Dynamic connectivity is a fundamental dynamic graph problem, and recent algorithmic breakthroughs on dynamic graph sketching have reshaped what is theoretically possible: by encoding the graph as per-vertex linear sketches, these algorithms solve dynamic connectivity in only $Θ(V \log^2 V)$ space, independent of the number of edges,outperforming lossless $Θ(V+E)$-space structures that grow as the graph becomes denser. Prior to this work, no practical dynamic connectivity algorithm has been able to translate these theoretical breakthroughs into space savings on real-world graphs. The main obstacle is that per-vertex sketches cost thousands of bytes per vertex, so sketching only pays off once the graph becomes extremely dense. We observe that sparse real-world graphs are often not uniformly sparse, these graphs can contain dense cores on a small subset of vertices that account for a large fraction of edges. We exploit this structure via hybrid sketching: sketch only the dense core, and store the sparse periphery losslessly. We design new hybrid algorithms for fully-dynamic and semi-streaming connectivity with space $O(\min\{V+E, V \log V \log(2+E/V)\})$ w.h.p., simultaneously matching the lossless bound on sparse graphs, the sketching bound on dense graphs, and improving on both in an intermediate regime. A key component is BalloonSketch, a new l0-sampler reducing per-vertex sketch sizes by up to 8x. We implement HybridSCALE, a modular system treating the lossless and sketch-based components as subroutines. HybridSCALE is the first sketch-based dynamic connectivity system to save space on common real-world graphs. Compared to the state-of-the-art lossless baseline, HybridSCALE saves up to 15% space on sparse graphs (average degree < 100), up to 92% on intermediate density graphs (average degree ~ 100-1000), and up to 97% on dense graphs (average degree > 1000).

翻译：动态连通性是一个基础的动态图问题，近年来动态图草图算法在理论上取得了重大突破：通过将图编码为每个顶点的线性草图，这些算法仅需$Θ(V \log^2 V)$空间即可解决动态连通性问题，且空间消耗与边数无关，性能优于随图密度增长而增大的无损$Θ(V+E)$空间结构。在此工作之前，尚未有实用的动态连通性算法能将上述理论突破转化为真实世界图上的空间节省。主要障碍在于每个顶点的草图需要数千字节存储，因此仅当图变得极为稠密时草图方法才能体现优势。我们观察到稀疏的真实世界图往往并非均匀稀疏，这些图在小规模顶点子集上可能包含稠密核心，而该核心占据了大部分边。我们通过混合图草方法来利用这一结构特性：仅对稠密核心进行草图编码，对稀疏外围部分采用无损存储。我们为全动态和半流式连通性设计了新的混合算法，其空间复杂度为$O(\min\{V+E, V \log V \log(2+E/V)\})$（以高概率成立），在稀疏图上匹配无损界，在稠密图上匹配草图界，并在中间密度区间同时优于两者。关键组件BalloonSketch是一种新型l0采样器，可将每个顶点的草图大小降低最多8倍。我们实现了模块化系统HybridSCALE，将无损和基于草图的组件作为子程序。HybridSCALE是首个能在常见真实世界图上节省空间的基于草图动态连通性系统。与当前最优的无损基线相比，HybridSCALE在稀疏图（平均度<100）上节省最多15%空间，在中等密度图（平均度约100-1000）上节省最多92%，在稠密图（平均度>1000）上节省最多97%。