Maintaining a dynamic $k$-core decomposition is an important problem that identifies dense subgraphs in dynamically changing graphs. Recent work by Liu et al. [SPAA 2022] presents a parallel batch-dynamic algorithm for maintaining an approximate $k$-core decomposition. In their solution, both reads and updates need to be batched, and therefore each type of operation can incur high latency waiting for the other type to finish. To tackle most real-world workloads, which are dominated by reads, this paper presents a novel hybrid concurrent-parallel dynamic $k$-core data structure where asynchronous reads can proceed concurrently with batches of updates, leading to significantly lower read latencies. Our approach is based on tracking causal dependencies between updates, so that causally related groups of updates appear atomic to concurrent readers. Our data structure guarantees linearizability and liveness for both reads and updates, and maintains the same approximation guarantees as prior work. Our experimental evaluation on a 30-core machine shows that our approach reduces read latency by orders of magnitude compared to the batch-dynamic algorithm, up to a $\left(4.05 \cdot 10^{5}\right)$-factor. Compared to an unsynchronized (non-linearizable) baseline, our read latency overhead is only up to a $3.21$-factor greater, while improving accuracy of coreness estimates by up to a factor of $52.7$.

翻译：维护动态 $k$-核分解是识别动态变化图中稠密子图的重要问题。Liu 等人近期的工作 [SPAA 2022] 提出了一种并行批量动态算法，用于维护近似的 $k$-核分解。在该方案中，读取和更新操作均需批量处理，因此每种操作类型都可能因等待另一种操作完成而引发高延迟。为解决以读取为主的真实工作负载问题，本文提出了一种新型混合并发并行动态 $k$-核数据结构，其中异步读取可与批量更新并发执行，从而显著降低读取延迟。我们的方法基于追踪更新之间的因果依赖关系，使得因果相关的更新组对并发读取者表现为原子操作。该数据结构保证了读取和更新的线性化与活性，并保持了与先前工作相同的近似保证。在 30 核机器上的实验评估表明，与批量动态算法相比，我们的方法将读取延迟降低了多个数量级，最高可达 $(4.05 \cdot 10^{5})$ 倍。与无同步（非线性化）基线相比，读取延迟开销仅增加最多 $3.21$ 倍，同时将核心度估计的准确性提升高达 $52.7$ 倍。