Global ETD Search

1	Coroutine-based combinatorial generation Saba, Sahand 29 January 2015 (has links) The two well-known approaches to designing combinatorial generation algorithms are the recursive approach and the iterative approach. In this thesis a third design approach using coroutines, introduced by Knuth and Ruskey, is explored further. An introduction to coroutines and their implementation in modern languages (in particular Python) is provided, and the coroutine-based approach is introduced using an example, and contrasted with the recursive and iterative approaches. The coroutine sum, coroutine product, and coroutine symmetric sum constructs are defined to create an algebra of coroutines, and used to give concise definitions of coroutine-based algorithms for generating ideals of chain and forest posets. Afterwards, new coroutine-based variations of several algorithms, including the Steinhaus-Johnson-Trotter algorithm for generating permutations in Gray order, the Varol-Rotem algorithm for generating linear extensions in Gray order, and the Pruesse-Ruskey algorithm for generating signed linear extensions of a poset in Gray order, are given. / Graduate / 0984 / saba@uvic.ca combinatorial generation coroutines python
2	Fusing Loopless Algorithms for Combinatorial Generation Violich, Stephen Scott January 2006 (has links) Loopless algorithms are an interesting challenge in the field of combinatorial generation. These algorithms must generate each combinatorial object from its predecessor in no more than a constant number of instructions, thus achieving theoretically minimal time complexity. This constraint rules out powerful programming techniques such as iteration and recursion, which makes loopless algorithms harder to develop and less intuitive than other algorithms. This thesis discusses a divide-and-conquer approach by which loopless algorithms can be developed more easily and intuitively: fusing loopless algorithms. If a combinatorial generation problem can be divided into subproblems, it may be possible to conquer it looplessly by fusing loopless algorithms for its subproblems. A key advantage of this approach is that is allows existing loopless algorithms to be reused. This approach is not novel, but it has not been generalised before. This thesis presents a general framework for fusing loopless algorithms, and discusses its implications. It then applies this approach to two combinatorial generation problems and presents two new loopless algorithms. The first new algorithm, MIXPAR, looplessly generates well-formed parenthesis strings comprising two types of parentheses. It is the first loopless algorithm for generating these objects. The second new algorithm, MULTPERM, generates multiset permutations in linear space using only arrays, a benchmark recently set by Korsh and LaFollette (2004). Algorithm MULTPERM is evaluated against Korsh and LaFollette's algorithm, and shown to be simpler and more efficient in both space and time. loopless algorithm algorithms combinatorial generation
3	Lace tessellations: a mathematical model for bobbin lace and an exhaustive combinatorial search for patterns Irvine, Veronika 29 August 2016 (has links) Bobbin lace is a 500-year-old art form in which threads are braided together in an alternating manner to produce a lace fabric. A key component in its construction is a small pattern, called a bobbin lace ground, that can be repeated periodically to fill a region of any size. In this thesis we present a mathematical model for bobbin lace grounds representing the structure as the pair (Δ(G), ζ (v)) where Δ(G) is a topological embedding of a 2-regular digraph, G, on a torus and ζ(v) is a mapping from the vertices of G to a set of braid words. We explore in depth the properties that Δ(G) must possess in order to produce workable lace patterns. Having developed a solid, logical foundation for bobbin lace grounds, we enumerate and exhaustively generate patterns that conform to that model. We start by specifying an equivalence relation and define what makes a pattern prime so that we can identify unique representatives. We then prove that there are an infinite number of prime workable patterns. One of the key properties identified in the model is that it must be possible to partition Δ(G) into a set of osculating circuits such that each circuit has a wrapping index of (1,0); that is, the circuit wraps once around the meridian of the torus and does not wrap around the longitude. We use this property to exhaustively generate workable patterns for increasing numbers of vertices in G by gluing together lattice paths in an osculating manner. Using a backtracking algorithm to process the lattice paths, we identify over 5 million distinct prime patterns. This is well in excess of the roughly 1,000 found in lace ground catalogues. The lattice paths used in our approach are members of a family of partially directed lattice paths that have not been previously reported. We explore these paths in detail, develop a recurrence relation and generating function for their enumeration and present a bijection between these paths and a subset of Motzkin paths. Finally, to draw out of the extremely large number of patterns some of the more aesthetically interesting cases for lacemakers to work on, we look for examples that have a high degree of symmetry. We demonstrate, by computational generation, that there are lace ground representatives from each of the 17 planar periodic symmetry groups. / Graduate / 0389 / 0984 / 0405 / veronikairvine@gmail.com exhaustive combinatorial generation lattice path bobbin lace graph theory planar symmetry tessellation computational lace alternating braid
4	Graph Distinguishability and the Generation of Non-Isomorphic Labellings Bird, William Herbert 26 August 2013 (has links) A distinguishing colouring of a graph G is a labelling of the vertices of G with colours such that no non-trivial automorphism of G preserves all colours. The distinguishing number of G is the minimum number of colours in a distinguishing colouring. This thesis presents a survey of the history of distinguishing colouring problems and proves new bounds and computational results about distinguishability. An algorithm to generate all labellings of a graph up to isomorphism is presented and compared to a previously published algorithm. The new algorithm is shown to have performance competitive with the existing algorithm, as well as being able to process automorphism groups far larger than the previous limit. A specialization of the algorithm is used to generate all minimal distinguishing colourings of a set of graphs with large automorphism groups and compute their distinguishing numbers. / Graduate / 0984 / 0405 / bbird@uvic.ca Distinguishing Number Sims Table Schreier-Sims Algorithm Computational Group Theory Graph Algorithms List-Distinguishing Colouring Distinguishing Colouring List-Distinguishing Number Combinatorial Generation Permutation Groups
5	Monomino-Domino Tatami Coverings Erickson, Alejandro 03 September 2013 (has links) We present several new results on the combinatorial properties of a locally restricted version of monomino-domino coverings of rectilinear regions. These are monomino-domino tatami coverings, and the restriction is that no four tiles may meet at any point. The global structure that the tatami restriction induces has numerous implications, and provides a powerful tool for solving enumeration problems on tatami coverings. Among these we address the enumeration of coverings of rectangles, with various parameters, and we develop algorithms for exhaustive generation of coverings, in constant amortised time per covering. We also con- sider computational complexity on two fronts; firstly, the structure shows that the space required to store a covering of the rectangle is linear in its longest dimension, and secondly, it is NP-complete to decide whether an arbitrary polyomino can be tatami-covered only with dominoes. / Graduate / 0984 / 0405 / alejandro.erickson@gmail.com combinatorics cstheory math.CO cs.CC computer science theory mathematics tiling covering tatami restriction monomino domino enumeration combinatorial algorithms combinatorial generation computational complexity
6	Shift gray codes Williams, 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). 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 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

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