{"paper":{"title":"Hybrid Sketching Methods for Dynamic Connectivity on Sparse Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Hybrid sketching algorithms achieve dynamic connectivity space that matches the best of lossless and sketch-based methods on all graph densities.","cross_cats":["cs.DB"],"primary_cat":"cs.DS","authors_text":"David Tench, Gilvir Gill, Laxman Dhulipala, Michael A. Bender, Quinten De Man","submitted_at":"2026-05-14T17:57:03Z","abstract_excerpt":"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 $\\Theta(V \\log^2 V)$ space, independent of the number of edges,outperforming lossless $\\Theta(V+E)$-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 "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Real-world sparse graphs contain dense cores on a small subset of vertices that account for a large fraction of edges, so that sketching only the core yields net space savings.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Hybrid sketching saves up to 97% space on dense graphs and 15% on sparse ones by sketching dense cores and storing sparse parts exactly, with new BalloonSketch reducing sketch sizes up to 8x.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Hybrid sketching algorithms achieve dynamic connectivity space that matches the best of lossless and sketch-based methods on all graph densities.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"842ecf3925a3c374969928580078d2620b696c64ac8921951d7ea2f25cc88611"},"source":{"id":"2605.15173","kind":"arxiv","version":1},"verdict":{"id":"e04909a5-a135-489d-a059-d0bdad82312e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:56:46.810947Z","strongest_claim":"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.","one_line_summary":"Hybrid sketching saves up to 97% space on dense graphs and 15% on sparse ones by sketching dense cores and storing sparse parts exactly, with new BalloonSketch reducing sketch sizes up to 8x.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Real-world sparse graphs contain dense cores on a small subset of vertices that account for a large fraction of edges, so that sketching only the core yields net space savings.","pith_extraction_headline":"Hybrid sketching algorithms achieve dynamic connectivity space that matches the best of lossless and sketch-based methods on all graph densities."},"references":{"count":66,"sample":[{"doi":"10.1145/3323165.3323196","year":2019,"title":"Acar, Daniel Anderson, Guy E","work_id":"aac71688-ed3e-43d8-9a69-d0b581c830b6","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.4230/lipics.esa.2020.2","year":2020,"title":"Acar, Daniel Anderson, Guy E","work_id":"00156186-ebd2-4163-8f6f-f696336c5c46","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. 2012. Analyzing graph structure via linear measurements. InProceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms. SIAM, 459–","work_id":"675c85e2-0a72-4b47-8531-ef30a02db4b5","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. 2012. Graph sketches: sparsification, spanners, and subgraphs. InPODS. 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