120 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
Generation, Ranking and Unranking of Ordered Trees with Degree Bounds
We study the problem of generating, ranking and unranking of unlabeled
ordered trees whose nodes have maximum degree of . This class of trees
represents a generalization of chemical trees. A chemical tree is an unlabeled
tree in which no node has degree greater than 4. By allowing up to
children for each node of chemical tree instead of 4, we will have a
generalization of chemical trees. Here, we introduce a new encoding over an
alphabet of size 4 for representing unlabeled ordered trees with maximum degree
of . We use this encoding for generating these trees in A-order with
constant average time and O(n) worst case time. Due to the given encoding, with
a precomputation of size and time O(n^2) (assuming is constant), both
ranking and unranking algorithms are also designed taking O(n) and O(nlogn)
time complexities.Comment: In Proceedings DCM 2015, arXiv:1603.0053
Two Reflected Gray Code based orders on some restricted growth sequences
We consider two order relations: that induced by the m-ary reflected Gray
code and a suffix partitioned variation of it. We show that both of them when
applied to some sets of restricted growth sequences still yield Gray codes.
These sets of sequences are: subexcedant or ascent sequences, restricted growth
functions, and staircase words. In each case we give efficient exhaustive
generating algorithms and compare the obtained results
Lattice path matroids: enumerative aspects and Tutte polynomials
Fix two lattice paths P and Q from (0,0) to (m,r) that use East and North
steps with P never going above Q. We show that the lattice paths that go from
(0,0) to (m,r) and that remain in the region bounded by P and Q can be
identified with the bases of a particular type of transversal matroid, which we
call a lattice path matroid. We consider a variety of enumerative aspects of
these matroids and we study three important matroid invariants, namely the
Tutte polynomial and, for special types of lattice path matroids, the
characteristic polynomial and the beta invariant. In particular, we show that
the Tutte polynomial is the generating function for two basic lattice path
statistics and we show that certain sequences of lattice path matroids give
rise to sequences of Tutte polynomials for which there are relatively simple
generating functions. We show that Tutte polynomials of lattice path matroids
can be computed in polynomial time. Also, we obtain a new result about lattice
paths from an analysis of the beta invariant of certain lattice path matroids.Comment: 28 pages, 11 figure
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