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12
A Survey of Combinatorial Gray Codes
 SIAM Review
, 1996
"... The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing nbit binary numbers so that ..."
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Cited by 132 (2 self)
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The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing nbit binary numbers so that successive numbers differ in exactly one bit position, as well as work in the 1960's and 70's on minimal change listings for other combinatorial families, including permutations and combinations. The area of combinatorial Gray codes was popularized by Herbert Wilf in his invited address at the SIAM Discrete Mathematics Conference in 1988 and his subsequent SIAM monograph in which he posed some open problems and variations on the theme. This resulted in much recent activity in the area and most of the problems posed by Wilf are now solved. In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems. ...
The Posterior Probability of Bayes Nets with Strong Dependences
 Soft Computing
, 1999
"... Stochastic independence is an idealized relationship located at one end of a continuum of values measuring degrees of dependence. Modeling real world systems, we are often not interested in the distinction between exact independence and any degree of dependence, but between weak ignorable and strong ..."
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Cited by 17 (1 self)
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Stochastic independence is an idealized relationship located at one end of a continuum of values measuring degrees of dependence. Modeling real world systems, we are often not interested in the distinction between exact independence and any degree of dependence, but between weak ignorable and strong substantial dependence. Good models map significant deviance from independence and neglect approximate independence or dependence weaker than a noise threshold. This intuition is applied to learning the structure of Bayes nets from data. We determine the conditional posterior probabilities of structures given that the degree of dependence at each of their nodes exceeds a critical noise level. Deviance from independence is measured by mutual information. Arc probabilities are determined by the amount of mutual information the neighbors contribute to a node, is greater than a critical minimum deviance from independence. A Ø 2 approximation for the probability density function of mutual info...
Shift Gray Codes
, 2009
"... 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 = s1s2⋯sn, the rightshift operation ��→ shift(s, i, j) replaces the substring sisi+1⋯sj by si+1⋯sjsi ..."
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Cited by 7 (4 self)
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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 = s1s2⋯sn, the rightshift operation ��→ shift(s, i, j) replaces the substring sisi+1⋯sj by si+1⋯sjsi. In other words, si is rightshifted into position j by applying the permutation (j j −1 ⋯ i) to the indices of s. Rightshifts include prefixshifts (i = 1) and adjacenttranspositions (j = i + 1). A fixedcontent language is a set of strings that contain the same multiset of symbols. Given a fixedcontent 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 prefixshift Gray code for multiset permutations is discovered, and it provides the first algorithm for generating multiset permutations in O(1)time while
BINARY BUBBLE LANGUAGES AND COOLLEX ORDER
"... A bubble language is a set of binary strings with a simple closure property: The first 01 of any string can be replaced by 10 to obtain another string in the set. Natural representations of many combinatorial objects are bubble languages. Examples include binary string representations of kary trees ..."
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A bubble language is a set of binary strings with a simple closure property: The first 01 of any string can be replaced by 10 to obtain another string in the set. Natural representations of many combinatorial objects are bubble languages. Examples include binary string representations of kary trees, unit interval graphs, linearextensions of Bposets, binary necklaces and Lyndon words, and feasible solutions to knapsack problems. In colexicographic order, fixeddensity binary strings are ordered so that their suffixes of the form 10i occur (recursively) in the order i = max,max −1,...,min +1,min for some values of max and min. In coollex order the suffixes occur (recursively) in the order max −1,..., min+1, min, max. This small change has significant consequences. We prove that the strings in any bubble language appear in a Gray code order when listed in coollex order. This Gray code may be viewed from two different perspectives. On the one hand, successive binary strings differ by one or two transpositions, and on the other hand, they differ by a shift of some substring one position to the right. This article also provides the theoretical foundation for many efficient generation algorithms, as well as the first
Loopless Generation of Schröder Trees
, 2003
"... ... This paper presents the first loopless algorithms for directly generating Schröder tree representations. They use a new loopless algorithm for generating kcompositions of n in inverse lexicographic order ..."
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... This paper presents the first loopless algorithms for directly generating Schröder tree representations. They use a new loopless algorithm for generating kcompositions of n in inverse lexicographic order
Fusing Loopless Algorithms for Combinatorial Generation
"... 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 ..."
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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 divideandconquer 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.
Ranking and Loopless Generation of kary Dyck Words in Coollex Order
"... Abstract. A binary string B of length n = kt is a kary Dyck word if it contains t copies of 1, and the number of 0s in every prefix of B is at most k−1 times the number of 1s. We provide two loopless algorithms for generating kary Dyck words in coollex order: (1) The first requires two index vari ..."
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Abstract. A binary string B of length n = kt is a kary Dyck word if it contains t copies of 1, and the number of 0s in every prefix of B is at most k−1 times the number of 1s. We provide two loopless algorithms for generating kary Dyck words in coollex order: (1) The first requires two index variables and assumes k is a constant; (2) The second requires t index variables and works for any k. We also efficiently rank kary Dyck words in coollex order. Our results generalize the “coolCat ” algorithm by Ruskey and Williams (Generating balanced parentheses and binary trees by prefix shifts in CATS 2008) and provide the first loopless and ranking applications of the general coollex Gray code by Ruskey, Sawada, and Williams (Binary bubble languages and coollex order under review). 1
Covering Linear Orders with Posets
"... Much research has been done on the combinatorial problem of generating the linear extensions of a given poset. This paper focuses on the reverse of that problem, where the input is a set of linear orders, and the goal is to construct a poset or set of posets that generates the input. Such a problem ..."
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Much research has been done on the combinatorial problem of generating the linear extensions of a given poset. This paper focuses on the reverse of that problem, where the input is a set of linear orders, and the goal is to construct a poset or set of posets that generates the input. Such a problem finds applications in computational neuroscience, systems biology, paleontology, and physical plant engineering. In this paper, several algorithms are presented for efficiently finding a single poset that generates the input set of linear orders. Several variations of the problem are addressed. Algorithms are presented for constructing posets whose set of linear extensions is a subset of the input. Finally, it is shown that the problem of finding the minimum set of posets that cover the input is polynomially solvable for one class of simple posets (kite(2)posets) but NPcomplete for a related class (hammock(2,2,2)posets).
Mining Posets from Linear Orders
"... There has been much research on the combinatorial problem of generating the linear extensions of a given poset. This paper focuses on the reverse of that problem, where the input is a set of linear orders, and the goal is to construct a poset or set of posets that generates the input. Such a problem ..."
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There has been much research on the combinatorial problem of generating the linear extensions of a given poset. This paper focuses on the reverse of that problem, where the input is a set of linear orders, and the goal is to construct a poset or set of posets that generates the input. Such a problem finds applications in computational neuroscience, systems biology, paleontology, and physical plant engineering. In this paper, two algorithms are presented for efficiently finding a single poset, if such a poset exists, whose linear extensions are exactly the same as the input set of linear orders. The variation of the problem where a minimum set of posets that cover the input is also explored. This variation is shown to be polynomially solvable for one class of simple posets (kite(2) posets) but NPcomplete for a related class (hammock(2,2,2) posets).