{"paper":{"title":"An Optimal-Time Construction of Euclidean Sparse Spanners with Tiny Diameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.CG","authors_text":"Shay Solomon","submitted_at":"2010-05-22T23:30:32Z","abstract_excerpt":"In STOC'95 \\cite{ADMSS95} Arya et al.\\ showed that for any set of $n$ points in $\\mathbb R^d$, a $(1+\\epsilon)$-spanner with diameter at most 2 (respectively, 3) and $O(n \\log n)$ edges (resp., $O(n \\log \\log n)$ edges) can be built in $O(n \\log n)$ time. Moreover, it was shown in \\cite{ADMSS95,NS07} that for any $k \\ge 4$, one can build in $O(n (\\log n) 2^k \\alpha_k(n))$ time a $(1+\\epsilon)$-spanner with diameter at most $2k$ and $O(n 2^k \\alpha_k(n))$ edges. The function $\\alpha_k$ is the inverse of a certain function at the $\\lfloor k/2 \\rfloor$th level of the primitive recursive hierarchy"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.4155","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":""},"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"}