The document presents an overview of a parallel n-body code implementation, comparing serial, OpenMP, and CUDA approaches. Key insights include the computational challenges of the n-body problem, with emphasis on efficiency improvements in force calculations and memory usage in GPU architecture. Experimental results demonstrate significant cost reductions achieved with the CUDA implementation compared to serial and OpenMP versions.

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