{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:HWCDQMKQEYF7PU2XXWR6YOYB3N","short_pith_number":"pith:HWCDQMKQ","schema_version":"1.0","canonical_sha256":"3d84383150260bf7d357bda3ec3b01db5606836b709eab9653de31cfc05cab24","source":{"kind":"arxiv","id":"1105.0654","version":2},"attestation_state":"computed","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. 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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. 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