{"paper":{"title":"A dynamic $(1+\\varepsilon)$-spanner for disk intersection graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Persistent data structures maintain a (1+ε)-spanner for dynamic disk intersection graphs with bounded diameters.","cross_cats":["cs.DS"],"primary_cat":"cs.CG","authors_text":"Eva Rotenberg, Ivor van der Hoog, Johanne M. Vistisen, Sampson Wong, Sarita de Berg","submitted_at":"2026-04-28T09:07:25Z","abstract_excerpt":"We maintain a $(1+\\varepsilon)$-spanner over the disk intersection graph of a dynamic set of disks. We restrict all disks to have their diameter in $[4,\\Psi]$ for some fixed and known $\\Psi$. The resulting $(1+\\varepsilon)$-spanner has size $O(n \\varepsilon^{-2} \\log \\Psi \\log (\\varepsilon^{-1}))$, where $n$ is the present number of disks.\n  We develop a novel use of persistent data structures to dynamically maintain our $(1+\\varepsilon)$-spanner. Our approach requires $O(\\varepsilon^{-2} n \\log^4 n \\log \\Psi)$ space and has an $O( \\left( \\frac{\\Psi}{\\varepsilon} \\right)^2 \\log^4 n \\log^2 \\Psi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We maintain a (1+ε)-spanner over the disk intersection graph of a dynamic set of disks with diameters in [4, Ψ]. The spanner has size O(n ε^{-2} log Ψ log(ε^{-1})), uses O(ε^{-2} n log^4 n log Ψ) space, and supports updates in O((Ψ/ε)^2 log^4 n log^2 Ψ log^2(ε^{-1})) expected amortized time. For constant ε and Ψ the bounds become near-linear size, near-linear space, and polylogarithmic updates. The same spanner supports connectivity queries.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"All disks have diameters restricted to the fixed known interval [4, Ψ]; without this bounded-diameter assumption the size and update bounds no longer hold and the persistent-structure approach may not apply directly.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A dynamic (1+ε)-spanner of size O(n ε^{-2} log Ψ log(ε^{-1})) with O((Ψ/ε)^2 log^4 n log^2 Ψ log^2(ε^{-1})) expected amortized update time for disk intersection graphs with bounded diameters.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Persistent data structures maintain a (1+ε)-spanner for dynamic disk intersection graphs with bounded diameters.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"af50a833922500faa7988ec527bfbcb3785d93942c8355e21fee7d4ef7a174f8"},"source":{"id":"2604.25397","kind":"arxiv","version":1},"verdict":{"id":"e059a77a-5dc2-421b-95ee-b9d06aca6d18","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T13:48:27.312915Z","strongest_claim":"We maintain a (1+ε)-spanner over the disk intersection graph of a dynamic set of disks with diameters in [4, Ψ]. The spanner has size O(n ε^{-2} log Ψ log(ε^{-1})), uses O(ε^{-2} n log^4 n log Ψ) space, and supports updates in O((Ψ/ε)^2 log^4 n log^2 Ψ log^2(ε^{-1})) expected amortized time. For constant ε and Ψ the bounds become near-linear size, near-linear space, and polylogarithmic updates. The same spanner supports connectivity queries.","one_line_summary":"A dynamic (1+ε)-spanner of size O(n ε^{-2} log Ψ log(ε^{-1})) with O((Ψ/ε)^2 log^4 n log^2 Ψ log^2(ε^{-1})) expected amortized update time for disk intersection graphs with bounded diameters.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"All disks have diameters restricted to the fixed known interval [4, Ψ]; without this bounded-diameter assumption the size and update bounds no longer hold and the persistent-structure approach may not apply directly.","pith_extraction_headline":"Persistent data structures maintain a (1+ε)-spanner for dynamic disk intersection graphs with bounded diameters."},"integrity":{"clean":false,"summary":{"advisory":1,"critical":0,"by_detector":{"doi_compliance":{"total":1,"advisory":1,"critical":0,"informational":0}},"informational":0},"endpoint":"/pith/2604.25397/integrity.json","findings":[{"note":"DOI in the printed bibliography is fragmented by whitespace or line breaks. A longer candidate (10.1016/J.COMGEO.2022.101979.48) was visible in the surrounding text but could not be confirmed against doi.org as printed.","detector":"doi_compliance","severity":"advisory","ref_index":14,"audited_at":"2026-05-19T21:12:15.013657Z","detected_doi":"10.1016/J.COMGEO.2022.101979.48","finding_type":"recoverable_identifier","verdict_class":"incontrovertible","detected_arxiv_id":null}],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T21:12:15.013657Z","status":"completed","version":"1.0.0","findings_count":1}],"snapshot_sha256":"c76144c57fc1b6f7a82e85537dfa06f7242cbcff618859fcc00dd3af2fbcc2fc"},"references":{"count":17,"sample":[{"doi":"10.4230/lipics.socg.2015.186","year":2015,"title":"Geometric spanners for points inside a polygonal domain","work_id":"0ededbb4-271e-423c-9d04-3ff5162f4961","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1109/sfcs.1998.743504","year":1998,"title":"4 Ingo Althöfer, Gautam Das, David P","work_id":"761aec79-405c-4087-82b5-4f9db1202504","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/bf02189308","year":2025,"title":"5 Shinwoo An, Eunjin Oh, and Jie Xue","work_id":"a59d4e23-f6fc-4acd-9015-9584ea95565e","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1137/19m1246493","year":null,"title":"34 A dynamic(1 +ε)-spanner for disk intersection graphs 21 Timothy M","work_id":"2aa90d24-5a25-4e4e-bea4-75e42b3ffacb","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.4230/lipics.socg.2023.23","year":2023,"title":"doi:10.4230/LIPIcs.SoCG.2023.23","work_id":"d4148f61-83b1-470d-98b3-989791f3c829","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":17,"snapshot_sha256":"8ad569cbdf07861b71cad2fc81e87c588c79e8c3b5c08d53088e303d0da2e547","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}