eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-07-04
66:1
66:14
10.4230/LIPIcs.ICALP.2018.66
article
Gray Codes and Symmetric Chains
Gregor, Petr
1
Jäger, Sven
2
Mütze, Torsten
2
Sawada, Joe
3
Wille, Kaja
2
Department of Theoretical Computer Science and Mathematical Logic, Charles University, Prague, Czech Republic
Institut für Mathematik, Technische Universität Berlin, Germany
School of Computer Science, University of Guelph, Canada
We consider the problem of constructing a cyclic listing of all bitstrings of length 2n+1 with Hamming weights in the interval [n+1-l,n+l], where 1 <= l <= n+1, by flipping a single bit in each step. This is a far-ranging generalization of the well-known middle two levels problem (l=1). We provide a solution for the case l=2 and solve a relaxed version of the problem for general values of l, by constructing cycle factors for those instances. Our proof uses symmetric chain decompositions of the hypercube, a concept known from the theory of posets, and we present several new constructions of such decompositions. In particular, we construct four pairwise edge-disjoint symmetric chain decompositions of the n-dimensional hypercube for any n >= 12.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol107-icalp2018/LIPIcs.ICALP.2018.66/LIPIcs.ICALP.2018.66.pdf
Gray code
Hamilton cycle
hypercube
poset
symmetric chain