Shift gray codes

• Main Author: Williams, Aaron Michael
• Other Authors: Ruskey, Frank
• Language: English; en
• Published: 2009
• Subjects: shorthand universal cycles; combinatorial generation; minimal-change order; loopless algorithm; efficient algorithm; combinations; multiset permutations; balanced parentheses; Dyck words; Catalan paths; Schroder paths; Motzkin words; linear-extensions; posets; connected unit interval graphs; inversions; binary trees; k-ary trees; ordered trees with fixed branching sequence; Lyndon words; pre-necklaces; theoretical computer science; discrete mathematics; combinatorics; brute forcs; de Bruijn cycles; bubble languages; cool-lex order; lexicographic order; combinatorial enumeration; stacker-crane problem; traveling salesman problem; middle levels; fixed-density de Bruijn cycle; fixed-content; UVic Subject Index::Sciences and Engineering::Applied Sciences::Computer science; UVic Subject Index::Sciences and Engineering::Mathematics
• Online Access: http://hdl.handle.net/1828/1966

Combinatorial objects can be represented by strings, such as 21534 for the permutation (1 2) (3 5 4), or 110100 for the binary tree corresponding to the balanced parentheses (()()). Given a string s = s1 s2 sn, the right-shift operation shift(s, i, j) replaces the substring si si+1..sj by si+1..sj si. In other words, si is right-shifted into position j by applying the permutation (j j−1 .. i) to the indices of s. Right-shifts include prefix-shifts (i = 1) and adjacent-transpositions (j = i+1). A fixed-content language is a set of strings that contain the same multiset of symbols. Given a fixed-content language, a shift Gray code is a list of its strings where consecutive strings differ by a shift. This thesis asks if shift Gray codes exist for a variety of combinatorial objects. This abstract question leads to a number of practical answers. The first prefix-shift Gray code for multiset permutations is discovered, and it provides the first algorithm for generating multiset permutations in O(1)-time while using O(1) additional variables. Applications of these results include more efficient exhaustive solutions to stacker-crane problems, which are natural NP-complete traveling salesman variants. This thesis also produces the fastest algorithm for generating balanced parentheses in an array, and the first minimal-change order for fixed-content necklaces and Lyndon words. These results are consequences of the following theorem: Every bubble language has a right-shift Gray code. Bubble languages are fixed-content languages that are closed under certain adjacent-transpositions. These languages generalize classic combinatorial objects: k-ary trees, ordered trees with fixed branching sequences, unit interval graphs, restricted Schr oder and Motzkin paths, linear-extensions of B-posets, and their unions, intersections, and quotients. Each Gray code is circular and is obtained from a new variation of lexicographic order known as cool-lex order. Gray codes using only shift(s, 1, n) and shift(s, 1, n−1) are also found for multiset permutations. A universal cycle that omits the last (redundant) symbol from each permutation is obtained by recording the first symbol of each permutation in this Gray code. As a special case, these shorthand universal cycles provide a new fixed-density analogue to de Bruijn cycles, and the first universal cycle for the "middle levels" (binary strings of length 2k + 1 with sum k or k + 1).