{"id":340259,"date":"2016-12-21T16:03:37","date_gmt":"2016-12-22T00:03:37","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&#038;p=340259"},"modified":"2018-10-16T20:31:16","modified_gmt":"2018-10-17T03:31:16","slug":"faster-generation-shorthand-universal-cycles-permutations","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/faster-generation-shorthand-universal-cycles-permutations\/","title":{"rendered":"Faster Generation of Shorthand Universal Cycles for Permutations"},"content":{"rendered":"<p>A universal cycle for the <em>k<\/em>-permutations of <<em>n<\/em>> = {1, 2, &#8230;, <em>n<\/em>} is a circular string of length (<em>n<\/em>)<sub>k<\/sub> that contains each k-permutation exactly once as a substring. Jackson (Discrete Mathematics, 149 (1996) 123\u2013129) proved their existence for all <em>k<\/em> \u2264 <em>n<\/em> \u2212 1. Knuth (<em>The Art of Computer Programming, Volume 4<\/em>, Fascicle 2, Addison-Wesley, 2005) pointed out the importance of the <em>k<\/em> = <em>n<\/em> \u2212 1 case, where each (<em>n<\/em> \u2212 1)- permutation is \u201cshorthand\u201d for exactly one permutation of <<em>n<\/em>>. RuskeyWilliams (ACM Transactions on Algorithms, in press) answered Knuth\u2019s request for an explicit construction of a shorthand universal cycle for permutations, and gave an algorithm that creates successive symbols in worst-case <em>O<\/em>(1)-time. This paper provides two new algorithmic constructions that create successive blocks of <em>n<\/em> symbols in <em>O<\/em>(1) amortized time within an array of length <em>n<\/em>. The constructions are based on: (a) an approach known to bell-ringers for over 300 years, and (b) the recent shift Gray code by Williams (SODA, (2009) 987-996). For (a), we show that the majority of changes between successive permutations are full rotations; asymptotically, the proportion of them is (<em>n<\/em> \u2212 2)\/<em>n<\/em>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A universal cycle for the k-permutations of = {1, 2, &#8230;, n} is a circular string of length (n)k that contains each k-permutation exactly once as a substring. Jackson (Discrete Mathematics, 149 (1996) 123\u2013129) proved their existence for all k \u2264 n \u2212 1. Knuth (The Art of Computer Programming, Volume 4, Fascicle 2, Addison-Wesley, [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Springer Berlin Heidelberg","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"Computing and Combinatorics, 16th Annual International Conference, COCOON 2010, Nha Trang, Vietnam, July 19-21, 2010. 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