Efficient generation of the ideals of a poset in Gray code order

Abstract

would like to thank Timothy Walsh, director of my research, for what he taught me, for his encouragement, for the time he spent answering my questions, for his valu able advice given to me when 1 was a teaching assistant for his course, and for his financial support. Without his supervision, this work would not have been completed. ln addition, 1 thank him for the concerts to which he invited me when he was celebrating his birthday and elsewhere. 1 would also like to thank Srecko Brlek for what he taught me and for his interest in students and especially for his valuable advice, encouragement and collaboration. 1 thank my father Abdulhamid and my mother Naima who supported and con stantly encouraged me to achieve my dreams in my studies. 1 will never forget my brother Ahmad who passed away in November 2009. He was my real friend and support, we were as twins. l'm glad that my wife was beside me while 1 was writing my thesis. 1 owe her a lot because she gives me the love and enthusiasm that 1 need. Thanks to Manon Gauthier, who provides ail the administrative advice and who shows much interest in students. Thanks to Gisèle Legault, who, with her knowledge of computers, allowed me to save much time. Thanks to Lise Tourigny, Jérôme Tremblay and ail the staff in the Departments of Mathematics and Computer Science. Finally, 1 am grateful to LaCIM, ISM, FARE, FQRNT and the Faculty of Sciences for their financial support during my studies for my M.Sc. and Ph.D. These scholarships were complementary to the financial support of my supervisor.

Efficient generation of the ideals of a poset in Gray code order

“…Theorem 1 below (Abdo, 2009) shows that a Hamiltonian cycle can be constructed for any poset. We cali x a maximal (minimal) element of a poset P if no element of P is strictly greater (less) than x.…”

“…8. listing all the ideals of a poset so that successive ideals differ in one or two elements (Abdo, 2009;Pruesse and Ruskey, 93;Koda and Ruskey, 93).…”

“…We generalize Squire's recurrence and use our generalized recurrence to list the elements of D(P), the down-sets of P, so that two ideals that are adjacent in the list differ by one or two elements, as does the algorithm of Pruesse and Ruskey (Pruesse and Ruskey,93). A preliminary version of our algorithm, described in Section 1.3 and published in (Abdo, 2009), is similar to the one in (Pruesse and Ruskey,93) except that it lists each ideal once instead of twice. The presentation here is simpler than the one in (Pruesse and Ruskey,93) because it deals only with posets, whereas the one in (Pruesse and Ruskey,93) is generalized to antimatroids.…”

“…We were recently informed that the proof in (Pruesse and Ruskey,93) was recast in terms of posets in Theorem 4.4 (Chow and Ruskey,09), published in the same year as (Abdo, 2009). Both algorithms have the same time-complexity.…”