4,211 research outputs found
A constant-time algorithm for middle levels Gray codes
For any integer a middle levels Gray code is a cyclic listing of
all -element and -element subsets of such that
any two consecutive subsets differ in adding or removing a single element. The
question whether such a Gray code exists for any has been the subject
of intensive research during the last 30 years, and has been answered
affirmatively only recently [T. M\"utze. Proof of the middle levels conjecture.
Proc. London Math. Soc., 112(4):677--713, 2016]. In a follow-up paper [T.
M\"utze and J. Nummenpalo. An efficient algorithm for computing a middle levels
Gray code. To appear in ACM Transactions on Algorithms, 2018] this existence
proof was turned into an algorithm that computes each new set in the Gray code
in time on average. In this work we present an algorithm for
computing a middle levels Gray code in optimal time and space: each new set is
generated in time on average, and the required space is
Efficient computation of middle levels Gray codes
For any integer a middle levels Gray code is a cyclic listing of
all bitstrings of length that have either or entries equal to
1 such that any two consecutive bitstrings in the list differ in exactly one
bit. The question whether such a Gray code exists for every has been
the subject of intensive research during the last 30 years, and has been
answered affirmatively only recently [T. M\"utze. Proof of the middle levels
conjecture. Proc. London Math. Soc., 112(4):677--713, 2016]. In this work we
provide the first efficient algorithm to compute a middle levels Gray code. For
a given bitstring, our algorithm computes the next bitstrings in the
Gray code in time , which is
on average per bitstring provided that
Gray code
У статті подано коди Грея послідовностями по 4 символи для цілих чисел від 0 до 15 включно.В статье представлены коды Грея последовательностями по 4 символа для целых чисел от 0 до 15 включительно.The article presents a Gray code sequences to 4 characters to integers from 0 to 15 inclusive
A Gray Code for the Shelling Types of the Boundary of a Hypercube
We consider two shellings of the boundary of the hypercube equivalent if one
can be transformed into the other by an isometry of the cube. We observe that a
class of indecomposable permutations, bijectively equivalent to standard double
occurrence words, may be used to encode one representative from each
equivalence class of the shellings of the boundary of the hypercube. These
permutations thus encode the shelling types of the boundary of the hypercube.
We construct an adjacent transposition Gray code for this class of
permutations. Our result is a signed variant of King's result showing that
there is a transposition Gray code for indecomposable permutations
Subset-lex: did we miss an order?
We generalize a well-known algorithm for the generation of all subsets of a
set in lexicographic order with respect to the sets as lists of elements
(subset-lex order). We obtain algorithms for various combinatorial objects such
as the subsets of a multiset, compositions and partitions represented as lists
of parts, and for certain restricted growth strings. The algorithms are often
loopless and require at most one extra variable for the computation of the next
object. The performance of the algorithms is very competitive even when not
loopless. A Gray code corresponding to the subset-lex order and a Gray code for
compositions that was found during this work are described.Comment: Two obvious errors corrected (indicated by "Correction:" in the LaTeX
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