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Showing 1 to 2 of 2 (0.109 seconds)Spelling suggestions: "subject:"dde bruijn cycle"" "subject:"dde bruijn eycle""
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Universal Cycles for Some Combinatorial ObjectsCampbell, Andre A 01 May 2013 (has links) (PDF)
A de Bruijn cycle commonly referred to as a universal cycle (u-cycle), is a complete and compact listing of a collection of combinatorial objects. In this paper, we show the power of de Bruijn's original theorem, namely that the cycles bearing his name exist for n-letter words on a k-letter alphabet for all values of k,n, to prove that we can create de Bruijn cycles for multi-sets using natural encodings and M-Lipschitz n-letter words and the assignment of elements of [n]={1,2,...,n} to the sets in any labeled subposet of the Boolean lattice; de Bruijn's theorem corresponds to the case when the subposet in question consists of a single ground element. In this paper, we also show that de Bruijn's cycles exist for words with weight between s and t, where these parameters are suitably restricted.
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Shift gray codesWilliams, Aaron Michael 11 December 2009 (has links)
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).
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