There has been significant recent interest in parallel graph processing due to the need to quickly analyze the large graphs available today. Many graph codes have been designed for distributed memory or external memory. However, today even the largest publicly-available real-world graph (the Hyperlink Web graph with over 3.5 billion vertices and 128 billion edges) can fit in the memory of a single commodity multicore server. Nevertheless, most experimental work in the literature report results on much smaller graphs, and the ones for the Hyperlink graph use distributed or external memory. Therefore, it is natural to ask whether we can efficiently solve a broad class of graph problems on this graph in memory. This paper shows that theoretically-efficient parallel graph algorithms can scale to the largest publicly-available graphs using a single machine with a terabyte of RAM, processing them in minutes. We give implementations of theoretically-efficient parallel algorithms for 13 important graph problems. We also present the optimizations and techniques that we used in our implementations, which were crucial in enabling us to process these large graphs quickly. We show that the running times of our implementations outperform existing state-of-the-art implementations on the largest real-world graphs. For many of the problems that we consider, this is the first time they have been solved on graphs at this scale. We provide a publicly-available benchmark suite containing our implementations.

https://dl.acm.org/doi/pdf/10.1145/3210377.3210414

Theoretically Efficient Parallel Graph Algorithms Can Be Fast and Scalable

We develop the first parallel graph coloring heuristics with strong theoretical guarantees on work and depth and coloring quality. The key idea is to design a relaxation of the vertex degeneracy order, a well-known graph theory concept, and to color vertices in the order dictated by this relaxation. This introduces a tunable amount of parallelism into the degeneracy ordering that is otherwise hard to parallelize. This simple idea enables significant benefits in several key aspects of graph coloring. For example, one of our algorithms ensures polylogarithmic depth and a bound on the number of used colors that is superior to all other parallelizable schemes, while maintaining work-efficiency. In addition to provable guarantees, the developed algorithms have competitive run-times for several real-world graphs, while almost always providing superior coloring quality. Our degeneracy ordering relaxation is of separate interest for algorithms outside the context of coloring.

High-Performance Parallel Graph Coloring with Strong Guarantees on Work, Depth, and Quality

Connected components is a fundamental kernel in graph applications. The fastest existing parallel multicore algorithms for connectivity are based on some form of edge sampling and/or linking and compressing trees. However, many combinations of these design choices have been left unexplored. In this paper, we design the ConnectIt framework, which provides different sampling strategies as well as various tree linking and compression schemes. ConnectIt enables us to obtain several hundred new variants of connectivity algorithms, most of which extend to computing spanning forest. In addition to static graphs, we also extend ConnectIt to support mixes of insertions and connectivity queries in the concurrent setting. 
We present an experimental evaluation of ConnectIt on a 72-core machine, which we believe is the most comprehensive evaluation of parallel connectivity algorithms to date. Compared to a collection of state-of-the-art static multicore algorithms, we obtain an average speedup of 12.4x (2.36x average speedup over the fastest existing implementation for each graph). Using ConnectIt, we are able to compute connectivity on the largest publicly-available graph (with over 3.5 billion vertices and 128 billion edges) in under 10 seconds using a 72-core machine, providing a 3.1x speedup over the fastest existing connectivity result for this graph, in any computational setting. For our incremental algorithms, we show that our algorithms can ingest graph updates at up to several billion edges per second. To guide the user in selecting the best variants in ConnectIt for different situations, we provide a detailed analysis of the different strategies. Finally, we show how the techniques in ConnectIt can be used to speed up two important graph applications: approximate minimum spanning forest and SCAN clustering.

ConnectIt: A Framework for Static and Incremental Parallel Graph Connectivity Algorithms

We develop the first parallel graph coloring heuristics with strong theoretical guarantees on work and depth and coloring quality. The key idea is to design a relaxation of the vertex degeneracy order, a well-known graph theory concept, and to color vertices in the order dictated by this relaxation. This introduces a tunable amount of parallelism into the degeneracy ordering that is otherwise hard to parallelize. This simple idea enables significant benefits in several key aspects of graph coloring. For example, one of our algorithms ensures polylogarithmic depth and a bound on the number of used colors that is superior to all other parallelizable schemes, while maintaining workefficiency. In addition to provable guarantees, the developed algorithms have competitive run-times for several real-world graphs, while almost always providing superior coloring quality. Our degeneracy ordering relaxation is of separate interest for algorithms outside the context of coloring.

SCAN (Structural Clustering Algorithm for Networks) is a well-studied, widely used graph clustering algorithm. For large graphs, however, sequential SCAN variants are prohibitively slow, and parallel SCAN variants do not effectively share work among queries with different SCAN parameter settings. Since users of SCAN often explore many parameter settings to find good clusterings, it is worthwhile to precompute an index that speeds up queries. This paper presents a practical and provably efficient parallel index-based SCAN algorithm based on GS*-Index, a recent sequential algorithm. Our parallel algorithm improves upon the asymptotic work of the sequential algorithm by using integer sorting. It is also highly parallel, achieving logarithmic span (parallel time) for both index construction and clustering queries. Furthermore, we apply locality-sensitive hashing (LSH) to design a novel approximate SCAN algorithm and prove guarantees for its clustering behavior. We present an experimental evaluation of our algorithms on large real-world graphs. On a 48-core machine with two-way hyper-threading, our parallel index construction achieves 50--151× speedup over the construction of GS*-Index. In fact, even on a single thread, our index construction algorithm is faster than GS*-Index. Our parallel index query implementation achieves 5--32× speedup over GS*-Index queries across a range of SCAN parameter values, and our implementation is always faster than ppSCAN, a state-of-the-art parallel SCAN algorithm. Moreover, our experiments show that applying LSH results in faster index construction while maintaining good clustering quality.

Parallel Index-Based Structural Graph Clustering and Its Approximation

In this demonstration paper, we present the Graph Based Benchmark Suite (GBBS), a suite of scalable, provably-efficient implementations of over 20 fundamental graph problems for shared-memory multicore machines. Our results are obtained using a graph processing interface written in C++, extending the Ligra interface with additional functional primitives that have clearly defined cost bounds. Our approach enables writing high-level codes that are simultaneously simple and high-performance by virtue of using highly-optimized primitives. Another benefit is that optimizations, such as graph compression, are implemented transparently to high-level user code, and can thus be utilized without changing the implementation. Our approach enables our codes to scale to the largest publicly-available real-world graph containing over 200 billion edges on a single multicore machine. We show how to use GBBS to process and perform a variety of tasks on real-world graphs. We present the high-level C++ APIs that enable us to write concise, high-performance implementations. We also introduce a Python interface to GBBS, which lets users easily prototype algorithms and pipelines in Python that significantly outperform NetworkX, a mature Python-based graph processing solution.

The Graph Based Benchmark Suite (GBBS)

SCAN (Structural Clustering Algorithm for Networks) is a well-studied, widely used graph clustering algorithm. For large graphs, however, sequential SCAN variants are prohibitively slow, and parallel SCAN variants do not effectively share work among queries with different SCAN parameter settings. Since users of SCAN often explore many parameter settings to find good clusterings, it is worthwhile to precompute an index that speeds up queries. 
This paper presents a practical and provably efficient parallel index-based SCAN algorithm based on GS*-Index, a recent sequential algorithm. Our parallel algorithm improves upon the asymptotic work of the sequential algorithm by using integer sorting. It is also highly parallel, achieving logarithmic span (parallel time) for both index construction and clustering queries. Furthermore, we apply locality-sensitive hashing (LSH) to design a novel approximate SCAN algorithm and prove guarantees for its clustering behavior. 
We present an experimental evaluation of our algorithms on large real-world graphs. On a 48-core machine with two-way hyper-threading, our parallel index construction achieves 50--151$\times$ speedup over the construction of GS*-Index. In fact, even on a single thread, our index construction algorithm is faster than GS*-Index. Our parallel index query implementation achieves 5--32$\times$ speedup over GS*-Index queries across a range of SCAN parameter values, and our implementation is always faster than ppSCAN, a state-of-the-art parallel SCAN algorithm. Moreover, our experiments show that applying LSH results in faster index construction while maintaining good clustering quality.

Tom Tseng

Papers

The Graph Based Benchmark Suite (GBBS)

Parallel Index-Based Structural Graph Clustering and Its Approximation

Parallel Index-Based Structural Graph Clustering and Its Approximation