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21
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 81 (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 56 (20 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,
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 generate all of t ..."
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Cited by 36 (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 14 (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 diagram is ..."
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Cited by 10 (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 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...
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 Cay(B ..."
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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.
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...
Generating Permutations With kDifferences
 SIAM Journal on Discrete Mathematics
, 1989
"... Given (n; k) with n k 2 and k 6= 3, we show how to generate all permutations of n objects (each exactly once) so that successive permutations differ in exactly k positions, as do the first and last permutations. This solution generalizes known results for the specific cases where k = 2 and k = ..."
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Cited by 2 (2 self)
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Given (n; k) with n k 2 and k 6= 3, we show how to generate all permutations of n objects (each exactly once) so that successive permutations differ in exactly k positions, as do the first and last permutations. This solution generalizes known results for the specific cases where k = 2 and k = n. When k = 3, we show that it is possible to generate all even (odd) permutations of n objects so that successive permutations differ in exactly 3 positions. Keywords. permutations, Gray codes, combinatorial algorithms, Cayley graphs, Hamilton cycles AMS(MOS) subject classifications. 05A15, 05C45 1 Introduction The problem of generating permutations of n distinct objects is of fundamental importance both in Computer Science and in Combinatorics. Many practical problems require for their solution a sampling of random permutations or, worse, a search through all n! permutations. In order for such a search to be possible, even for moderate size n, permutation generation methods must be e...
Finding Parity Difference by Involutions
, 2003
"... Parity difference equal to 0 or ±1 is a necessary condition for the existence of minimal change generation algorithms for many combinatorial objects. We prove, that finding parity difference for linear extensions of posets is #Pcomplete. We also show a new method of finding parity difference ..."
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Cited by 1 (0 self)
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Parity difference equal to 0 or ±1 is a necessary condition for the existence of minimal change generation algorithms for many combinatorial objects. We prove, that finding parity difference for linear extensions of posets is #Pcomplete. We also show a new method of finding parity difference for strings representing forests and a combinatorial interpretation of this result as well as all cases when this value is equal 0 or ±1 (see [5]).