{"paper":{"title":"Parameterized dynamic data structure for Split Completion","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Anna Zych-Pawlewicz, Konrad Majewski, Micha{\\l} Pilipczuk","submitted_at":"2024-02-13T21:46:53Z","abstract_excerpt":"We design a randomized data structure that, for a fully dynamic graph $G$ updated by edge insertions and deletions and integers $k, d$ fixed upon initialization, maintains the answer to the Split Completion problem: whether one can add $k$ edges to $G$ to obtain a split graph. The data structure can be initialized on an edgeless $n$-vertex graph in time $n \\cdot (k d \\cdot \\log n)^{\\mathcal{O}(1)}$, and the amortized time complexity of an update is $5^k \\cdot (k d \\cdot \\log n)^{\\mathcal{O}(1)}$. The answer provided by the data structure is correct with probability $1-\\mathcal{O}(n^{-d})$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.08816","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2402.08816/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"}