
• Assume that in the parallel formulation of
  Djikstras single source shortest path
  algorithm, the number of processor is m.
• The time complexity is
• In source-parallel formulation, m p/n

m p/n
31
Source-Parallel Speedup and Efficiency

• Speedup
• Efficiency
• For cost-optimal