{"id":339992,"date":"2016-12-21T11:30:54","date_gmt":"2016-12-21T19:30:54","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&#038;p=339992"},"modified":"2018-10-16T21:19:28","modified_gmt":"2018-10-17T04:19:28","slug":"shorthand-universal-cycles-permutations","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/shorthand-universal-cycles-permutations\/","title":{"rendered":"Shorthand Universal Cycles for Permutations"},"content":{"rendered":"<p>The set of permutations of <<em>n<\/em>> = {1,&#8230;,<em>n<\/em>} in one-line notation is \u03a0(<em>n<\/em>). The shorthand encoding of a<sub>1<\/sub> \u00b7\u00b7\u00b7a<sub>n<\/sub> \u2208 \u03a0(<em>n<\/em>) is a<sub>1<\/sub> \u00b7\u00b7\u00b7a<sub>n\u22121<\/sub>. A shorthand universal cycle for permutations (SP-cycle) is a circular string of length <em>n<\/em>! whose substrings of length <em>n<\/em>\u22121 are the shorthand encodings of \u03a0(<em>n<\/em>). When an SP-cycle is decoded, the order of \u03a0(<em>n<\/em>) is a Gray code in which successive permutations differ by the prefix-rotation \u03c3i = (1 2 <em>cdots<\/em> <em>i<\/em>) for i \u2208 {<em>n<\/em>\u22121,<em>n<\/em>}. Thus, SP-cycles can be represented by <em>n<\/em>! bits. We investigate SP-cycles with maximum and minimum \u2018weight\u2019 (number of \u03c3<sub><em>n<\/em>\u22121<\/sub>s in the Gray code). An SP-cycle <em>n<\/em><strong>a<\/strong><em>n<\/em><strong>b<\/strong>\u00b7\u00b7\u00b7<em>n<\/em><strong>z<\/strong> is \u2018periodic\u2019 if its \u2018sub-permutations\u2019 <strong>a<\/strong>,<strong>b<\/strong>,&#8230;,<strong>z<\/strong> equal \u03a0(<em>n<\/em>\u22121). We prove that periodic min-weight SP-cycles correspond to spanning trees of the (<em>n<\/em>\u22121)-permutohedron. We provide two constructions: B(<em>n<\/em>) and C(<em>n<\/em>). In B(<em>n<\/em>) the spanning trees use \u2018half-hunts\u2019 from bell-ringing, and in C(<em>n<\/em>) the subpermutations use cool-lex order by Williams (SODA (2009) 987-996). Algorithmic results are: 1) memoryless decoding of B(<em>n<\/em>) and C(<em>n<\/em>), 2) <em>O<\/em>((<em>n<\/em>\u22121)!)-time generation of B(<em>n<\/em>) and C(<em>n<\/em>) using sub-permutations, 3) loopless generation of B(<em>n<\/em>)\u2019s binary representation n bits at a time, and 4) O(<em>n<\/em> + \u03bd(<em>n<\/em>))-time ranking of B(<em>n<\/em>)\u2019s permutations where \u03bd(<em>n<\/em>) is the cost of computing a permutation\u2019s inversion vector. Results 1)-4) improve on those for the previous SP-cycle construction D(<em>n<\/em>) by Ruskey and Williams (ACM Transactions on Algorithms, Vol. 6 No. 3 Art. 45 (2010)), which we characterize here using \u2018recycling\u2019.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The set of permutations of = {1,&#8230;,n} in one-line notation is \u03a0(n). The shorthand encoding of a1 \u00b7\u00b7\u00b7an \u2208 \u03a0(n) is a1 \u00b7\u00b7\u00b7an\u22121. A shorthand universal cycle for permutations (SP-cycle) is a circular string of length n! whose substrings of length n\u22121 are the shorthand encodings of \u03a0(n). When an SP-cycle is decoded, the order [&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 Science+Business Media, 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