Lexicographic Order Research Articles

A lexicographical order induced by Schauder bases

In this paper, we show that every Banach space with a Schauder basis can be seen as a totally ordered vector space. Indeed, this order can be considered as a lexicographical order since it is a generalization of lexicographical order in $\mathbb{R}^{n}.$ We also provide order structural properties of the order by approaching geometrical (cone) sense.

Koszul multi-Rees algebras of principal L-Borel ideals

Given a monomial m in a polynomial ring and a subset L of the variables of the polynomial ring, the principal L-Borel ideal generated by m is the ideal generated by all monomials which can be obtained from m by successively replacing variables of m by those which are in L and have smaller index. Given a collection I={I1,…,Ir} where Ii is Li-Borel for i=1,…,r (where the subsets L1,…,Lr may be different for each ideal), we prove in essence that if the bipartite incidence graph among the subsets L1,…,Lr is chordal bipartite, then the defining equations of the multi-Rees algebra of I has a Gröbner basis of quadrics with squarefree lead terms under a lexicographic order. Thus the multi-Rees algebra of such a collection of ideals is Koszul, Cohen-Macaulay, and normal. This significantly generalizes a theorem of Ohsugi and Hibi on Koszul bipartite graphs. As a corollary we obtain that the multi-Rees algebra of a collection of principal Borel ideals is Koszul. To prove our main result we use a fiber-wise Gröbner basis criterion for the kernel of a toric map and we introduce a modification of Sturmfels' sorting algorithm.

Lexicographic Unranking of Combinations Revisited

In the context of combinatorial sampling, the so-called “unranking method” can be seen as a link between a total order over the objects and an effective way to construct an object of given rank. The most classical order used in this context is the lexicographic order, which corresponds to the familiar word ordering in the dictionary. In this article, we propose a comparative study of four algorithms dedicated to the lexicographic unranking of combinations, including three algorithms that were introduced decades ago. We start the paper with the introduction of our new algorithm using a new strategy of computations based on the classical factorial numeral system (or factoradics). Then, we present, in a high level, the three other algorithms. For each case, we analyze its time complexity on average, within a uniform framework, and describe its strengths and weaknesses. For about 20 years, such algorithms have been implemented using big integer arithmetic rather than bounded integer arithmetic which makes the cost of computing some coefficients higher than previously stated. We propose improvements for all implementations, which take this fact into account, and we give a detailed complexity analysis, which is validated by an experimental analysis. Finally, we show that, even if the algorithms are based on different strategies, all are doing very similar computations. Lastly, we extend our approach to the unranking of other classical combinatorial objects such as families counted by multinomial coefficients and k-permutations.

Arbitrary-term-absent max-product fuzzy relation inequalities and its lexicographic minimal solution

Recently the max-product fuzzy relation inequalities were applied to describe the wireless communication station system. However, when an arbitrary line fault appears between the emission basic station and the signal testing point, the classical max-product system is no longer effective. Considering an arbitrary line fault in the wireless communication station system, we introduce and investigate the so-called arbitrary-term-absent (ATA) max-product fuzzy relation inequalities in this paper. In order to reduce the damage caused by the electromagnetic radiation from the basic stations, a minimal solution is usually desired. However, the structure of solution set to the ATA max-product fuzzy relation inequalities shows that the minimal solutions are not unique in most cases. Hence, we further introduce the lexicographic order and define the relevant concept of lexicographic minimal solution. A lexicographic minimal solution is indeed an optimal scheduling, minimizing the electromagnetic radiations in a lexicographic order. An efficient algorithm with polynomial complexity is developed for obtaining the unique lexicographic minimal solution of the ATA max-product system. At last, our proposed resolution algorithm is compared to the existing one and illustrated by a numerical example.

Method and information technology to research the component architecture of products to justify investments of high-tech enterprise

The development of enterprises in strategic industries depends on funding made innovative products that are in demand in the markets for high-tech products. The interest of investors depends on the innovation and competitiveness of the products that the enterprise can produce. The enterprise should make a new, diversified portfolio of orders to attract funding from potential investors. The innovativeness of the product is determined by the novelty of the components in its composition. Therefore, the pressing challenge is to study the innovation of high-tech products based on their component architecture. It makes it possible for investors to assess the possibility of enterprise financing while making a promising diversified portfolio of orders. The study develops a method to justify investments into the new orders that are based on the research of the component architecture of the complex product. The tasks analyzed the product component architecture innovation and investment attractiveness, justify and select the diversified portfolio of orders, simulate and assess orders portfolio feasibility are stated and solved. The paper proposes the component method that makes it possible to evaluate the architecture of the new product in terms of innovation and investment attractiveness. The research of innovation is conducted depending on the composition of the components in the architecture of the whole product. These components can be either new that require a new cycle of creation, or “old” ones, taken from previous experience with the possible adaptation to the technical requirements of the new product. By using the proposed multifactor planning of the experiment, the possible options are considered and the main indicators of the new product are assessed: investment attractiveness, costs, timelines, and risks of order fulfillment. Using lexicographic ordering of alternatives the compromised selection of the optimal option in terms of limited capabilities of the enterprise is conducted. To optimize the diversified portfolio of orders the method of integer (Boolean) programming is used. Investment attractiveness is used as a target function. The restrictions consider allowable costs, timelines, and risks of the orders portfolio fulfillment. In the last part of the paper, the method of simulation agent modeling in a form of applied information technology is used to assess the timeline for order fulfillment and the impact of risks on the feasibility of the diversified portfolio of orders. The novelty of the results is related to the justification of the choice of a diversified portfolio of orders, which in contrast to the already existing approaches, is based on the advanced component architecture of complex products and the simulation of orders portfolio selection considering innovation and investor interests. The proposed method and information technology are planned for the future development of an enterprise that makes it possible to assess the competitiveness of products, as well as the possibility to attract funding.

A matheuristic for the stochastic facility location problem

In this paper, we describe a matheuristic to solve the stochastic facility location problem which determines the location and size of storage facilities, the quantities of various types of supplies stored in each facility, and the assignment of demand locations to the open facilities, which minimize unmet demand and response time in lexicographic order. We assume uncertainties about demands, inventory spoilage, and transportation network availability. A good example where such a formulation makes sense is the the problem of pre-positioning emergency supplies, which aims to increase disaster preparedness by making the relief items readily available to people in need. The matheuristic employs iterated local search techniques to look for good location and inventory configurations, and uses CPLEX to optimize the assignments. Numerical experiments on a number of case studies and random instances for the pre-positioning problem demonstrate the effectiveness and efficiency of the matheuristic, which is shown to be particularly useful for tackling larger instances that are intractable for exact solvers. The matheuristic is therefore a contribution to the literature on heuristic approaches to solving facility location under uncertainties, can be used to further study the particular variant of the facility location problem, and can also support humanitarian logisticians in their planning of pre-positioning strategies.

Levitin–Polyak well-posedness for equilibrium problems with the lexicographic order

The aim of this work is to investigate optimization-related problems with the objective spaces ordered by the lexicographic cones, including parametric lexicographic equilibrium problems and optimization problems with lexicographic equilibrium constraints. We introduce concepts of Levitin–Polyak well-posedness for these problems and establish a number of sufficient conditions for such properties. The assumptions are imposed directly on the data of the problems and really verifiable. We do not need to suppose the existence (and/or convexity, compactness) of the solution set because it is proved using the mentioned assumptions on the data. Moreover, our assumptions are more relaxed than those which are usually imposed.

Christoffel words and weak Markoff theory

We call weak Markoff theory the theory of Markoff restricted to integral quadratic forms (instead of real ones), and to quadratic real numbers (instead of general real numbers). We show that weak Markoff theory may be reduced to combinatorial properties of Christoffel words, avoiding the use of bi-infinite sequences. These properties are two new characterizations of Christoffel words, one using the lexicographical order, the other being of arithmetic nature, based on values of continuant polynomials.

On infinite prefix normal words

Prefix normal words are binary words with the property that no factor has more 1s than the prefix of the same length. Finite prefix normal words were introduced in Fici and Lipták (2011) [18]. In this paper, we study infinite prefix normal words and explore their relationship to some known classes of infinite binary words. In particular, we establish a connection between prefix normal words and Sturmian words, between prefix normal words and abelian complexity, and between prefix normality and lexicographic order.1

Notice of Removal: A Novel Representation for Permutations

In this paper, we present a variable-length binary code for permutations of degree n, where n is a power of 2. The Lehmer code and its variants provide a bijection between permutations and the indexes in lexicographic ordering. They provide compression of permutations approaching Shannon bound at the expense of structural information. The variable-length code proposed in this paper has asymptotically optimal compression capability while preserving structural information of permutations. In other words, the code encodes the way a permutation is organized, and is optimal in the sense that the ratio of average codeword length and the entropy of permutations approaches 1 as n tends to infinity. The encoding and decoding are efficient. Like the Lehmer code and other enumerative codes, it is not necessary to construct look-up tables. The code is complete and satisfies the prefix condition. Furthermore, the codeword length indicates how “well shuffled” a permutation is. Permutations with longer codewords seem more “random.” An enumeration method of the variable-length code is also given by using T. Cover's enumerative coding scheme and Schalkwijk code. This gives a different indexing from the Lehmer code. This work can be extended to the case of arbitrary n in a straightforward way.

New Algorithms for Generation of Combinations in Lexicographical Order

New Algorithms for Generation of Combinations in Lexicographical Order

Analysis of the CCFD method for MC-based image denoising problems

Image denoising using mean curvature leads to the problem of solving a nonlinear fourth-order integro-differential equation. The nonlinear fourth-order term comes from the mean curvature regularization functional. In this paper, we treat this high-order nonlinearity by reducing the nonlinear fourth-order integro-differential equation to a system of first-order equations. Then a cell-centered finite difference scheme is applied to this system. With a lexicographical ordering of the unknowns, the discretization of the mean curvature functional leads to a block pentadiagonal matrix. Our contributions are fourfold: (i) we give a new method for treating the high-order nonlinearity term; (ii) we express the discretization of this term in terms of simple matrices; (iii) we give an analysis for this new method and establish that the error is of first order; and (iv) we verify this theoretical result by illustrating the convergence rates in numerical experiments.

Quasi-stability and quasi-synchronization control of quaternion-valued fractional-order discrete-time memristive neural networks

A discrete-time fractional-order quaternion-valued memristive system is considered in this note. By utilizing contraction mapping theory, a sufficient condition for the existence and uniqueness of the equilibrium point for the considered system is derived. Via the comparison principle of linear fractional difference system, the quasi-stability condition of the given system is obtained, subsequently, the quasi-synchronization conclusion is derived through Lyapunov method and a proper controller, which can well handle the quasi-synchronization problem in the process of implementing the controller. Applying the lexicographical order method to the quaternion-valued memristive neural networks, the closed convex hull consisted by the connection weights is meaningful. One example is given to substantiate the obtained conclusions.

Exhaustive generation of some lattice paths and their prefixes

We refer to positive lattice paths as to paths in the discrete plane constituted by different kinds of steps (north-east, east and south-east), starting from the origin and never going under the x-axis. They have been deeply studied both from a combinatorial and an algorithmic point of view. We propose some algorithms for the exhaustive generation of positive paths which are Motzkin and Schröder paths and their prefixes, according to their length. For each kind of paths we define a recursive algorithm as well as an iterative one, specifying which path follows a given one in the lexicographic order. Furthermore we study the complexity of these algorithms by using the relations between the number of paths of a given size and the number of final north-east or south-east steps.

Existence of solutions and solving method of lexicographic problem of convex optimization with the linear criteria functions

Among vector problems, the lexicographic ones constitute a broad significant class of problems of optimization. Lexicographic ordering is applied to establish rules of subordination and priority. Hence, a lot of problems including the ones of complex system optimization, of stochastic programming under a risk, of the dynamic character, etc. may be presented in the form of lexicographic problems of optimization. We have revealed the conditions of existence of solutions of multicriteria of lexicographic optimization problems with an unbounded set of feasible solutions on the basis of applying the properties of a recession cone of a con vex feasible set, the cone which puts it in order lexicographically with respect to optimization criteria. The obtained conditions may be successfully used while developing algorithms for finding the optimal solutions of the mentioned problems of lexicographic optimization. A method of finding the optimal solutions of convex lexico graphic problems with the linear functions of criteria is built and grounded on the basis of ideas of the method of linearization and the Kelley cutting plane method.

A study of right zero divisor graph of 3 × 3 matrices over ℤ2

In [10] we have computed left zero divisor graph of the ring R, where R is the set of all 3 × 3 matrices over the ring ℤ2 of integers modulo 2 is constructed. We have extended our result for right zero divisor graph of the ring R. Thus we have listed vertices of RZDG by means of lexicographic order. Which enables us to present algorithm that computes the edges of this graph. In addition, it helps to analyze some of its important properties.

Some problems and algorithms related to the weight order relation on the n-dimensional Boolean cube

The problem “Given a Boolean function [Formula: see text] of [Formula: see text] variables by its truth table vector. Find (if exists) a vector [Formula: see text] of maximal (or minimal) weight, such that [Formula: see text].” is considered here. It is closely related to the problem of computing the algebraic degree of Boolean functions which is an important cryptographic parameter. To solve this problem efficiently, we explore the orders of the vectors of the [Formula: see text]-dimensional Boolean cube [Formula: see text] according to their weights. The notion of “[Formula: see text]th layer” of [Formula: see text] is involved in the definition and examination of the “weight order” relation. It is compared with the known relation “precedes”. Several enumeration problems concerning these relations are solved and the relevant notes were added to three sequences in the on-line encyclopedia of integer sequences (OEIS). One special weight order is defined and examined in detail. In it, the lexicographic order is a second criterion for an ordinance of the vectors of equal weights. So a total order called weight-lexicographic order (WLO) is obtained. Two algorithms for generating the WLO sequence and two algorithms for generating the characteristic vectors of the layers are proposed. The results obtained by them were used in creating two new sequences: A294648 and A305860 in the OEIS. Two algorithms for solving the problem considered are developed — the first one works in a byte-wise manner and uses the WLO sequence, and the second one works in a bitwise manner and uses the characteristic vector as masks. The experimental results from numerous tests confirm the efficiency of these algorithms. Other applications of the obtained algorithms are also discussed — when representing, generating and ranking other combinatorial objects.

Constraint programming approaches to disassembly line balancing problem with sequencing decisions

Recovery of products has received much attention in the last decade due to the increase in both environmental awareness and regulations enacted by governments. In product recovery, disassembly of a product into its constituent parts on a line is one of the most significant operations. This paper deals with a disassembly line balancing and sequencing (DLBS) problem subject to balancing issues, hazardousness of parts, demand quantities and direction changes considered in a lexicographic order. Due to the combinatorial nature of this problem, exact methods, e.g., mixed integer linear programming (MILP), are able to solve only small and medium size problems. Therefore, various metaheuristic algorithms are proposed in literature to find near-optimal solutions. In this paper, constraint programming (CP), which is a suitable technique especially for highly-constrained discrete problems, is used to develop models and solution approaches. To the best of author’s knowledge, this study is the first that uses CP for the disassembly line balancing problems. For the DLBS problem, first, a generic CP model is developed. This CP model provides efficient results for small/medium size disassembly problems and benchmark instances. Observing that the generic CP model could not produce even feasible sequence of tasks for some large-sized benchmark instances, a CP-based solution approach is proposed. This approach generates a feasible sequence subject to a fixed assignment of tasks to the workstations by using a CP model and uses this sequence as an initial feasible solution within a warm-start context in CP sequencing models. The computational results show that the proposed CP model improves the several best solutions of medium-sized benchmark instances, while the proposed CP-based solution approach produces excellent results in all large test instances by either improving the best solutions (found so far) or establishing new benchmark solutions.

Staff scheduling in blood collection problems

To improve the blood collection volume and reduce the staff costs, this study investigates the staff scheduling problem faced by blood donation center under variant situations derived from reality. Staff scheduling for blood donation center is a complex task due to the stochastic arriving of blood donors. In this study, we propose the deterministic demand model and stochastic demand model of blood donors according to the number of donors arriving at donation sites. Then, based on the deterministic demand and the stochastic demand models, we consider the number of staff in the blood donation center and group blood donation. Five blood collection scenarios are proposed using the combination of donors and staff. To solve the proposed models, linear transformation and lexicographic order optimal method are applied. To verify the effectiveness of the proposed models, both large-scale and small-scale numerical experiments are conducted and its the stability is also validated using the 10 times of random numerical experiments. All of the experimental results show that the proposed models need few staff when facing adequate staff scenarios, and reduce the number of donors when facing inadequate staff scenarios.

A loopless algorithm for generating (k, m)-ary trees in Gray code order

A family of (k, m)-ary trees was firstly introduced by Du and Liu when they studied hook length polynomial for plane trees. Recently, Amani and Nowzari-Dalini presented a generation algorithm to produce (k, m)-ary trees of order n encoding by Z-sequences in reverse lexicographic order. In this paper, we propose a loopless algorithm to generate all such Z-sequences in Gray code order. Hence, the worst-case time complexity of generating one Z-sequence is $${\mathcal {O}}(1)$$ , and the space requirement of our algorithm is $$2n+{\mathcal {O}}(1)$$ . Moreover, based on this ordering, we also provide ranking and unranking algorithms. The ranking algorithm can be run in $${\mathcal {O}}(\max \{kmn,n^2\})$$ time and $${\mathcal {O}}(kmn)$$ space, whereas the unranking algorithm requires $${\mathcal {O}}(kmn^2)$$ time and space.