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43
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 130 (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. ...
Optimal Partitioners and Endcase Placers for Standardcell Layout
 IEEE TRANS. ON CAD
, 2000
"... We study alternatives to classic FMbased partitioning algorithms in the context of endcase processing for topdown standardcell placement. While the divide step in the topdown divide and conquer is usually performed heuristically, we observe that optimal solutions can be found for many su cientl ..."
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Cited by 63 (22 self)
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We study alternatives to classic FMbased partitioning algorithms in the context of endcase processing for topdown standardcell placement. While the divide step in the topdown divide and conquer is usually performed heuristically, we observe that optimal solutions can be found for many su ciently small partitioning instances. Our main motivation is that small partitioning instances frequently contain multiple cells that are larger than the prescribed partitioning tolerance, and that cannot be moved iteratively while preserving the legality ofa solution. To sample the suboptimality of FMbased partitioning algorithms, we focus on optimal partitioning and placement algorithms based on either enumeration or branchandbound that are invoked for instances below prescribed size thresholds,
Configuration space analysis of the traveling salesman problem
 J. de Physique
, 1985
"... Résumé. 2014 Le problème du voyageur de commerce (TSP) et le modèle d’Ising d’un verre de spin sont, respectivement, des archétypes pour les problèmes d’optimisation combinatoire en informatique et pour les systèmes désordonnés frustrés en physique de la matière condensée. Il a été suggéré récemme ..."
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Cited by 49 (1 self)
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Résumé. 2014 Le problème du voyageur de commerce (TSP) et le modèle d’Ising d’un verre de spin sont, respectivement, des archétypes pour les problèmes d’optimisation combinatoire en informatique et pour les systèmes désordonnés frustrés en physique de la matière condensée. Il a été suggéré récemment que ces deux domaines ont beaucoup de phénomènes en commun. Pour voir si, de fait, les problèmes d’optimisation combinatoire peuvent être des verres de spin, nous définissons un TSP à distance aléatoire aussi semblable que possible au modèle idéalisé, à portée infinie, des verres de spin. A la lumière des résultats récents pour les verres de spin, nous analysons les observables thermodynamiques et les corrélations internes entre configurations localement stables. L’hypothèse d’un gel dû à la frustration et d’une structure hiérarchique ultramétrique dans l’espace des configurations est solidement argumentée, pour ce problème de voyageur de commerce. Abstract 2014 The travelling salesman problem (TSP) and the Ising model of a spin glass are archetypes, respectively, of the combinatorial optimization problems of computer science and of the frustrated disordered systems studied in condensed matter physics. It has recently been proposed that these two fields have many phenomena in common. To see if, in fact, combinatorial optimization problems may be spin glasses, we define a random distance TSP as similar as possible to the idealized infiniteranged model of spin glasses. Thermodynamic observables and internal correlations among locally stable configurations are analysed in the light of recent results for spin glasses. Evidence for freezing due to frustration and for a hierarchical, ultrametric structure of configuration space in this TSP is presented.
Generating Linear Extensions Fast
"... One of the most important sets associated with a poset P is its set of linear extensions, E(P) . "ExtensionFast.html" 87 lines, 2635 characters One of the most important sets associated with a poset P is its set of linear extensions, E(P) . In this paper, we present an algorithm to generat ..."
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Cited by 48 (6 self)
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One of the most important sets associated with a poset P is its set of linear extensions, E(P) . "ExtensionFast.html" 87 lines, 2635 characters One of the most important sets associated with a poset P is its set of linear extensions, E(P) . In this paper, we present an algorithm to generate all of the linear extensions of a poset in constant amortized time; that is, in time O(e(P)) , where e ( P ) =  E(P) . The fastest previously known algorithm for generating the linear extensions of a poset runs in time O(n e(P)) , where n is the number of elements of the poset. Our algorithm is the first constant amortized time algorithm for generating a ``naturally defined'' class of combinatorial objects for which the corresponding counting problem is #Pcomplete. Furthermore, we show that linear extensions can be generated in constant amortized time where each extension differs from its predecessor by one or two adjacent transpositions. The algorithm is practical and can be modified to efficiently count linear extensions, and to compute P(x < y) , for all pairs x,y , in time O( n^2 + e ( P )).
Offline Algorithms for The List Update Problem
, 1996
"... Optimum offline algorithms for the list update problem are investigated. The list update problem involves implementing a dictionary of items as a linear list. Several characterizations of optimum algorithms are given; these lead to optimum algorithm which runs in time \Theta2 n (n \Gamma 1)!m, wh ..."
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Cited by 17 (2 self)
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Optimum offline algorithms for the list update problem are investigated. The list update problem involves implementing a dictionary of items as a linear list. Several characterizations of optimum algorithms are given; these lead to optimum algorithm which runs in time \Theta2 n (n \Gamma 1)!m, where n is the length of the list and m is the number of requests. The previous best algorithm, an adaptation of a more general algorithm due to Manasse et al. [9], runs in time \Theta(n!) 2 m. 1 Introduction A dictionary is an abstract data type that stores a collection of keyed items and supports the operations access, insert, and delete. In the sequential search or list update problem, a dictionary is implemented as simple linear list, either stored as a linked collection of items or as an array. An access is done by starting at the front of the list and examining each succeeding item until either finding the item desired or reaching the end of the list and reporting the item not present...
Gray Codes for Reflection Groups
 Graphs and Combinatorics
, 1989
"... This paper appeared in "Graphs and Combinatorics", vol. 5 (1989), pp. 315325.  2  We call (1) a Gray code for G. It is wellknown that any group generated by reflections can be described by a Coxeter diagram ([7], [14], [15], [31]). The finite reflection groups for which the Coxeter d ..."
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Cited by 9 (0 self)
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This paper appeared in "Graphs and Combinatorics", vol. 5 (1989), pp. 315325.  2  We call (1) a Gray code for G. It is wellknown that any group generated by reflections can be described by a Coxeter diagram ([7], [14], [15], [31]). The finite reflection groups for which the Coxeter diagram is a connected graph are ([7], p. 193, Theorem 1) the groups ! n (n ³ 1),
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 8 (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...
Hamilton Cycles which Extend Transposition Matchings in Cayley Graphs of Sn
 SIAM J. DISCRETE MATHEMATICS
, 1993
"... Let B be a basis of transpositions for S n and let Cay(B : S n ) be the Cayley graph of S n with respect to B. It was shown by Kompel'makher and Liskovets that Cay(B : S n ) is hamiltonian. We extend this result as follows. Note that every transposition b in B induces a perfect matching M b in ..."
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Cited by 5 (2 self)
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Let B be a basis of transpositions for S n and let Cay(B : S n ) be the Cayley graph of S n with respect to B. It was shown by Kompel'makher and Liskovets that Cay(B : S n ) is hamiltonian. We extend this result as follows. Note that every transposition b in B induces a perfect matching M b in Cay(B : S n ). We show here when n ? 4 that for any b 2 B, there is a Hamilton cycle in Cay(B : S n ) which includes every edge of M b . That is, for n ? 4, for any basis B of transpositions of S n , and for any b 2 B, it is possible to generate all permutations of 1; 2; : : : ; n by transpositions in B so that every other transposition is b.