1. Network coding on the GPU Péter Vingelmann Supervisor: Frank H.P. Fitzek

2. What is network coding? Traditional routing in packet networks: data is simply forwarded by the intermediate nodes Network coding breaks with this principle: nodes may recombine several input packets into one or several output packets Linear network coding: form linear combinations of incoming packets

3. What are the benefits of network coding? Throughtput The butterfly network: Robustness Each encoded packet is “equally important” Complexity Less complex protocols (e.g. for content distribution) Security It is more difficult to “overhear” anything that makes sense

4. What is the problem? The computational overhead introduced by network coding operations is not negligible There is no dedicated network coding hardware yet A possible solution: use the Graphics Processing Unit (GPU) to perform the necessary calculations

5. Overview of network coding Definition: coding at a node in a packet network All operations are performed over a Galois Field GF(2 s ), packets are divided into s bit-long symbols The process of network coding can be divided into two separate parts: K bits s bits Symbol 1 s bits Symbol 2 Symbol K/s

6. Encoding Encoded packets are linear combinations of the original packets, where addition and multiplication are performed over GF(2 s ) We can use random coding coefficients

7. Decoding Assume a node has received M encoded packets (together with its coefficients) Linear system with M equations and N unknowns We need M ≥ N to have a chance of solving this system of equations using standard Gaussian elimination At least N linearly independent encoded packets must be received in order to recover all the original data packets

8. CPU implementation A simple C++ console application with some customizable parameters: L: packet length N: generation size Object-oriented implementation: Encoder and Decoder classes Addition and subtraction over the Galois Field are simply XOR operations on the CPU Galois multiplication and division tables are pre-calculated and stored in arrays: both operations can be performed by array lookups Gauss-Jordan elimination is used for decoding: “on-the-fly” version of the standard Gaussian elimination It is used as a reference implementation

9. Graphics card Originally designed for real-time rendering of 3D graphics The past: fixed-function pipeline They evolved into programmable parallel processors with enormous computing power The present: programmable pipeline Now they can even perform general-purpose computations with some restrictions The future: General Purpose Graphics Processing Unit (GPGPU)

10. OpenGL & CG implementation OpenGL is a standard cross-platform API for computer graphics It cannot be used on its own, a shader language is also necessary to implement custom algorithms A shader is a short program which is used to program certain stages of the rendering pipeline I chose NVIDIA’s CG toolkit as a shader language The developer is forced to think with the traditional concepts of 3D graphics (e.g. vertices, pixels, triangles, lines and points)

11. Encoder shader in CG A regular bitmap image serves as input data Coefficients and data packets are stored in textures (2D arrays of bytes in graphics memory that can be accessed efficiently) The XOR operation and Galois multiplication are also implemented by texture look-ups: a 256x256-sized black&white texture is necessary for each The encoded packets are rendered (computed) line- by-line onto the screen and they are saved into a texture

12. Decoder shaders in CG The decoding algorithm is more complex It must be decomposed into 3 different shaders These shaders correspond to the 3 consecutive phases of the Gauss-Jordan elimination: 1. Forward substitution: reduce the new packet by the existing rows 2. Finding the pivot element in the reduced packet 3. Backward substitute the reduced and normalized packet into the existing rows

13. NVIDIA’s CUDA toolkit Compute Unified Device Architecture (CUDA) Parallel computing applications in the C language Modern GPUs have many processor cores and they can launch thousands of threads with zero scheduling overhead Terminology: host = CPU device = GPU kernel = a function executed on the GPU A kernel is executed in the Single Program Multiple Data (SPMD) model, meaning that a user- specified number of threads execute the same program.

14. CUDA implementation A CUDA-capable device is required! NVIDIA GeForce 8 series at minimum This is a more native approach, we have fewer restrictions A large number of threads must be launched to achieve the GPU’s peak performance All data structures are stored in CUDA arrays, which are bound to texture references if necessary Computations are visualized using an OpenGL GUI

15. Encoder kernel in CUDA Encoding is a matrix multiplication in the GF domain, and can be considered as a highly parallel computation problem We can achieve a very fine granularity by launching a thread for every single byte to be computed Galois multiplication is implemented by array look-ups, but we have a native XOR operator The encoder kernel is quite simple

16. Decoder kernels in CUDA Gauss-Jordan elimination means that the decoding of each coded packet can only start after the decoding of the previous coded packets has finished => we have a sequential algorithm!!! Parallelization is only possible within the decoding of the current coded packet We need 2 separate kernels for forward and backward substitution A search for the first non-zero element must be performed on the CPU side, because synchronization is not possible between all GPU threads => the CPU must assist the GPU!

17. Graphical User Interface

18. Performance evaluation It is difficult to compare the actual performance of these implementations A lot of factors has to be taken into consideration:  Shader/kernel execution times  Memory transfers between host and device memory  Shader/kernel initialization & parameter setup  CPU-GPU synchronization Measurement results are not uniform, because we cannot have exclusive control over the GPU: other applications may have a negative impact

19. CPU implementation

20. OpenGL & CG implementation

21. CUDA implementation

22. Achievements It has been shown that the GPU is capable of performing network coding calculations What’s more, it can outperform the CPU by a significant margin in some cases We have a submitted and accepted paper at European Wireless ’09 by the title: Implementation of Random Linear Network Coding on OpenGL-enabled Graphics Cards

23. Demonstration CPU implementation OpenGL & CG implementation CUDA implementation

24. Questions??? Thank you for your kind attention!