Results 1 
4 of
4
BINARY BUBBLE LANGUAGES AND COOLLEX ORDER
"... A bubble language is a set of binary strings with a simple closure property: The first 01 of any string can be replaced by 10 to obtain another string in the set. Natural representations of many combinatorial objects are bubble languages. Examples include binary string representations of kary trees ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
(Show Context)
A bubble language is a set of binary strings with a simple closure property: The first 01 of any string can be replaced by 10 to obtain another string in the set. Natural representations of many combinatorial objects are bubble languages. Examples include binary string representations of kary trees, unit interval graphs, linearextensions of Bposets, binary necklaces and Lyndon words, and feasible solutions to knapsack problems. In colexicographic order, fixeddensity binary strings are ordered so that their suffixes of the form 10i occur (recursively) in the order i = max,max −1,...,min +1,min for some values of max and min. In coollex order the suffixes occur (recursively) in the order max −1,..., min+1, min, max. This small change has significant consequences. We prove that the strings in any bubble language appear in a Gray code order when listed in coollex order. This Gray code may be viewed from two different perspectives. On the one hand, successive binary strings differ by one or two transpositions, and on the other hand, they differ by a shift of some substring one position to the right. This article also provides the theoretical foundation for many efficient generation algorithms, as well as the first
Efficient Oracles for Generating Binary Bubble Languages
"... A simple metaalgorithm is provided to efficiently generate a wide variety of combinatorial objects that can be represented by binary strings with a fixed number of 1s. Such objects include: kary Dyck words, connected unit interval graphs, binary strings lexicographically larger than ω, those avoid ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
A simple metaalgorithm is provided to efficiently generate a wide variety of combinatorial objects that can be represented by binary strings with a fixed number of 1s. Such objects include: kary Dyck words, connected unit interval graphs, binary strings lexicographically larger than ω, those avoiding 10 k for fixed k, reversible strings and feasible solutions to knapsack problems. Each object requires only a very simple objectspecific subroutine (oracle) that plugs into the generic coollex framework introduced by Williams. The result is that each object can be generated in amortized O(1)time. Moreover, the strings can be listed in either a conventional colexicographic order, or in the coollex Gray code order.
Enumeration of three term arithmetic progressions in fixed density sets
, 2014
"... Additive combinatorics is built around the famous theorem by Szemerédi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different techniques. Szemerédi’s theorem is an existence statement, wherea ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Additive combinatorics is built around the famous theorem by Szemerédi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different techniques. Szemerédi’s theorem is an existence statement, whereas the ultimate goal in combinatorics is always to make enumeration statements. In this article we develop new methods based on real algebraic geometry to obtain several quantitative statements on the number of arithmetic progressions in fixed density sets. We further discuss the possibility of a generalization of Szemerédi’s theorem using
De Bruijn Sequences for the Binary Strings with Maximum Specified Density
"... Abstract. A de Bruijn sequence is a circular binary string of length 2 n that contains each binary string of length n exactly once as a substring. A maximumdensity de Bruijn sequence is a circular binary string of length ( ) ( ) ( ) () n n n n + + + · · · + that contains each binary string of ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Abstract. A de Bruijn sequence is a circular binary string of length 2 n that contains each binary string of length n exactly once as a substring. A maximumdensity de Bruijn sequence is a circular binary string of length ( ) ( ) ( ) () n n n n + + + · · · + that contains each binary string of length n with density 0 1 2 m (number of 1s) between 0 and m, inclusively. In this paper we efficiently generate maximumdensity de Bruijn sequences for all values of n and m. When n = 2m + 1 our result gives a “complementfree de Bruijn sequence ” which is a circular binary string of length 2 n−1 that contains each binary string of length n or its complement exactly once as a substring.