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Accelerating Graph Betweenness Centrality with CUDA

Graph analysis is a fundamental tool for domains as diverse as social networks, computational biology, and machine learning. Real-world applications of graph algorithms involve tremendously large networks that cannot be inspected manually. Betweenness Centrality (BC) is a popular analytic that determines vertex influence in a graph. It has many practical use cases, including finding the best locations for stores within cities, power grid contingency analysis, and community detection. Unfortunately, the fastest known algorithm for computing betweenness centrality has O(mn) time complexity for graphs with n vertices and m edges, making the analysis of large networks challenging.

This post describes how we used CUDA and NVIDIA GPUs to accelerate the BC computation, and how choosing efficient parallelization strategies results in an average speedup of 2.7x, and more than 10x speedup for road networks and meshes versus a naïve edge-parallel strategy.

Example Betweenness Centrality scores for a small graph
Fig. 1. Example Betweenness Centrality scores for a small graph

Betweenness Centrality determines the importance of vertices in a network by measuring the ratio of shortest paths passing through a particular vertex to the total number of shortest paths between all pairs of vertices. Intuitively, this ratio determines how well a vertex connects pairs of vertices in the network. Formally, the Betweenness Centrality of a vertex v is defined as:

BC(v) = \sum_{s \neq t \neq v} \frac{\sigma_{st}(v)}{\sigma_{st}}

where \sigma_{st} is the number of shortest paths between vertices s and t and \sigma_{st}(v) is the number of those shortest paths that pass through v. Consider Figure 1 above. Vertex 4 is the only vertex that lies on paths from its left (vertices 5 through 9) to its right (vertices 1 through 3). Hence vertex 4 lies on all the shortest paths between these pairs of vertices and has a high BC score. In contrast, vertex 9 does not belong on a path between any pair of the remaining vertices and thus it has a BC score of 0. Continue reading

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CUDA Pro Tip: Minimize the Tail Effect

When I work on the optimization of CUDA kernels, I sometimes see a discrepancy between Achieved and Theoretical Occupancies. The Theoretical Occupancy is the ratio between the number of threads which may run on each multiprocessor (SM) and the maximum number of executable threads per SM (2048 on the Kepler architecture). This value is estimated from the size of the blocks and the amount of resources (registers and shared memory) used by those blocks for a particular GPU and is computed without running the kernel on the GPU. The Achieved Occupancy, on the other hand, is measured from the execution of the kernel (as the number of active warps divided by the number of active cycles compared to the maximum number of executable warps).

Recently, while working on a kernel for a finance benchmark, I could see an Achieved Occupancy of 41.52% whereas the Theoretical Occupancy was 50%. In NVIDIA Nsight Visual Studio Edition, the Instruction per Clock (IPC) showed a lot of load imbalance between the different SMs with respect to the number of executed instructions by the kernel (see the left graph in the figure below).

Instruction per Clock (IPC) with tail effect (Left) and without (Right)
Instruction per Clock (IPC) with tail effect (Left) and without (Right)

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CUDACasts Episode 19: CUDA 6 Guided Performance Analysis with the Visual Profiler

One of the main reasons for accelerating code on an NVIDIA GPU is for an increase in application performance. This is why it’s important to use the best tools available to help you get the performance you’re looking for. CUDA 6 includes great improvements to the guided analysis tool in the NVIDIA Visual Profiler. Watch today’s CUDACast to see how to use guided analysis to locate potential optimizations for your GPU code.

You can find the code used in this video in the CUDACasts GitHub repository.

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