Maintaining a $k$-core decomposition quickly in a dynamic graph has important applications in network analysis. The main challenge for designing efficient exact algorithms is that a single update to the graph can cause significant global changes. Our paper focuses on \emph{approximation} algorithms with small approximation factors that are much more efficient than what exact algorithms can obtain. We present the first parallel, batch-dynamic algorithm for approximate $k$-core decomposition that is efficient in both theory and practice. Our algorithm is based on our novel parallel level data structure, inspired by the sequential level data structures of Bhattacharya et al [STOC '15] and Henzinger et al [2020]. Given a graph with $n$ vertices and a batch of updates $\mathcal{B}$, our algorithm provably maintains a $(2 + \varepsilon)$-approximation of the coreness values of all vertices (for any constant $\varepsilon > 0$) in $O(|\mathcal{B}|\log^2 n)$ amortized work and $O(\log^2 n \log\log n)$ depth (parallel time) with high probability. As a by-product, our $k$-core decomposition algorithm also gives a batch-dynamic algorithm for maintaining an $O(\alpha)$ out-degree orientation, where $\alpha$ is the current arboricity of the graph. We demonstrate the usefulness of our low out-degree orientation algorithm by presenting a new framework to formally study batch-dynamic algorithms in bounded-arboricity graphs. Our framework obtains new provably-efficient parallel batch-dynamic algorithms for maximal matching, clique counting, and vertex coloring. We implemented and experimentally evaluated our $k$-core decomposition algorithm on a 30-core machine with two-way hyper-threading on $11$ graphs of varying densities and sizes. [...]
翻译:快速在动态图形中保持 $k$ 核心分解在网络分析中具有重要的应用。 设计高效精确算法的主要挑战是, 对图形的单次更新能够带来巨大的全球变化。 我们的文件侧重于 emph{ approximation} 算法, 其小的近似因素比精确算法所能获得的要效率高得多。 我们展示了第一个平行的、 批量动态算法, 其大约是 $k美元 核心分解, 在理论和实践上都是有效的。 我们的算法是基于我们新的平行水平数据结构, 受 Bhattachary etal [STOC'15] 和 Henchanger 等连续水平数据结构的启发。 鉴于一个以美元为顶值和一系列更新 $malcol_ calal- licalalalalalalalalal liversal 框架的图表 $ 2+ oral- dirioal- disalal- demoalal- exmoal- exisalal- exisal- excialal- exal- extial- exal- exal- exal- exal- exal- we- exal- exlental- exal- exal- exlational- exal- exal- exal- exal- exal- a a a a a a a lemental- exlational- 美元, 美元, 美元, le- lautal- laut- ex- ex- ex- ex- ex- ex- 美元, 美元=______________al- la- ial- exal- ial- ial- exal- exal- dal- le- ex- exlation- dal- dal- mo- ex- ex- ex- exl- ex- ex- la- dal- dal- le- exal- dal- dal- dal- dal- le- le- le- ex- ex