Parallel K-Clique Counting on GPUs (2104.13209v2)

Abstract: Counting k-cliques in a graph is an important problem in graph analysis with many applications such as community detection and graph partitioning. Counting k-cliques is typically done by traversing search trees starting at each vertex in the graph. Parallelizing k-clique counting has been well-studied on CPUs and many solutions exist. However, there are no performant solutions for k-clique counting on GPUs. Parallelizing k-clique counting on GPUs comes with numerous challenges such as the need for extracting fine-grain multi-level parallelism, sensitivity to load imbalance, and constrained physical memory capacity. While there has been work on related problems such as finding maximal cliques and generalized sub-graph matching on GPUs, k-clique counting in particular has yet to be explored in depth. In this paper, we present the first parallel GPU solution specialized for the k-clique counting problem. Our solution supports both graph orientation and pivoting for eliminating redundant clique discovery. It incorporates both vertex-centric and edge-centric parallelization schemes for distributing work across thread blocks, and further partitions work within each thread block to extract fine-grain multi-level parallelism while tolerating load imbalance. It also includes optimizations such as binary encoding of induced sub-graphs and sub-warp partitioning to limit memory consumption and improve the utilization of execution resources. Our evaluation shows that our best GPU implementation outperforms the best state-of-the-art parallel CPU implementation by a geometric mean of 12.39x, 6.21x, and 18.99x for k=4, 7, and 10, respectively. We also perform a detailed evaluation of the trade-offs involved in the choice of parallelization scheme, and the incremental speedup of each optimization to provide an in-depth understanding of the optimization space. ...

• The paper introduces the first parallel GPU solution for k-clique counting, reducing redundant computations through graph orientation and pivoting.
• The methodology leverages fine-grained parallelism with vertex- and edge-centric schemes, enhanced by sub-warp partitioning and binary encoding.
• Empirical results demonstrate up to 18.99x speedup over CPU methods, highlighting significant improvements in graph analysis performance.

An Expert Review of "Parallel K-clique Counting on GPUs"

The paper "Parallel K-clique Counting on GPUs" tackles the challenge of efficiently counting k-cliques in graphs using GPU architectures. The authors introduce the first parallel GPU solution dedicated to this problem, which is pertinent for graph analysis tasks such as community detection and graph partitioning. This discussion provides an expert overview of the methodologies, results, and implications of this research.

Methodology and Innovations

The authors highlight several key challenges in adapting k-clique counting to GPUs, including extracting fine-grained parallelism, addressing load imbalance, and navigating the constraints of GPU memory capacity. To overcome these challenges, the paper presents a GPU solution that leverages both graph orientation and pivoting to eliminate redundant clique discoveries. The solution employs sophisticated parallelization schemes—both vertex-centric and edge-centric—alongside optimizations such as binary encoding and sub-warp partitioning.

1. Graph Orientation and Pivoting: These two approaches serve to minimize unnecessary computations by ensuring each k-clique is counted once. Graph orientation converts the graph to a directed version where cliques are discovered from predetermined vertices, reducing redundancy. Conversely, pivoting, drawn from techniques in maximal clique finding, aids in minimizing search space by focusing on the largest cliques first.
2. Fine-Grained Parallelism: The paper outlines a strategy for dividing computation into manageable sections by assigning GPU thread blocks to different vertices or edges. This not only helps in distribution but also in optimizing the workload across the available computing resources, especially via innovative sub-warp partitioning.
3. Memory Optimization: The use of binary encoding for the induced sub-graphs is a particularly noteworthy optimization. By representing adjacency lists as bit vectors, memory consumption is substantially reduced, enabling efficient bitwise operations for set intersections, which are fundamental to k-clique counting.

Performance Evaluation

The empirical results decisively showcase the performance superiority of the proposed GPU solution. The authors report substantial speedup over state-of-the-art CPU implementations, such as ARB-COUNT and Pivoter, with geometric mean improvements of up to 18.99 times for various k-values on large datasets. This stark improvement underscores the effectiveness of GPUs in handling computationally intensive graph tasks, especially when coupled with the paper's innovative approaches to parallelism and memory management.

Implications and Future Directions

The implications of this research are manifold. Practically, the proposed GPU-based k-clique counting algorithm could significantly expedite analysis in domains where graph-based methods are prevalent, such as bioinformatics, social network analysis, and computer vision. Theoretically, the paper's methods open avenues for further exploration in optimizing parallel algorithms for other complex graph problems on GPU architectures.

Future work could explore extending these optimizations to distributed environments or integrating them into broader graph processing frameworks. Additionally, as GPU architectures evolve, there could be new opportunities to refine these strategies further.

In conclusion, this research makes a significant contribution to the field of graph analysis by demonstrating how GPUs can be harnessed to handle the specific challenges of k-clique counting. Through thoughtful parallelization and innovative memory optimizations, this work sets a new performance standard and paves the way for future advancements in high-performance graph processing.