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The k-core of a graph is the largest induced sub-graph with minimum degree k. The problem of k-core decomposition finds the k-cores of a graph for all valid values of k, and it has many applications such as network analysis, computational biology and graph visualization. Currently, there are two types of parallel algorithms for k-core decomposition: (1) degree-based vertex peeling, and (2) iterative h-index refinement. There is, however, few studies on accelerating k-core decomposition using GPU. In this paper, we propose a highly optimized peeling algorithm on a GPU, and compare it with possible implementations on top of think-like-a-vertex graph-parallel GPU systems as well as existing serial and parallel k-core decomposition algorithms on CPUs. Extensive experiments show that our GPU algorithm is the overall winner in both time and space. Our source code is released at https://github.com/akhlaqueak/KCoreGPU. 

Given a user-specified minimum degree threshold γ, a γ-quasi-clique is a subgraph where each vertex connects to at least γ fraction of the other vertices. Quasi-clique is a natural definition for dense structures, so finding large and hence statistically significant quasi-cliques is useful in applications such as community detection in social networks and discovering significant biomolecule structures and pathways. However, mining maximal quasi-cliques is notoriously expensive, and even a recent algorithm for mining large maximal quasi-cliques is flawed and can lead to a lot of repeated searches. This paper proposes a parallel solution for mining maximal quasi-cliques that is able to fully utilize CPU cores. Our solution utilizes divide and conquer to decompose the workloads into independent tasks for parallel mining, and we addressed the problem of (i) drastic load imbalance among different tasks and (ii) difficulty in predicting the task running time and the time growth with task subgraph size, by (a) using a timeout-based task decomposition strategy, and by (b) utilizing a priority task queue to schedule long-running tasks earlier for mining and decomposition to avoid stragglers. Unlike our conference version in PVLDB 2020 where the solution was built on a distributed graph mining framework called G-thinker, this paper targets a single-machine multi-core environment which is more accessible to an average end user. A general framework called T-thinker is developed to facilitate the programming of parallel programs for algorithms that adopt divide and conquer, including but not limited to our quasi-clique mining algorithm. Additionally, we consider the problem of directly mining large quasi-cliques from dense parts of a graph, where we identify the repeated search issue of a recent method and address it using a carefully designed concurrent trie data structure. Extensive experiments verify that our parallel solution scales well with the number of CPU cores, achieving 26.68× runtime speedup when mining a graph with 3.77M vertices and 16.5M edges with 32 mining threads. Additionally, mining large quasi-cliques from dense parts can provide an additional speedup of up to 89.46×. 

A frequent pattern is a substructure that appears in a database with frequency (aka. support) no less than a user-specified threshold, while a closed pattern is one that has no super-pattern that has the same support. Here, a substructure can refer to different structural forms, such as itemsets, subsequences, subtrees, and subgraphs, and mining such substructures is important in many real applications such as product recommendation and feature extraction. Currently, there lacks a general programming framework that can be easily customized to mine different types of patterns, and existing parallel and distributed solutions are IO-bound rendering CPU cores underutilized. Since mining frequent and/or closed patterns are NP-hard, it is important to fully utilize the available CPU cores. This paper presents such a general-purpose framework called PrefixFPM. The framework is based on the idea of prefix projection which allows a divide-and-conquer mining paradigm. PrefixFPM exposes a unified programming interface to users who can readily customize it to mine their desired patterns. We have adapted the state-of-the-art serial algorithms for mining patterns including subsequences, subtrees, and subgraphs on top of PrefixFPM, and extensive experiments demonstrate an excellent speedup ratio of PrefixFPM with the number of CPU cores. 

Decision trees and tree ensembles are popular supervised learning models on tabular data. Two recent research trends on tree models stand out: (1) bigger and deeper models with many trees, and (2) scalable distributed training frameworks. However, existing implementations on distributed systems are IO-bound leaving CPU cores underutilized. They also only find best node-splitting conditions approximately due to row-based data partitioning scheme. In this paper, we target the exact training of tree models by effectively utilizing the available CPU cores. The resulting system called TreeServer adopts a column-based data partitioning scheme to minimize communication, and a node-centric task-based engine to fully explore the CPU parallelism. Experiments show that TreeServer is up to 10x faster than models in Spark MLlib. We also showcase TreeServer's high training throughput by using it to build big "deep forest" models. 

Quasi-cliques are a type of dense subgraphs that generalize the notion of cliques, important for applications such as community/module detection in various social and biological networks. However, the existing quasi-clique definition and algorithms are only applicable to undirected graphs. In this paper, we generalize the concept of quasi-cliques to directed graphs by proposing $(\gamma_1, \gamma_2)$-quasi-cliques which have density requirements in both inbound and outbound directions of each vertex in a quasi-clique subgraph. An efficient recursive algorithm is proposed to find maximal $(\gamma_1, \gamma_2)$-quasi-cliques which integrates many effective pruning rules that are validated by ablation studies. We also study the finding of top-$k$ large quasi-cliques directly by bootstrapping the search from more compact quasi-cliques, to scale the mining to larger networks. The algorithms are parallelized with effective load balancing, and we demonstrate that they can scale up effectively with the number of CPU cores. 

Finding from a big graph those subgraphs that satisfy certain conditions is useful in many applications such as community detection and subgraph matching. These problems have a high time complexity, but existing systems that attempt to scale them are all IO-bound in execution. We propose the first truly CPU-bound distributed framework called G-thinker for subgraph finding algorithms, which adopts a task-based computation model, and which also provides a user-friendly subgraph-centric vertex-pulling API for writing distributed subgraph finding algorithms that can be easily adapted from existing serial algorithms. To utilize all CPU cores of a cluster, G-thinker features (1) a highly concurrent vertex cache for parallel task access and (2) a lightweight task scheduling approach that ensures high task throughput. These designs well overlap communication with computation to minimize the idle time of CPU cores. To further improve load balancing on graphs where the workloads of individual tasks can be drastically different due to biased graph density distribution, we propose to prioritize the scheduling of those tasks that tend to be long running for processing and decomposition, plus a timeout mechanism for task decomposition to prevent long-running straggler tasks. The idea has been integrated into a novelty algorithm for maximum clique finding (MCF) that adopts a hybrid task decomposition strategy, which significantly improves the running time of MCF on dense and large graphs: The algorithm finds a maximum clique of size 1,109 on a large and dense WikiLinks graph dataset in 70 minutes. Extensive experiments demonstrate that G-thinker achieves orders of magnitude speedup compared even with the fastest existing subgraph-centric system, and it scales well to much larger and denser real network data. G-thinker is open-sourced at http://bit.ly/gthinker with detailed documentation. 

Mining frequent subtree patterns in a tree database (or, forest) is useful in domains such as bioinformatics and mining semi-structured data. We consider the problem of mining embedded subtrees in a database of rooted, labeled, and ordered trees. We compare two existing serial mining algorithms, PrefixTreeSpan and TreeMiner, and adapt them for parallel execution using PrefixFPM, our general-purpose framework for frequent pattern mining that is designed to effectively utilize the CPU cores in a multicore machine. Our experiments show that TreeMiner is faster than its successor PrefixTreeSpan when a limited number of CPU cores are used, as the total mining workloads is smaller; however, PrefixTreeSpan has a much higher speedup ratio and can beat TreeMiner when given enough CPU cores. 

Frequent pattern mining (FPM) has been a focused theme in data mining research for decades, but there lacks a general programming framework that can be easily customized to mine different kinds of frequent patterns, and existing solutions to FPM over big transaction databases are IO-bound rendering CPU cores underutilized even though FPM is NP-hard. This paper presents, PrefixFPM, a general-purpose framework for FPM that is able to fully utilize the CPU cores in a multicore machine. PrefixFPM follows the idea of prefix projection to partition the workloads of PFM into independent tasks by divide and conquer. PrefixFPM exposes a unified programming interface to users who can customize it to mine their desired patterns, and the parallel execution engine is transparent to end-users and can be reused for mining all kinds of patterns. We have adapted the state-of-the-art serial algorithms for mining frequent patterns including subsequences, subtrees, and subgraphs on top of PrefixFPM, and extensive experiments demonstrate an excellent speedup ratio of PrefixFPM with the number of cores. A demo is available at https://youtu.be/PfioC0GDpsw; the code is available at https://github.com/yanlab19870714/PrefixFPM. 

Canonical polyadic (CP) decomposition of a tensor is a basic operation in a lot of applications such as data mining and video foreground/background separation. However, existing algorithms for CP decomposition require users to provide a rank of the target tensor data as part of the input and finding the rank of a tensor is an NP-hard problem. Currently, to perform CP decomposition, users are required to make an informed guess of a proper tensor rank based on the data at hand, and the result may still be suboptimal. In this paper, we propose to conduct CP decomposition and tensor rank approximation together, so that users do not have to provide the proper rank beforehand, and the decomposition algorithm will find the proper rank and return a high-quality result. We formulate an optimization problem with an objective function consisting of a least-squares Tikhonov regularization and a sparse L1-regularization term. We also test its effectiveness over real applications with moving object videos. 

Given a user-specified minimum degree threshold γ, a γ-quasi-clique is a subgraph g = (Vg, Eg) where each vertex ν ∈ Vg connects to at least γ fraction of the other vertices (i.e., ⌈γ · (|Vg|- 1)⌉ vertices) in g. Quasi-clique is one of the most natural definitions for dense structures useful in finding communities in social networks and discovering significant biomolecule structures and pathways. However, mining maximal quasi-cliques is notoriously expensive. In this paper, we design parallel algorithms for mining maximal quasi-cliques on G-thinker, a distributed graph mining framework that decomposes mining into compute-intensive tasks to fully utilize CPU cores. We found that directly using G-thinker results in the straggler problem due to (i) the drastic load imbalance among different tasks and (ii) the difficulty of predicting the task running time. We address these challenges by redesigning G-thinker's execution engine to prioritize long-running tasks for execution, and by utilizing a novel timeout strategy to effectively decompose long-running tasks to improve load balancing. While this system redesign applies to many other expensive dense subgraph mining problems, this paper verifies the idea by adapting the state-of-the-art quasi-clique algorithm, Quick, to our redesigned G-thinker. Extensive experiments verify that our new solution scales well with the number of CPU cores, achieving 201× runtime speedup when mining a graph with 3.77M vertices and 16.5M edges in a 16-node cluster.