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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 38 (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 )).
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...
The Graph Of Linear Extensions Revisited
 SIAM J. Disc. Math
"... The graph of linear extensions G(P ) of a poset P has as vertices the linear extensions of P , and two vertices are adjacent if they differ only by an adjacent transposition. This graph has so far been investigated mainly with respect to Hamilton paths and intrinsic geodesic convexity. Especially th ..."
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The graph of linear extensions G(P ) of a poset P has as vertices the linear extensions of P , and two vertices are adjacent if they differ only by an adjacent transposition. This graph has so far been investigated mainly with respect to Hamilton paths and intrinsic geodesic convexity. Especially the work on the latter topic showed how the graphtheoretic properties of G(P ) reflect the ordertheoretic structure of P . The aim of this paper is to study the graph G(P ) with respect to topics from classical graph theory, e.g., connectivity, cycle space, isometric embeddings, and nd ordertheoretic interpretations of these notions. The main theorems in this paper are the equality of the connectivity of G(P ) and the jump number of P , the existence of a certain generating system for the cycle space and a relationship of P and its subposets obtained via an embedding of the graph of linear extensions.
Gray Codes from Antimatroids
, 1993
"... We show three main results concerning Hamiltonicity of graphs derived from antimatroids. These results provide Gray codes for the feasible sets and basic words of antimatroids. For antimatroid (E; F), let J(F) denote the graph whose vertices are the sets of F , where two vertices are adjacent if the ..."
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We show three main results concerning Hamiltonicity of graphs derived from antimatroids. These results provide Gray codes for the feasible sets and basic words of antimatroids. For antimatroid (E; F), let J(F) denote the graph whose vertices are the sets of F , where two vertices are adjacent if the corresponding sets differ by one element. Define J(F ; k) to be the subgraph of J(F) 2 induced by the sets in F with exactly k elements. Both graphs J(F) and J(F ; k) are connected, and the former is bipartite. We show that there is a Hamiltonian cycle in J(F) \Theta K 2 . As a consequence, the ideals of any poset P may be listed in such a way that successive ideals differ by at most two elements. We also show that J(F ; k) has a Hamilton path if (E; F) is the poset antimatroid of a seriesparallel poset. Similarly, we show that G(L) \Theta K 2 is Hamiltonian, where G(L) is the "basic word graph" of a language antimatroid (E; L). This result was known previously for poset antimatroids. K...
Code Enumeration of Families of Integer Partitions
 JOURNAL OF COMBINATORIAL THEORY, SERIES A
"... In this paper we show that the elements of certain families of integer partitions can be listed in a minimal change, or Gray code, order. In particular, we construct Gray code listings for the classes Pδ(n,k) and D(n,k) of partitions of n into parts of size at most k in which, for Pδ(n,k), the parts ..."
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In this paper we show that the elements of certain families of integer partitions can be listed in a minimal change, or Gray code, order. In particular, we construct Gray code listings for the classes Pδ(n,k) and D(n,k) of partitions of n into parts of size at most k in which, for Pδ(n,k), the parts are congruent to one modulo δ and, for D(n,k), the parts are distinct. The change required between successive partitions is the increase of one part by δ (or the addition of δ ones) and the decrease of one part by δ (or the removal of δ ones), where, in the case of D(n,k), δ = 1.
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 differ ..."
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Parity difference equal to 0 or &plusmn;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 &plusmn;1 (see [5]).