Rigorous Upper Bound for the Discrete Bak–Sneppen Model

Fix some $$p\in [0,1]$$ p ∈ [ 0 , 1 ] and a positive integer n. The discrete Bak–Sneppen model is a Markov chain on the space of zero-one sequences of length n with periodic boundary conditions. At each moment of time a minimum element (typically, zero) is chosen with equal probability, and it is then replaced alongside both its neighbours by independent Bernoulli(p) random variables. Let $$\nu ^{(n)}(p)$$ ν ( n ) ( p ) be the probability that an element of this sequence equals one under the stationary distribution of this Markov chain. It was shown in Barbay and Kenyon (in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (Washington, DC, 2001), pp. 928–933, SIAM, Philadelphia, PA, 2001) that $$\nu ^{(n)}(p)\rightarrow 1$$ ν ( n ) ( p ) → 1 as $$n\rightarrow \infty $$ n → ∞ when $$p>0.54\dots $$ p > 0.54 ⋯ ; the proof there is, alas, not rigorous. The complimentary fact that $$\displaystyle \limsup _{n\rightarrow \infty } \nu ^{(n)}(p)< 1$$ lim sup n → ∞ ν ( n ) ( p ) < 1 for $$p\in (0,p')$$ p ∈ ( 0 , p ′ ) for some $$p'>0$$ p ′ > 0 is much harder; this was eventually shown in Meester and Znamenski (J Stat Phys 109:987–1004, 2002). The purpose of this note is to provide a rigorous proof of the result from Barbay and Kenyon (in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (Washington, DC, 2001), pp. 928–933, SIAM, Philadelphia, PA, 2001), as well as to improve it, by showing that $$\nu ^{(n)}(p)\rightarrow 1$$ ν ( n ) ( p ) → 1 when $$p>0.45$$ p > 0.45 . (Our method, in fact, shows that with some finer tuning the same is true for $$p>0.419533$$ p > 0.419533 .)

Property Testing on Graphs and Games

Constant-time algorithms are powerful tools, since they run by reading only a constant-sized part of each input. Property testing is the most popular research framework for constant-time algorithms. In property testing, an algorithm determines whether a given instance satisfies some predetermined property or is far from satisfying the property with high probability by reading a constant-sized part of the input. A property is said to be testable if there is a constant-time testing algorithm for the property. This chapter covers property testing on graphs and games. The fields of graph algorithms and property testing are two of the main streams of research on discrete algorithms and computational complexity. In the section on graphs in this chapter, we present some important results, particularly on the characterization of testable graph properties. At the end of the section, we show results that we published in 2020 on a complete characterization (necessary and sufficient condition) of testable monotone or hereditary properties in the bounded-degree digraphs. In the section on games, we present results that we published in 2019 showing that the generalized chess, Shogi (Japanese chess), and Xiangqi (Chinese chess) are all testable. We believe that this is the first results for testable EXPTIME-complete problems.

Feature Selection and Knapsack Problem Resolution Based on a Discrete Backtracking Optimization Algorithm

Backtracking search optimization algorithm is a recent stochastic-based global search algorithm for solving real-valued numerical optimization problems. In this paper, a binary version of backtracking algorithm is proposed to deal with 0-1 optimization problems such as feature selection and knapsack problems. Feature selection is the process of selecting a subset of relevant features for use in model construction. Irrelevant features can negatively impact model performances. On the other hand, knapsack problem is a well-known optimization problem used to assess discrete algorithms. The objective of this research is to evaluate the discrete version of backtracking algorithm on the two mentioned problems and compare obtained results with other binary optimization algorithms using four usual classifiers: logistic regression, decision tree, random forest, and support vector machine. Empirical study on biological microarray data and experiments on 0-1 knapsack problems show the effectiveness of the binary algorithm and its ability to achieve good quality solutions for both problems.

Optimal fast Johnson–Lindenstrauss embeddings for large data sets

Johnson–Lindenstrauss embeddings are widely used to reduce the dimension and thus the processing time of data. To reduce the total complexity, also fast algorithms for applying these embeddings are necessary. To date, such fast algorithms are only available either for a non-optimal embedding dimension or up to a certain threshold on the number of data points. We address a variant of this problem where one aims to simultaneously embed larger subsets of the data set. Our method follows an approach by Nelson et al. (New constructions of RIP matrices with fast multiplication and fewer rows. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1515-1528, 2014): a subsampled Hadamard transform maps points into a space of lower, but not optimal dimension. Subsequently, a random matrix with independent entries projects to an optimal embedding dimension. For subsets whose size scales at least polynomially in the ambient dimension, the complexity of this method comes close to the number of operations just to read the data under mild assumptions on the size of the data set that are considerably less restrictive than in previous works. We also prove a lower bound showing that subsampled Hadamard matrices alone cannot reach an optimal embedding dimension. Hence, the second embedding cannot be omitted.

Quantitative stability and error estimates for optimal transport plans

Optimal transport maps and plans between two absolutely continuous measures $\mu$ and $\nu$ can be approximated by solving semidiscrete or fully discrete optimal transport problems. These two problems ensue from approximating $\mu$ or both $\mu$ and $\nu$ by Dirac measures. Extending an idea from Gigli (2011, On Hölder continuity-in-time of the optimal transport map towards measures along a curve. Proc. Edinb. Math. Soc. (2), 54, 401–409), we characterize how transport plans change under the perturbation of both $\mu$ and $\nu$. We apply this insight to prove error estimates for semidiscrete and fully discrete algorithms in terms of errors solely arising from approximating measures. We obtain weighted $L^2$ error estimates for both types of algorithms with a convergence rate $O(h^{1/2})$. This coincides with the rate in Theorem 5.4 in Berman (2018, Convergence rates for discretized Monge–Ampère equations and quantitative stability of optimal transport. Preprint available at arXiv:1803.00785) for semidiscrete methods, but the error notion is different.

An improved path discretization method for automated fiber placement

In the automated fiber placement process, the continuous placement paths need to be discretized into a finite number of path points because the laying head cannot continuously trace the predetermined curved path. However, the discretization of the placement path, which is a spatial curve, will inevitably introduce error. In this paper, an improved path discretization algorithm is proposed for the fiber placement of complex double-curved structures. Firstly, the discrete error was decomposed into normal direction and binormal direction, and they are correlated with the laying process and their influences on the laying quality are discussed, respectively. Secondly, the relationship between the binormal error and the overlap of the tow is analyzed with differential geometry, and the influence of the normal error on laying force is discussed by the pressure experiment and the finite element method. Finally, the improved path discretization algorithm has been verified on double-curved surface and compared with the traditional path discrete algorithms. The results showed that the number of discrete path points decreases by 45.8% on average compared with the chordal deviation discretization algorithm and by 63.1% compared with the equal-arc discretization algorithm.