Binary Strings Of Length Research Articles

D-magic labelings of the halved n-cube

Let G=(V,E) be a graph with path-length distance function ∂ and diameter d. Let D⊆{0,1,…,d} be a set of distances in G and let ND(x)={y|∂(x,y)∈D} for a fixed vertex x∈V. A bijection φ:V→{1,2,…,|V|} is called a D-magic labeling of G if there exists a constant k such that ∑y∈ND(x)f(y)=k for any x∈V. In this paper, we will study D-magic labelings of the halved n-cube (n≥2) that is on all binary strings of length n with even number of 1s as vertices and edges between any two strings of Hamming distance 2. We prove that the halved n-cube is {1}-magic if and only if n=m2 where m≥2 and m≢0(mod4), and is {0,1}-magic if and only if n=m2+2 where m≥0 and m≢2(mod4).

On a Combinatorial Generation Problem of Knuth

The well-known middle levels conjecture asserts that for every integer $n\geq 1$, all binary strings of length $2(n+1)$ with exactly $n+1$ many 0s and 1s can be ordered cyclically so that any two consecutive strings differ in swapping the first bit with a complementary bit at some later position. In his book `The Art of Computer Programming Vol. 4A' Knuth raised a stronger form of this conjecture (Problem 56 in Chapter 7, Section 2.1.3), which requires that the sequence of positions with which the first bit is swapped in each step of such an ordering has $2n+1$ blocks of the same length, and each block is obtained by adding $s=1$ (modulo $2n+1$) to the previous block. In this work, we prove Knuth's conjecture in a more general form, allowing for arbitrary shifts $s\geq 1$ that are coprime to $2n+1$. We also present an algorithm to compute this ordering, generating each new bitstring in $\mathcal{O}(n)$ time, using $\mathcal{O}(n)$ memory in total.

Finding binary words with a given number of subsequences

We relate binary words with a given number of subsequences to continued fractions of rational numbers with a given denominator. We deduce that there are binary strings of length O(log⁡nlog⁡log⁡n) with exactly n subsequences; this can be improved to O(log⁡n) under assumption of Zaremba's conjecture.

Sizes of simultaneous core partitions

There is a well-studied correspondence by Jaclyn Anderson between partitions that avoid hooks of length s or t and certain binary strings of length s+t. Using this map, we prove that the total size of a random partition of this kind converges in law to Watson's U2 distribution, as conjectured by Doron Zeilberger.

Construction of One-Dimensional Nonuniform Number Conserving Elementary Cellular Automata Rules

An effort to study one-dimensional nonuniform elementary number conserving cellular automata (NCCA) rules from an exponential order rule space of cellular automata is an excellent computational task. To perform this task effectively, a mathematical heritage under the number of conserving functions over binary strings of length [Formula: see text] has been highlighted along with their number conserving cellular automata rules (either uniform or nonuniform). A basic approach for the construction of some feasible nonuniform NCCA rules of any finite configuration with the assistance of nine uniform elementary CA rules has been investigated. From our construction procedure, recurrence equations are formulated as suitably solved to ascertain the actual range of NCCA rules. The state transition diagrams (STDs) of NCCA rules are analyzed. While classifying the binary strings through STDs, we found a fascinating optical insight that equal weight strings from a class whose cardinality is the same as the binomial coefficient [Formula: see text] where [Formula: see text] is the length and [Formula: see text] is the weight of the binary string.

Upper Bounds on the Domination and Total Domination Number of Fibonacci Cubes

One of the basic model for interconnection networks is the $n$-dimensional hypercube graph $Q_n$ and the vertices of $Q_n$ are represented by all binary strings of length $n$. The Fibonacci cube $\Gamma_n$ of dimension $n$ is a subgraph of $Q_n$, where the vertices correspond to those without two consecutive 1s in their string representation. In this paper, we deal with the domination number and the total domination number of Fibonacci cubes. First we obtain upper bounds on the domination number of $\Gamma_n$ for $n\ge 13$. Then using these result we obtain upper bounds on the total domination number of $\Gamma_n$ for $n\ge 14$ and we see that these upper bounds improve the bounds given in [1].

System Reliability Evaluation of a Stochastic - Flow Network using Spanning Trees

In this study, a new method is presented to evaluate the system reliability of a flow network using spanning trees with flow. The proposed method includes two main tasks: 1. Identifying the spanning trees without flow by representing each network link as a binary string of length N (N is the number of nodes) and performing all possible combinations between N − 1 links, and 2. Using the generated spanning trees without flow to find the spanning trees with flow by calculating the total flow carried by the links. The proposed method is tested on different examples from the literature to illustrate its efficiency in generating the spanning trees with flow and calculating the system reliability.

The Number of Moves of the Largest Disc in Shortest Paths on Hanoi Graphs

In contrast to the widespread interest in the Frame-Stewart conjecture (FSC) about the optimal number of moves in the classical Tower of Hanoi task with more than three pegs, this is the first study of the question of investigating shortest paths in Hanoi graphs $H_p^n$ in a more general setting. Here $p$ stands for the number of pegs and $n$ for the number of discs in the Tower of Hanoi interpretation of these graphs. The analysis depends crucially on the number of largest disc moves (LDMs). The patterns of these LDMs will be coded as binary strings of length $p-1$ assigned to each pair of starting and goal states individually. This will be approached both analytically and numerically. The main theoretical achievement is the existence, at least for all $n\geqslant p(p-2)$, of optimal paths where $p-1$ LDMs are necessary. Numerical results, obtained by an algorithm based on a modified breadth-first search making use of symmetries of the graphs, lead to a couple of conjectures about some cases not covered by our ascertained results. These, in turn, may shed some light on the notoriously open FSC.

Calculating Kolmogorov complexity from the output frequency distributions of small Turing machines.

Drawing on various notions from theoretical computer science, we present a novel numerical approach, motivated by the notion of algorithmic probability, to the problem of approximating the Kolmogorov-Chaitin complexity of short strings. The method is an alternative to the traditional lossless compression algorithms, which it may complement, the two being serviceable for different string lengths. We provide a thorough analysis for all binary strings of length and for most strings of length by running all Turing machines with 5 states and 2 symbols ( with reduction techniques) using the most standard formalism of Turing machines, used in for example the Busy Beaver problem. We address the question of stability and error estimation, the sensitivity of the continued application of the method for wider coverage and better accuracy, and provide statistical evidence suggesting robustness. As with compression algorithms, this work promises to deliver a range of applications, and to provide insight into the question of complexity calculation of finite (and short) strings.Additional material can be found at the Algorithmic Nature Group website at http://www.algorithmicnature.org. An Online Algorithmic Complexity Calculator implementing this technique and making the data available to the research community is accessible at http://www.complexitycalculator.com.

On the adjacency matrix of a threshold graph

A threshold graph on n vertices is coded by a binary string of length n−1. We obtain a formula for the inertia of (the adjacency matrix of) a threshold graph in terms of the code of the graph. It is shown that the number of negative eigenvalues of the adjacency matrix of a threshold graph is the number of ones in the code, whereas the nullity is given by the number of zeros in the code that are preceded by either a zero or a blank. A formula for the determinant of the adjacency matrix of a generalized threshold graph and the inverse, when it exists, of the adjacency matrix of a threshold graph are obtained. Results for antiregular graphs follow as special cases.

Counting Runs of Ones and Ones in Runs of Ones in Binary Strings

Consider a binary string (a symmetric Bernoulli sequence) of length . For a positive integer , we exactly enumerate, in all possible binary strings of length , the number of all runs of 1s of length (equal, at least) and the number of 1s in all runs of 1s of length at least . To solve these counting problems, we use probability theory and we obtain simple and easy to compute explicit formulae as well as recursive schemes, for these potential useful in engineering numbers.

Social interaction as a heuristic for combinatorial optimization problems

We investigate the performance of a variant of Axelrod's model for dissemination of culture--the Adaptive Culture Heuristic (ACH)--on solving an NP-Complete optimization problem, namely, the classification of binary input patterns of size F by a Boolean Binary Perceptron. In this heuristic, N agents, characterized by binary strings of length F which represent possible solutions to the optimization problem, are fixed at the sites of a square lattice and interact with their nearest neighbors only. The interactions are such that the agents' strings (or cultures) become more similar to the low-cost strings of their neighbors resulting in the dissemination of these strings across the lattice. Eventually the dynamics freezes into a homogeneous absorbing configuration in which all agents exhibit identical solutions to the optimization problem. We find through extensive simulations that the probability of finding the optimal solution is a function of the reduced variable F/N(¼) so that the number of agents must increase with the fourth power of the problem size, N∝F⁴, to guarantee a fixed probability of success. In this case, we find that the relaxation time to reach an absorbing configuration scales with F⁶ which can be interpreted as the overall computational cost of the ACH to find an optimal set of weights for a Boolean binary perceptron, given a fixed probability of success.

On the distribution of runs of ones in binary strings

In this paper, we derive the number of binary strings which contain, for a given i k , exactly i k runs of 1’s of length k in all possible binary strings of length n , 1 ≤ k ≤ n . Such a knowledge about the distribution pattern of runs of 1’s in binary strings is useful in many engineering applications — for example, data compression, bus encoding techniques to reduce crosstalk in VLSI chip design, computer arithmetic using redundant binary number system and design of energy-efficient communication schemes in wireless sensor networks by transformation of runs of 1’s into compressed information patterns, among others. We present, here, a generating function based approach to derive a solution to this counting problem. Our experimental results demonstrate that, for most commonly used file formats, the observed distributions of exactly i k runs of length k , 1 ≤ k ≤ n , closely follow the theoretically derived distributions, for a given n . For n = 8 , we find that the experimentally obtained values for most file formats agree within ± 5 % of the theoretically obtained values for all i k runs of length k , 1 ≤ k ≤ n . Also, the root mean square (RMS) values of these deviations across all file types studied in this paper are less than 5% for n = 8 . In view of these facts, the results presented in this paper could be useful in various application domains, like the ones mentioned above.

Gray codes avoiding matchings

Graphs and Algorithms A (cyclic) n-bit Gray code is a (cyclic) ordering of all 2(n) binary strings of length n such that consecutive strings differ in a single bit. Equivalently, an n-bit Gray code can be viewed as a Hamiltonian path of the n-dimensional hypercube Q(n), and a cyclic Gray code as a Hamiltonian cycle of Q(n). In this paper we study (cyclic) Gray codes avoiding a given set of faulty edges that form a matching. Given a matching M and two vertices u, v of Q(n), n >= 4, our main result provides a necessary and sufficient condition, expressed in terms of forbidden configurations for M, for the existence of a Gray code between u and v that avoids M. As a corollary. we obtain a similar characterization for a cyclic Gray code avoiding M. In particular, in the case that M is a perfect matching, Q(n) has a (cyclic) Gray code that avoids M if and only if Q(n) - M is a connected graph. This complements a recent result of Fink, who proved that every perfect matching of Q(n) can be extended to a Hamiltonian cycle. Furthermore, our results imply that the problem of Hamilionicity of Q(n) with faulty edges, which is NP-complete in general, becomes polynomial for up to 2(n-1) edges provided they form a matching.

The Structure of a Full-length Response Regulator from Mycobacterium tuberculosis in a Stabilized Three-dimensional Domain-swapped, Activated State

The full-length, two-domain response regulator RegX3 from Mycobacterium tuberculosis is a dimer stabilized by three-dimensional domain swapping. Dimerization is known to occur in the OmpR/PhoB subfamily of response regulators upon activation but has previously only been structurally characterized for isolated receiver domains. The RegX3 dimer has a bipartite intermolecular interface, which buries 2357 A(2) per monomer. The two parts of the interface are between the two receiver domains (dimerization interface) and between a composite receiver domain and the effector domain of the second molecule (interdomain interface). The structure provides support for the importance of threonine and tyrosine residues in the signal transduction mechanism. These residues occur in an active-like conformation stabilized by lanthanum ions. In solution, RegX3 exists as both a monomer and a dimer in a concentration-dependent equilibrium. The dimer in solution differs from the active form observed in the crystal, resembling instead the model of the inactive full-length response regulator PhoB.

Gray-ordered Binary Necklaces

A $k$-ary necklace of order $n$ is an equivalence class of strings of length $n$ of symbols from $\{0,1,\ldots,k-1\}$ under cyclic rotation. In this paper we define an ordering on the free semigroup on two generators such that the binary strings of length $n$ are in Gray-code order for each $n$. We take the binary necklace class representatives to be the least of each class in this ordering. We examine the properties of this ordering and in particular prove that all binary strings factor canonically as products of these representatives. We conjecture that stepping from one representative of length $n$ to the next in this ordering requires only one bit flip, except at easily characterized steps.

How many strings are easy to predict?

It is well known in the theory of Kolmogorov complexity that most strings cannot be compressed; more precisely, only exponentially few (Θ (2 n − m )) binary strings of length n can be compressed by m bits. This paper extends the ‘incompressibility’ property of Kolmogorov complexity to the ‘unpredictability’ property of predictive complexity. The ‘unpredictability’ property states that predictive complexity (defined as the loss suffered by a universal prediction algorithm working infinitely long) of most strings is close to a trivial upper bound (the loss suffered by a trivial minimax constant prediction strategy). We show that only exponentially few strings can be successfully predicted and find the base of the exponent.

DICTIONARY LOOK-UP WITHIN SMALL EDIT DISTANCE

Let [Formula: see text] be a dictionary consisting of n binary strings of length m each, represented as a trie. The usual d-query asks if there exists a string in [Formula: see text] within Hamming distance d of a given binary query string q. We present a simple algorithm to determine if there is a member in [Formula: see text] within edit distanced of a given query string q of length m. The method takes time O(dmd+1) in the RAM model, independent of n, and requires O(dm) additional space. We also generalize the results for the case of the problem over a larger alphabet.

Approximation algorithms for Hamming clustering problems

We study Hamming versions of two classical clustering problems. The Hamming radius p- clustering problem (HRC) for a set S of k binary strings, each of length n, is to find p binary strings of length n that minimize the maximum Hamming distance between a string in S and the closest of the p strings; this minimum value is termed the p- radius of S and is denoted by ϱ. The related Hamming diameter p-clustering problem (HDC) is to split S into p groups so that the maximum of the Hamming group diameters is minimized; this latter value is called the p- diameter of S. We provide an integer programming formulation of HRC which yields exact solutions in polynomial time whenever k is constant. We also observe that HDC admits straightforward polynomial-time solutions when k=O(log n) and p=O(1), or when p=2. Next, by reduction from the corresponding geometric p-clustering problems in the plane under the L 1 metric, we show that neither HRC nor HDC can be approximated within any constant factor smaller than two unless P=NP. We also prove that for any ε>0 it is NP-hard to split S into at most pk 1/7− ε clusters whose Hamming diameter does not exceed the p-diameter, and that solving HDC exactly is an NP-complete problem already for p=3. Furthermore, we note that by adapting Gonzalez' farthest-point clustering algorithm [T. Gonzalez, Theoret. Comput. Sci. 38 (1985) 293–306], HRC and HDC can be approximated within a factor of two in time O( pkn). Next, we describe a 2 O( pϱ/ ε) k O( p/ ε) n 2-time (1+ ε)-approximation algorithm for HRC. In particular, it runs in polynomial time when p=O(1) and ϱ=O(log( k+ n)). Finally, we show how to find in O(( n ε +kn logn+k 2 logn)(2 ϱk) 2/ε) time a set L of O( plog k) strings of length n such that for each string in S there is at least one string in L within distance (1+ ε) ϱ, for any constant 0< ε<1.

The observability of the Fibonacci and the Lucas cubes

The Fibonacci cube Γ n is the graph whose vertices are binary strings of length n without two consecutive 1's and two vertices are adjacent when their Hamming distance is exactly 1. If the binary strings do not contain two consecutive 1's nor a 1 in the first and in the last position, we obtain the Lucas cube L n . We prove that the observability of Γ n and L n is n, where the observability of a graph G is the minimum number of colors to be assigned to the edges of G so that the coloring is proper and the vertices are distinguished by their color sets.