This is a brief overview of a nbody code, including the sequential and parallel versions (OpenMP and CUDA) and its computational complexity.

1. Brief overview of a
parallel nbody code
Implementation and analysis
Filipo Novo Mór
Graduate Program in Computer Science UFRGS
Prof. Nicollas Maillard
2013, December

2. Overview
• About the nbody problem
• The Serial Implementation
• The OpenMP Implementation
• The CUDA Implementation
• Experimental Results
• Conclusion

3. About the nbody problem
Features:
 Force calculation between all particles.
 Complexity O(N2)
 Energy should be constant.
 The brute force algorithm demands huge
computational power.

4. The Serial Implementation
NAIVE!
• Clearly N2
• Each pair is evaluated twice
• Acceleration has to be adjusted at the end.

5. The Serial Implementation
• It stills under N2 domain, but:
• Each pair is evaluated once only.
• Acceleration it’s OK at the end!

6. The OpenMP Implementation
• MUST be based on the “naive” version.
• We lost the “/2”, but we gain the “/p”!
• OBS: the static schedule seems to be slightly faster than dynamic schedule.

7. Analysis
*****
*****
*****
*****
*****
for (i=0; i<N; i++)
{
for(j=i+1; j<N; j++)
{
printf(“*”);
}
printf(“n”);
}

8. The CUDA Implementation
Basic CUDA GPU architecture

9. Global Memory
N = 15
K=3
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Shared
Memory
Bank

10. Global Memory
Active Tasks
Active Transfers
Shared
Memory
Bank
BARRIER
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

11. Global Memory
Active Tasks
Active Transfers
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1
2
Shared
Memory
Bank

12. Global Memory
Active Tasks
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Shared
Memory
Bank
BARRIER
Active Transfers

13. Global Memory
Active Tasks
Active Transfers
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
3
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
4
5
Shared
Memory
Bank

14. Global Memory
Active Tasks
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Shared
Memory
Bank
BARRIER
Active Transfers

15. Global Memory
Active Tasks
Active Transfers
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
6
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
7
8
Shared
Memory
Bank

16. Global Memory
Active Tasks
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Shared
Memory
Bank
BARRIER
Active Transfers

17. Global Memory
Active Tasks
Active Transfers
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
9 10 11
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Shared
Memory
Bank

18. Global Memory
Active Tasks
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Shared
Memory
Bank
BARRIER
Active Transfers

19. Global Memory
Active Tasks
Active Transfers
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
12 13 14
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Shared
Memory
Bank

20. Global Memory
Active Tasks
Active Transfers
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Shared
Memory
Bank

21. Global Memory
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Shared
Memory
Bank

22. Analysis
C : cost of the CalculateForce
function.
M : transfer cost between global and
shared memories.
T : transfer cost between CPU and
device memories.
 Access to shared memory is
around 100X faster than to the
global memory.

23. Experimental Results
How much would it cost???
Testing Environment:
 Dell PowerEdge R610
 2 Intel Xeon Quad-Core E5520 2.27 GHz Hyper-Threading
 8 physical cores, 16 threads.
 RAM 16GB
 NVIDIA Tesla S2050
 Ubuntu Server 10.0.4 LTS
 GCC 4.4.3
 CUDA 5.0
Version
Cost
Naive
$
0.49
Smart
$
0.33
OMP
$
0.08
CUDA
$
0.05

Amazon EC2:

General Purpose - m1.large plan

GPU Instances - g2.2xlarge plan

24. Conclusions
• PRAM is OK for sequential and OpenMP.
• But for CUDA, we need a better model!
– Considering block threads, warps and latency.
Thanks!

25. Additional Slides

26. About the nbody problem
• Calculations
Force (acceleration)
𝑚 𝑗 𝑟𝑖𝑗
𝑓𝑖 ≈ 𝐺𝑚 𝑖
1<𝑗<𝑁
𝑗 ≠𝑖
𝑟𝑖𝑗
2
+
3
2
𝜀2
Energy (kinetic and potential)
𝐸 = 𝐸𝑘 + 𝐸𝑝
𝑁
𝐺𝑚 𝑖 𝑚𝑗
𝐸𝑝 = −
1<𝑗 <𝑁
𝑖≠𝑗
𝑁
𝐸𝑘 =
1<𝑖<𝑁
𝑟𝑖𝑗
𝑚 𝑖 𝑣2
𝑖
2
Softening Factor
collisionless system
virtual particles

27. About the nbody problem