19 research outputs found
The K-Centre Problem for Necklaces
In graph theory, the objective of the k-centre problem is to find a set of vertices for which the largest distance of any vertex to its closest vertex in the -set is minimised. In this paper, we introduce the -centre problem for sets of necklaces, i.e. the equivalence classes of words under the cyclic shift. This can be seen as the k-centre problem on the complete weighted graph where every necklace is represented by a vertex, and each edge has a weight given by the overlap distance between any pair of necklaces. Similar to the graph case, the goal is to choose necklaces such that the distance from any word in the language and its nearest centre is minimised. However, in a case of k-centre problem for languages the size of associated graph maybe exponential in relation to the description of the language, i.e., the length of the words l and the size of the alphabet q. We derive several approximation algorithms for the -centre problem on necklaces, with logarithmic approximation factor in the context of l and k, and within a constant factor for a more restricted case
Combinatorial generation via permutation languages
In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations.
This approach provides a unified view on many known results and allows us to prove many new ones.
In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an -element set by adjacent transpositions; the binary reflected Gray code to generate all -bit strings by flipping a single bit in each step; the Gray code for generating all -vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an -element ground set by element exchanges due to Kaye.
We present two distinct applications for our new framework:
The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others.
We thus also obtain new Gray code algorithms for the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into rectangles subject to certain restrictions.
The second main application of our framework are lattice congruences of the weak order on the symmetric group~.
Recently, Pilaud and Santos realized all those lattice congruences as -dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc.
Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian.
We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope
Combinatorial Algorithms for Multidimensional Necklaces
A necklace is an equivalence class of words of length over an alphabet under the cyclic shift (rotation) operation. As a classical object, there have been many algorithmic results for key operations on necklaces, including counting, generating, ranking, and unranking. This paper generalises the concept of necklaces to the multidimensional setting. We define multidimensional necklaces as an equivalence classes over multidimensional words under the multidimensional cyclic shift operation. Alongside this definition, we generalise several problems from the one dimensional setting to the multidimensional setting for multidimensional necklaces with size over an alphabet of size including: providing closed form equations for counting the number of necklaces; an time algorithm for transforming some necklace to the next necklace in the ordering; an time algorithm to rank necklaces (determine the number of necklaces smaller than in the set of necklaces); an time algorithm to unrank multidimensional necklace (determine the necklace in the set of necklaces). Our results on counting, ranking, and unranking are further extended to the fixed content setting, where every necklace has the same Parikh vector, in other words every necklace shares the same number of occurrences of each symbol. Finally, we study the -centre problem for necklaces both in the single and multidimensional settings. We provide strong approximation algorithms for solving this problem in both the one dimensional and multidimensional settings
Upper bounds on the growth rates of hard squares and related models via corner transfer matrices
We study the growth rate of the hard squares lattice gas, equivalent to the
number of independent sets on the square lattice, and two related models -
non-attacking kings and read-write isolated memory. We use an assortment of
techniques from combinatorics, statistical mechanics and linear algebra to
prove upper bounds on these growth rates. We start from Calkin and Wilf's
transfer matrix eigenvalue bound, then bound that with the Collatz-Wielandt
formula from linear algebra. To obtain an approximate eigenvector, we use an
ansatz from Baxter's corner transfer matrix formalism, optimised with Nishino
and Okunishi's corner transfer matrix renormalisation group method. This
results in an upper bound algorithm which no longer requires exponential memory
and so is much faster to calculate than a direct evaluation of the Calkin-Wilf
bound. Furthermore, it is extremely parallelisable and so allows us to make
dramatic improvements to the previous best known upper bounds. In all cases we
reduce the gap between upper and lower bounds by 4-6 orders of magnitude.Comment: Also submitted to FPSAC 2015 conferenc
Upper bounds on the growth rates of hard squares and related models via corner transfer matrices
International audienceWe study the growth rate of the hard squares lattice gas, equivalent to the number of independent sets on the square lattice, and two related models â non-attacking kings and read-write isolated memory. We use an assortment of techniques from combinatorics, statistical mechanics and linear algebra to prove upper bounds on these growth rates. We start from Calkin and Wilfâs transfer matrix eigenvalue bound, then bound that with the Collatz-Wielandt formula from linear algebra. To obtain an approximate eigenvector, we use an ansatz from Baxterâs corner transfer matrix formalism, optimised with Nishino and Okunishiâs corner transfer matrix renormalisation group method. This results in an upper bound algorithm which no longer requires exponential memory and so is much faster to calculate than a direct evaluation of the Calkin-Wilf bound. Furthermore, it is extremely parallelisable and so allows us to make dramatic improvements to the previous best known upper bounds. In all cases we reduce the gap between upper and lower bounds by 4-6 orders of magnitude.Nous Ă©tudions le taux de croissance du systĂšme de particules dur sur un rĂ©seau carrĂ©. Ce taux est Ă©quivalent au nombre dâensembles indĂ©pendants sur le rĂ©seau carrĂ©. Nous Ă©tudions Ă©galement deux modĂšles qui lui sont reliĂ©s : les rois non-attaquants et la mĂ©moire isolĂ©e dâĂ©criture-rĂ©Ă©criture. Nous utilisons techniques diverses issues de la combinatoire, de la mĂ©canique statistique et de lâalgĂšbre linĂ©aire pour prouver des bornes supĂ©rieures sur ces taux de croissances. Nous partons de la borne de Calkin et Wilf sur les valeurs propres des matrices de transfert, que nous bornons Ă lâaide de la formule de Collatz-Wielandt issue de lâalgĂšbre linĂ©aire. Pour obtenir une valeur approchĂ©e dâun vecteur propre, nous utilisons un ansatz du formalisme de Baxter sur les matrices de transfert de coin, que nous optimisons avec la mĂ©thode de Nishino et Okunishi qui exploite ces matrices. Il en rĂ©sulte un algorithme pour calculer la borne supĂ©rieure qui nâest plus exponentiel en mĂ©moire et est ainsi beaucoup plus rapide quâune Ă©valuation directe de la borne de Calkin-Wilf. De plus, cet algorithme est extrĂȘmement parallĂ©lisable et permet ainsi une nette amĂ©lioration des meilleurs bornes supĂ©rieures existantes. Dans tous les cas lâĂ©cart entre les bornes supĂ©rieures et infĂ©rieures sâen trouve rĂ©duit de 4 Ă 6 ordres de grandeur
Combinatorial generation via permutation languages. I. Fundamentals
In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations.
This approach provides a unified view on many known results and allows us to prove many new ones.
In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an -element set by adjacent transpositions; the binary reflected Gray code to generate all -bit strings by flipping a single bit in each step; the Gray code for generating all -vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an -element ground set by element exchanges due to Kaye.
We present two distinct applications for our new framework:
The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, boxed patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others.
We also obtain new Gray codes for all the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into rectangles subject to certain restrictions.
The second main application of our framework are lattice congruences of the weak order on the symmetric group~.
Recently, Pilaud and Santos realized all those lattice congruences as -dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc.
Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian.
We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope