Williams, Aaron Michael2009-12-112009-12-1120092009-12-11http://hdl.handle.net/1828/1966Combinatorial 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).enAvailable to the World Wide Webshorthand universal cyclescombinatorial generationminimal-change orderloopless algorithmefficient algorithmcombinationsmultiset permutationsbalanced parenthesesDyck wordsCatalan pathsSchroder pathsMotzkin wordslinear-extensionsposetsconnected unit interval graphsinversionsbinary treesk-ary treesordered trees with fixed branching sequenceLyndon wordspre-necklacestheoretical computer sciencediscrete mathematicscombinatoricsbrute forcsde Bruijn cyclesbubble languagescool-lex orderlexicographic ordercombinatorial enumerationstacker-crane problemtraveling salesman problemmiddle levelsfixed-density de Bruijn cyclefixed-contentUVic Subject Index::Sciences and Engineering::Applied Sciences::Computer scienceUVic Subject Index::Sciences and Engineering::MathematicsThesisA. Williams. Loopless generation of multiset permutations using a constant number of variables by pre x shifts. In SODA '09: The Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, New York, New York, USA, 2009.F. Ruskey and A. Williams. The coolest way to generate combinations. Discrete Mathematics, 309(17):5305{5320, September 2009.F. Ruskey and A. Williams. Generating balanced parentheses and binary trees by pre x shifts. In CATS '08: Fourteenth Computing: The Australasian Theory Symposium, volume 77 of CRPIT, Wollongong, Australia, 2008. ACS.F. Ruskey and A. Williams. An explicit universal cycle for the (n − 1)- permutations of an n-set. ACM Transactions on Algorithms, (accepted), 2008.F. Ruskey and A. Williams. Generating combinations by pre x shifts. In COCOON '05: Computing and Combinatorics, 11th Annual International Conference, volume 3595 of Lecture Notes in Computer Science, pages 570{576, Kunming, China, 2005. Springer-Verlag.