Hybrid Sketching Methods for Dynamic Connectivity on Sparse Graphs

A digest of the paper by De Man, Gill, Bender, Dhulipala, and Tench (2026)

Quinten De Man; Gilvir Gill; Michael A. Bender; Laxman Dhulipala; David Tench· 2026· DOI 10.48550/arXiv.2605.15173

The core problem

Dynamic connectivity is a fundamental problem in dynamic graph algorithms, with applications in network analysis, social networks, and databases. Recent breakthroughs in dynamic graph sketching have shown that by encoding the graph as per-vertex linear sketches, dynamic connectivity can be solved in only space, independent of the number of edges . This outperforms lossless -space structures as graphs become denser. However, 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. The authors observe that sparse real-world graphs are often not uniformly sparse; they can contain dense cores on a small subset of vertices that account for a large fraction of edges. This motivates the hybrid sketching approach: sketch only the dense core, and store the sparse periphery losslessly.

The authors evaluate HybridSCALE against state-of-the-art lossless baselines. The results show significant space savings: up to 15% 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). These savings are achieved without compromising query performance. The experiments were conducted on a variety of real-world graphs, demonstrating the practical applicability of the hybrid sketching approach. The BalloonSketch component alone reduces per-vertex sketch sizes by up to 8x, contributing to the overall space efficiency. The following table summarizes the space savings: | Graph Type | Average Degree | Space Savings | |------------|----------------|---------------| | Sparse | < 100 | up to 15% | | Intermediate | ~100-1000 | up to 92% | | Dense | > 1000 | up to 97% | These results confirm that hybrid sketching effectively translates theoretical advances into practical space savings.

Dynamic connectivity is a fundamental problem in dynamic graph algorithms, with applications in network analysis, social networks, and databases. Recent breakthroughs in dynamic graph sketching have shown that by encoding the graph as per-vertex linear sketches, dynamic connectivity can be solved in only space, independent of the number of edges . This outperforms lossless -space structures as graphs become denser. However, 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. The authors observe that sparse real-world graphs are often not uniformly sparse; they can contain dense cores on a small subset of vertices that account for a large fraction of edges. This motivates the hybrid sketching approach: sketch only the dense core, and store the sparse periphery losslessly.

The authors design new hybrid algorithms for fully-dynamic and semi-streaming connectivity with space O (min {V + E, V lo g V lo g (2 + E / V)}) with high probability. This simultaneously matches the lossless bound on sparse graphs, the sketching bound on dense graphs, and improves on both in an intermediate regime. A key component is BalloonSketch, a new ℓ_{0} -sampler that reduces per-vertex sketch sizes by up to 8x. The hybrid approach partitions the graph into a dense core and a sparse periphery. The dense core is sketched using linear sketches, while the sparse periphery is stored losslessly. The algorithm dynamically adapts to the graph's density, ensuring optimal space usage. The system, HybridSCALE, is implemented as a modular system treating the lossless and sketch-based components as subroutines. The architecture is illustrated below:

The hybrid sketching approach addresses the limitations of both lossless and sketch-based methods. By exploiting the non-uniform density of real-world graphs, it achieves space savings across a wide range of graph densities. The theoretical space bound O (min {V + E, V lo g V lo g (2 + E / V)}) demonstrates that the algorithm adapts to the graph's structure, providing optimal space usage. The BalloonSketch ℓ_{0} -sampler is a key innovation, reducing sketch sizes significantly. HybridSCALE is the first sketch-based dynamic connectivity system to save space on common real-world graphs. The modular design allows for easy integration with existing systems. Future work could explore further optimizations and applications to other dynamic graph problems. The authors note that the hybrid approach is particularly beneficial for graphs with dense cores, which are common in social networks and web graphs. This work bridges the gap between theory and practice in dynamic graph sketching.

CS practitioners and researchers

Opening member content…