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34
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 83 (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. ...
Diamond: A storage architecture for early discard in interactive search
, 2004
"... Permission is granted for noncommercial reproduction of the work for educational or research purposes. ..."
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Cited by 62 (19 self)
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Permission is granted for noncommercial reproduction of the work for educational or research purposes.
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 14 (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...
Inferring a possibility distribution from empirical data
 Fuzzy Sets and Systems
"... Several transformations from probabilities to possibilities have been proposed. In particular, Dubois and Prade’s procedure produces the most specific possibility distribution among the ones dominating a given probability distribution. In this paper, this method is generalized to the case where the ..."
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Cited by 11 (2 self)
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Several transformations from probabilities to possibilities have been proposed. In particular, Dubois and Prade’s procedure produces the most specific possibility distribution among the ones dominating a given probability distribution. In this paper, this method is generalized to the case where the probabilities are unknown, the only information being a data sample represented by a histogram. It is proposed to characterize the probabilities of the different classes by simultaneous confidence intervals with a given confidence level 1 − α. From this imprecise specification, a procedure for constructing a possibility distribution is described, insuring that the resulting possibility distribution will dominate the true probability distribution in at least 100(1 − α) % of the cases. Finally, a simple efficient algorithm is given which makes the computations tractable even if the number of classes is high.
Evolution on distributive lattices
 J Theor Biol
"... Abstract. We consider the directed evolution of a population after an intervention that has significantly altered the underlying fitness landscape. We model the space of genotypes as a distributive lattice; the fitness landscape is a realvalued function on that lattice. The risk of escape from inte ..."
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Cited by 10 (5 self)
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Abstract. We consider the directed evolution of a population after an intervention that has significantly altered the underlying fitness landscape. We model the space of genotypes as a distributive lattice; the fitness landscape is a realvalued function on that lattice. The risk of escape from intervention, i.e., the probability that the population develops an escape mutant before extinction, is encoded in the risk polynomial. Tools from algebraic combinatorics are applied to compute the risk polynomial in terms of the fitness landscape. In an application to the development of drug resistance in HIV, we study the risk of viral escape from treatment with the protease inhibitors ritonavir and indinavir.
A Gray Code for Necklaces of Fixed Density
 SIAM J. Discrete Math
, 1997
"... A necklace is an equivalence class of binary strings under rotation. In this paper, we present a Gray code listing of all nbit necklaces with d ones so that (i) each necklace is listed exactly once by a representative from its equivalence class and (ii) successive representatives, including the las ..."
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Cited by 7 (0 self)
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A necklace is an equivalence class of binary strings under rotation. In this paper, we present a Gray code listing of all nbit necklaces with d ones so that (i) each necklace is listed exactly once by a representative from its equivalence class and (ii) successive representatives, including the last and the first in the list, differ only by the transposition of two bits. The total time required is O(nN (n; d)), where N (n; d) denotes the number of nbit binary necklaces with d ones. This is the first algorithm for generating necklaces of fixed density which is known to achieve this time bound. 1 Introduction In a combinatorial family, a Gray code is an exhaustive listing of the objects in the family so that successive objects differ only in a small way [Wil]. The classic example is the binary reflected Gray code [Gra], which is a list of all nbit binary strings in which each string differs from its successor in exactly one bit. By applying the binary Gray code, a variety of problems...
Generating and characterizing the perfect elimination orderings of a chordal graph
, 2003
"... WedevelV a constant time transposition"oraclo for the set of perfectelectdVU orderings of chordal graphs. Using thisoracl; we can generate a Gray code ofal perfectelectd;; orderings in constant amortized time using knownresul: about antimatroids. Using clngd trees, we show how theinitialdW:A: o ..."
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Cited by 6 (2 self)
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WedevelV a constant time transposition"oraclo for the set of perfectelectdVU orderings of chordal graphs. Using thisoracl; we can generate a Gray code ofal perfectelectd;; orderings in constant amortized time using knownresul: about antimatroids. Using clngd trees, we show how theinitialdW:A: of theald:::Eq can be performedinlerfo time. Weal develB two new characterizations of perfectelectd::qq orderings: one in terms ofchordlqd paths, and the other in terms uv separators.
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 4 (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
Gray Code Results for Acyclic Orientations
 Congressus Numerantium
, 1993
"... Given a graph G, the acyclic orientation graph of G, denoted AO(G), is the graph whose vertices are the acyclic orientations of G, and two acyclic orientations are joined by an edge in AO(G) iff they differ by the reversal of a single edge. A hamilton cycle in AO(G) gives a Gray code listing of t ..."
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Cited by 3 (1 self)
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Given a graph G, the acyclic orientation graph of G, denoted AO(G), is the graph whose vertices are the acyclic orientations of G, and two acyclic orientations are joined by an edge in AO(G) iff they differ by the reversal of a single edge. A hamilton cycle in AO(G) gives a Gray code listing of the acyclic orientations of G. We prove that for certain graphs G, AO(G) is hamiltonian, and give explicit constructions of hamilton cycles or paths. This work includes Gray codes for listing the acyclic orientations of trees, complete graphs, odd cycles, chordal graphs, odd ladder graphs, and odd wheel graphs. We also give examples of graphs whose acyclic orientation graph is not hamiltonian. We show that the acyclic orientations of even cycles, some complete bipartite graphs, even ladder graphs, and even wheel graphs cannot be listed by the defined Gray code. 1 Introduction A Gray code for a set S is a listing of the elements of S such that successive elements differ by a small am...
Ant colony optimizations for resource and timingconstrained operation scheduling,” ComputerAided Design of Integrated Circuits and Systems
 IEEE Transactions
, 2007
"... Abstract—Operation scheduling (OS) is a fundamental problem in mapping an application to a computational device. It takes a behavioral application specification and produces a schedule to minimize either the completion time or the computing resources required to meet a given deadline. The OS problem ..."
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Cited by 3 (1 self)
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Abstract—Operation scheduling (OS) is a fundamental problem in mapping an application to a computational device. It takes a behavioral application specification and produces a schedule to minimize either the completion time or the computing resources required to meet a given deadline. The OS problem is NPhard; thus, effective heuristic methods are necessary to provide qualitative solutions. We present novel OS algorithms using the ant colony optimization approach for both timingconstrained scheduling (TCS) and resourceconstrained scheduling (RCS) problems. The algorithms use a unique hybrid approach by combining the MAX–MIN ant system metaheuristic with traditional scheduling heuristics. We compiled a comprehensive testing benchmark set from realworld applications in order to verify the effectiveness and efficiency of our proposed algorithms. For TCS, our algorithm achieves better results compared with forcedirected scheduling on almost all the testing cases with a maximum 19.5 % reduction of the number of resources. For RCS, our algorithm outperforms a number of different listscheduling heuristics with better stability and generates better results with up to 14.7 % improvement. Our algorithms outperform the simulated annealing method for both scheduling problems in terms of quality, computing time, and stability. Index Terms—Forcedirected scheduling (FDS), list scheduling, operation scheduling (OS), MAX–MIN ant system (MMAS). I.