Shorthand Universal Cycles for Permutations

• Published in: Algorithmica Vol. 64; no. 2; pp. 215 - 245
• Main Authors: Holroyd, Alexander E., Ruskey, Frank, Williams, Aaron
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.10.2012; Springer
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Applied sciences; Computer Science; Computer science; control theory; systems; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Exact sciences and technology; Information retrieval. Graph; Mathematics of Computing; Theoretical computing; Theory of Computation; Algorithms; Ucycles; Cayley graphs; Gray codes; Permutohedron; Polytope; Costs; Gray code; Hierarchical classification; Permutation; Algorithmics; Inversion; Decoding; Cayley graph; N order; Cayley algebra; Ranking; Character string; Coding; Spanning tree; Recycling
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-011-9544-z

The set of permutations of 〈 n 〉={1,…, n } in one-line notation is Π ( n ). The shorthand encoding of a 1 ⋯ a n ∈ Π ( n ) is a 1 ⋯ a n −1 . A shorthand universal cycle for permutations (SP-cycle) is a circular string of length n ! whose substrings of length n −1 are the shorthand encodings of Π ( n ). When an SP-cycle is decoded, the order of Π ( n ) is a Gray code in which successive permutations differ by the prefix-rotation σ i =(1 2 ⋯ i ) for i ∈{ n −1, n }. Thus, SP-cycles can be represented by n ! bits. We investigate SP-cycles with maximum and minimum ‘weight’ (number of σ n −1 s in the Gray code). An SP-cycle n a n b ⋯ n z is ‘periodic’ if its ‘sub-permutations’ a , b ,…, z equal Π ( n −1). We prove that periodic min-weight SP-cycles correspond to spanning trees of the ( n −1)-permutohedron. We provide two constructions: B( n ) and C( n ). In B( n ) the spanning trees use ‘half-hunts’ from bell-ringing, and in C( n ) the sub-permutations use cool-lex order by Williams (SODA, 987–996, 2009 ). Algorithmic results are: (1) memoryless decoding of B( n ) and C( n ), (2)  O (( n −1)!)-time generation of B( n ) and C( n ) using sub-permutations, (3) loopless generation of B( n )’s binary representation n bits at a time, and (4)  O ( n + ν ( n ))-time ranking of B( n )’s permutations where ν ( n ) is the cost of computing a permutation’s inversion vector. Results (1)–(4) improve on those for the previous SP-cycle construction D( n ) by Ruskey and Williams (ACM Trans. Algorithms 6(3):Art. 45, 2010 ), which we characterize here using ‘recycling’.