Vincent Vajnovszki

Discrete Mathematics and Theoretical Computer Science

Two philosophical applications of the concept of programsize complexity are discussed. First, we consider the light program-size complexity sheds on whether mathematics is invented or discovered, i.e., is empirical or is a priori. Second, we propose that the notion of algorithmic independence sheds light on the question of being and how the world of our experience can be partitioned into separate entities.

On the loopless generation of binary tree sequences

Weight sequences were introduced by Pallo in 1986 for coding binary trees and he presented a constant amortized time algorithm for their generation in lexicographic order. A year later, Roelants van Baronaigien and Ruskey developed a recursive constant amortized time algorithm for generating Gray code for binary trees in Pallos representation. It is common practice to find a loopless generating algorithm for a combinatorial object when enunciating a Gray code for this object. In this paper we regard weight sequences as variations and apply a Williamson algorithm in order to obtain a loopless generating algorithm for the Roelants van Baronaigien and Ruskeys Gray code for weight sequences.

Gray code for derangements

We give a Gray code and constant average time generating algorithm for derangements, i.e., permutations with no fixed points. In our Gray code, each derangement is transformed into its successor either via one or two transpositions or a rotation of three elements. We generalize these results to permutations with number of fixed points bounded between two constants.

A loopless algorithm for generating the permutations of a multiset

Many combinatorial structures can be constructed from simpler components. For example, a permutation can be constructed from cycles, or a Motzkin word from a Dyck word and a combination. In this paper we present a constructor for combinatorial structures, called shuffle on trajectories (defined previously in a non-combinatorial context), and we show how this constructor enables us to obtain a new loopless generating algorithm for multiset permutations from similar results for simpler objects.

A Loopless Generation of Bitstrings without p Consecutive Ones

Let F n (p) be the set of all n-length bitstrings such that there are no p consecutive ls. F n (p) is counted with the pth order Fibonacci numbers and it may be regarded as the subsets of {1, 2,…, n} without p consecutive elements and bitstrings in F n (p) code a particular class of trees or compositions of an integer. In this paper we give a Gray code for F n (p) which can be implemented in a recursive generating algorithm, and finally in a loopless generating algorithm.

Generating a Gray Code for P-Sequences

P-sequences are used for coding binary trees and they are also an alternative representation for well-formed parentheses strings. We present here the first Gray code and loopless generating algorithm for P-sequences, and extend them in a Gray code and a new loopless generating algorithm for well-formed parentheses strings. Ranking and unranking algorithms are also discussed.

Gray visiting Motzkins

Abstract. We present the first Gray code for Motzkin words and their generalizations: k colored Motzkin words and Schröder words. The construction of these Gray codes is based on the observation that a k colored Motzkin word is the shuffle of a Dyck word by a k-ary variation on a trajectory which is a combination. In the final part of the paper we give some algorithmic considerations and other possible applications of the techniques introduced here.

A loop-free two-close Gray-code algorithm for listing k-ary Dyck words

P. Chase and F. Ruskey each published a Gray code for length n binary strings with m occurrences of 1, coding m-combinations of n objects, which is two-close-that is, in passing from one binary string to its successor a single 1 exchanges positions with a 0 which is either adjacent to the 1 or separated from it by a single 0. If we impose the restriction that any suffix of a string contains at least k-1 times as many 0s as 1s, we obtain k-suffixes: suffixes of k-ary Dyck words. Combinations are retrieved as special case by setting k=1 and k-ary Dyck words are retrieved as a special case by imposing the additional condition that the entire string has exactly k-1 times as many 0s as 1s. We generalize Ruskeys Gray code to the first two-close Gray code for k-suffixes and we provide a loop-free implementation for k>=2. For k=1 we use a simplified version of Chases loop-free algorithm for generating his two-close Gray code for combinations. These results are optimal in the sense that there does not always exist a Gray code, either for combinations or Dyck words, in which the 1 and the 0 that exchange positions are adjacent.

Minimal change list for Lucas strings and some graph theoretic consequences

We give a minimal change list for the set of order p length-n Lucas strings, i.e., the set of length-n binary strings with no p consecutive 1s nor a 1l prefix and a 1m suffix with l+m ≥ p. The construction of this list proves also that the order p n-dimensional Lucas cube has a Hamiltonian path if and only if n is not a multiple of p + 1, and its second power always has a Hamiltonian path.

SYSTOLIC GENERATION OF k-ARY TREES

The only parallel generating algorithms for k-ary trees are those of Akl and Stojmenovic in 1996 and of Vajnovszki and Phillips in 1997. In the first of them, trees are represented by an inversion table and the processor model is a linear aray multicomputer. In the second, trees are represented by bitstrings and the algorithm executes on a shared memory multiprocessor. In this paper we give a parallel generating algorithm for k-ary trees represented by generalized P–sequences for execution on a linear array multicomputer.