Results 1 
3 of
3
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 ..."
Abstract

Cited by 49 (6 self)
 Add to MetaCart
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 )).
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 ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
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.
STEINHAUS ’ LATTICEPOINT PROBLEM FOR BANACH SPACES
"... Abstract. Given a positive integer n, one may find a circle on the Euclidean plane surrounding exactly n points of the integer lattice. This classical geometric fact due to Steinhaus has been recently extended to Hilbert spaces by Zwoleński, who replaced the integer lattice by any infinite set whic ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. Given a positive integer n, one may find a circle on the Euclidean plane surrounding exactly n points of the integer lattice. This classical geometric fact due to Steinhaus has been recently extended to Hilbert spaces by Zwoleński, who replaced the integer lattice by any infinite set which intersects every ball in at most finitely many points. We investigate the Banach spaces satisfying this property, which we call (S), and show that all strictly convex Banach spaces have (S). Nonetheless, we construct a norm in dimension three which has (S) but fails to be strictly convex. 1.