On the snake in the box problem

AbstractA snake in a graph is a simple cycle without chords. Denote by s(d) the length of a longest snake in the d-dimensional unit cube. We give a new proof of the theorem of Evdokimov that s(d) > λ2d, where λ is a positive constant

Constructions of Snake-in-the-Box Codes for Rank Modulation

Snake-in-the-box code is a Gray code which is capable of detecting a single error. Gray codes are important in the context of the rank modulation scheme which was suggested recently for representing information in flash memories. For a Gray code in this scheme the codewords are permutations, two consecutive codewords are obtained by using the "push-to-the-top" operation, and the distance measure is defined on permutations. In this paper the Kendall's $\tau$-metric is used as the distance measure. We present a general method for constructing such Gray codes. We apply the method recursively to obtain a snake of length $M_{2n+1}=((2n+1)(2n)-1)M_{2n-1}$ for permutations of $S_{2n+1}$, from a snake of length $M_{2n-1}$ for permutations of~$S_{2n-1}$. Thus, we have $\lim\limits_{n\to \infty} \frac{M_{2n+1}}{S_{2n+1}}\approx 0.4338$, improving on the previous known ratio of $\lim\limits_{n\to \infty} \frac{1}{\sqrt{\pi n}}$. By using the general method we also present a direct construction. This direct construction is based on necklaces and it might yield snakes of length $\frac{(2n+1)!}{2} -2n+1$ for permutations of $S_{2n+1}$. The direct construction was applied successfully for $S_7$ and $S_9$, and hence $\lim\limits_{n\to \infty} \frac{M_{2n+1}}{S_{2n+1}}\approx 0.4743$.Comment: IEEE Transactions on Information Theor

Long induced paths in expanders

We prove that any bounded degree regular graph with sufficiently strong spectral expansion contains an induced path of linear length. This is the first such result for expanders, strengthening an analogous result in the random setting by Dragani\'c, Glock and Krivelevich. More generally, we find long induced paths in sparse graphs that satisfy a mild upper-uniformity edge-distribution condition.Comment: 7 page

Further results on snakes in powers of complete graphs

AbstractBy a snake in a finite graph G is meant a cycle without chords. Denoted by S(G) the length of a longest snake in G. In this paper we obtain a new lower bound for S(G) in the case where G is the product of d copies of the complete graph on n vertices

Separator-based graph embedding into multidimensional grids with small edge-congestion

We study the problem of embedding a guest graph with minimum edge-congestion into a multidimensional grid with the same size as that of the guest graph. Based on a well-known notion of graph separators, we show that an embedding with a smaller edge-congestion can be obtained if the guest graph has a smaller separator, and if the host grid has a higher but constant dimension. Specifically, we prove that any graph with NN nodes, maximum node degree ΔΔ, and with a node-separator of size ss, where ss is a function such that s(n)=O(nα)s(n)=O(nα) with 0≤α1/(1−α)d>1/(1−α), O(ΔlogN)O(ΔlogN) if d=1/(1−α)d=1/(1−α), and View the MathML sourceO(ΔNα−1+1d) if d1/(1−α)d>1/(1−α), and matches an existential lower bound within a constant factor if d≤1/(1−α)d≤1/(1−α). Our result implies that if the guest graph has an excluded minor of a fixed size, such as a planar graph, then we can obtain an edge-congestion of O(ΔlogN)O(ΔlogN) for d=2d=2 and O(Δ)O(Δ) for any fixed d≥3d≥3. Moreover, if the guest graph has a fixed treewidth, such as a tree, an outerplanar graph, and a series–parallel graph, then we can obtain an edge-congestion of O(Δ)O(Δ) for any fixed d≥2d≥2. To design our embedding algorithm, we introduce edge-separators bounding extension , such that in partitioning a graph into isolated nodes using edge-separators recursively, the number of outgoing edges from a subgraph to be partitioned in a recursive step is bounded. We present an algorithm to construct an edge-separator with extension of O(Δnα)O(Δnα) from a node-separator of size O(nα)O(nα)

Optimization of parameters for binary genetic algorithms.

In the GA framework, a species or population is a collection of individuals or chromosomes, usually initially generated randomly. A predefined fitness function guides selection while operators like crossover and mutation are used probabilistically in order to emulate reproduction.Genetic Algorithms (GAs) belong to the field of evolutionary computation which is inspired by biological evolution. From an engineering perspective, a GA is an heuristic tool that can approximately solve problems in which the search space is huge in the sense that an exhaustive search is not tractable. The appeal of GAs is that they can be parallelized and can give us "good" solutions to hard problems.One of the difficulties in working with GAs is choosing the parameters---the population size, the crossover and mutation probabilities, the number of generations, the selection mechanism, the fitness function---appropriate to solve a particular problem. Besides the difficulty of the application problem to be solved, an additional difficulty arises because the quality of the solution found, or the sum total of computational resources required to find it, depends on the selection of the parameters of the GA; that is, finding a correct fitness function and appropriate operators and other parameters to solve a problem with GAs is itself a difficult problem. The contributions of this dissertation, then, are: to show that there is not a linear correlation between diversity in the initial population and the performance of GAs; to show that fitness functions that use information from the problem itself are better than fitness functions that need external tuning; and to propose a relationship between selection pressure and the probabilities of crossover and mutation that improve the performance of GAs in the context of of two extreme schema: small schema, where the building block in consideration is small (each bit individually can be considered as part of the general solution), and long schema, where the building block in consideration is long (a set of interrelated bits conform part of the general solution).Theoretical and practical problems like the one-max problem and the intrusion detection problem (considered as problems with small schema) and the snake-in-the-box problem (considered as a problem with long schema) are tested under the specific hypotheses of the Dissertation.The Dissertation proposes three general hypotheses. The first one, in an attempt to measure the impact of the input over the output, study that there is not a linear correlation between diversity in the initial population and performance of GAs. The second one, proposes the use of parameters that belong to the problem itself to joint objective and constraint in fitness functions, and the third one use Holland's Schema Theorem for finding an interrelation between selection pressure and the probabilities of crossover and mutation that, if obeyed, is expected to result in better performance of the GA in terms of the solution quality found within a given number of generations and/or the number of generations to find a solution of a given quality than if the interrelation is not obeyed

An iterated local search with . . . IN THE BOX PROBLEM

A snake-in-the box code, first described in [Kautz, 1958], is an achordal open path in a hypercube. Finding bounds for the longest snake at each dimension has proven to be a difficult problem in mathematics and computer science. Evolutionary techniques have succeeded in tightening the bounds of longest snakes in several dimensions [Potter, 1994] [Casella, 2005]. This thesis utilizes an Iterated Local Search heuristic with adaptive memory on the snake-in-the-box problem. The results match the best published results for problem instances up to dimension 8. The lack of implicit parallelism segregates this experiment from previous heuristics applied to this problem. As a result, this thesis provides insight into those problems in which evolutionary methods dominate