{"paper":{"title":"Constructions of asymptotically shortest k-radius sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jerzy W. Jaromczyk, Miroslaw Truszczynski, Zbigniew Lonc","submitted_at":"2011-05-03T19:11:54Z","abstract_excerpt":"Let k be a positive integer. A sequence s over an n-element alphabet A is called a k-radius sequence if every two symbols from A occur in s at distance of at most k. Let f_k(n) denote the length of a shortest k-radius sequence over A. We provide constructions demonstrating that (1) for every fixed k and for every fixed e>0, f_k(n) = n^2/(2k) +O(n^(1+e)) and (2) for every k, where k is the integer part of n^a for some fixed real a such that 0 < a <1, f_k(n) = n^2/(2k) +O(n^b), for some b <2-a. Since f_k(n) >= n^2/(2k) - n/(2k), the constructions give asymptotically optimal k-radius sequences. F"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.0654","kind":"arxiv","version":2},"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"}