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Convolutional neural networks (CNNs) have proven to be very successful in learning task-specific computer vision features. To integrate features from different layers in standard CNNs, we present a fusing framework of shortcut convolutional neural networks (S-CNNs). This framework can fuse arbitrary scale features by adding weighted shortcut connections to the standard CNNs. Besides the framework, we propose a shortcut indicator (SI) of binary string to stand for a specific S-CNN shortcut style. Additionally, we design a learning algorithm for the proposed S-CNNs. Comprehensive experiments are conducted to compare its performances with standard CNNs on multiple benchmark datasets for different visual tasks. Empirical results show that if we choose an appropriate fusing style of shortcut connections with learnable weights, S-CNNs can perform better than standard CNNs regarding accuracy and stability in different activation functions and pooling schemes initializations, and occlusions. Moreover, S-CNNs are competitive with ResNets and can outperform GoogLeNet, DenseNets, Multi-scale CNN, and DeepID.

In this paper, we study the self-assembly of rectangles in a non-cooperative, 3D version of Winfree's abstract Tile Assembly Model. We prove two results. First, we give the full details for the proof of a general construction for the efficient self-assembly of thin rectangles that was first reported by Furcy, Summers and Wendlandt (2019). This construction is non-cooperative and “just-barely” 3D in the sense that it places tiles at most one step into the third dimension. Second, we give a non-cooperative, just-barely 3D optimal encoding construction that self-assembles the bits of a given binary string along the perimeter of a thin rectangle of constant height.

The Fibonacci cube is the subgraph of the hypercube induced by the vertices whose binary string representations do not contain two consecutive 1s. These cubes were presented as an alternative interconnection network. In this paper, we calculate the number of induced paths and cycles of small length in Fibonacci cubes by using the recursive structure of these graphs.

When a pairing $$e: {\mathbb {G}}_1 \times {\mathbb {G}}_2 \rightarrow {\mathbb {G}}_{T}$$ e : G 1 × G 2 → G T , on an elliptic curve E defined over a finite field $${\mathbb {F}}_q$$ F q , is exploited for an identity-based protocol, there is often the need to hash binary strings into $${\mathbb {G}}_1$$ G 1 and $${\mathbb {G}}_2$$ G 2 . Traditionally, if E admits a twist $${\tilde{E}}$$ E ~ of order d, then $${\mathbb {G}}_1=E({\mathbb {F}}_q) \cap E[r]$$ G 1 = E ( F q ) ∩ E [ r ] , where r is a prime integer, and $${\mathbb {G}}_2={\tilde{E}}({\mathbb {F}}_{q^{k/d}}) \cap {\tilde{E}}[r]$$ G 2 = E ~ ( F q k / d ) ∩ E ~ [ r ] , where k is the embedding degree of E w.r.t. r. The standard approach for hashing into $${\mathbb {G}}_2$$ G 2 is to map to a general point $$P \in {\tilde{E}}({\mathbb {F}}_{q^{k/d}})$$ P ∈ E ~ ( F q k / d ) and then multiply it by the cofactor $$c=\#{\tilde{E}}({\mathbb {F}}_{q^{k/d}})/r$$ c = # E ~ ( F q k / d ) / r . Usually, the multiplication by c is computationally expensive. In order to speed up such a computation, two different methods—by Scott et al. (International conference on pairing-based cryptography. Springer, Berlin, pp 102–113, 2009) and by Fuentes-Castaneda et al. (International workshop on selected areas in cryptography)—have been proposed. In this paper we consider these two methods for BLS pairing-friendly curves having $$k \in \{12,24,30,42,48\}$$ k ∈ { 12 , 24 , 30 , 42 , 48 } , providing efficiency comparisons. When $$k=42,48$$ k = 42 , 48 , the application of Fuentes et al. method requires expensive computations which were infeasible for the computational power at our disposal. For these cases, we propose hashing maps that we obtained following Fuentes et al. idea.

We study codes that can detect the exact number of deletions and insertions in concatenated binary strings. We construct optimal codes for the case of detecting up to $\del$ deletions. We prove the optimality of these codes by deriving a converse result which shows that the redundancy of our codes is asymptotically optimal in $\del$ among all families of deletion detecting codes, and particularly optimal among all block-by-block decodable codes. For the case of insertions, we construct codes that can detect up to $2$ insertions in each concatenated binary string., Comment: Shorter version accepted in ISIT 2021

Distribution matchers for finite alphabets are shown to have informational divergences that grow logarithmically with the block length, generalizing a basic result for binary strings., 2021 IEEE International Symposium on Information Theory

In the paper, a study of detection probabilities and experimental tests of an intrusion detection system (IDS) utilizing the artificial immune system (AIS) based on the negative selection approach are presented. The algorithm uses binary strings called receptors to detect intrusions in programs in the operating system. In the work, a statistical approach is used to calculate the probabilities of detecting the intrusions and the results are compared with the experimental tests. New experiments are conducted using a virtual machine as the environment to check the effectiveness of the proposed IDS. For this purpose, real-world malware samples are launched in the virtual machine. The research of the IDS is then presented, analyzed and concluded.

We execute the average running time of the various binary strings of different length and evaluate the other useful characteristics of a hash function which is to be Pseudo-Randomness of a ...

The reduced Collatz map (Syracuse function) can be stated as “for any odd positive integer x, calculate 3x+1 and then divide by 2 until the result is odd.” We calculate the change in bit string len...

The generalized Fibonacci cube Q h ( f ) is the graph obtained from the h -cube Q h by removing all vertices that contain a given binary string f as a substring. If G is an induced subgraph of Q h , then the cube-complement of G is the graph induced by the vertices of Q h which are not in G . In particular, the cube-complement of a generalized Fibonacci cube Q h ( f ) is the subgraph of Q h induced by the set of all vertices that contain f as a substring. The questions whether a cube-complement of a generalized Fibonacci cube is (i) connected, (ii) an isometric subgraph of a hypercube or (iii) a median graph are studied. Questions (ii) and (iii) are completely solved, i.e. the sets of binary strings that allow a graph of this class to be an isometric subgraph of a hypercube or a median graph are given. The cube-complement of a daisy cube is also studied.

The following metric dimension of join two paths $P_2 + P_t$ is determined as follows. For every $k = 1, 2, 3, ...$ and $t = 2 + 5k$ or $t = 3 + 5k$, the dimension of $P_2 + P_t$ is $2 + 2k$ whereas for $t = 4 + 5k, t = 5(k+1)$ or $t = 1 + 5(k+1)$, the dimension is $3 + 2k$. In case $t \geq 7$, the dimension is determined by a chosen (maximal) ordered basis for $P_2 + P_t$ in which the integers 1, 2 are the two consecutive vertices of $P_2$ and the next integers $3, 4, ..., t + 2$ are the $t$ consecutive vertices of $P_t$. If $t \geq 10$, the ordered binary string contains repeated substrings of length 5. For $t < 7$, the dimension is easily found using a computer search, or even just using hand computations.

Kolmogorov complexity is an essential tool in the study of algorithmic information theory, and is used in the fields of Artificial Intelligence, cryptography, and coding theory. The formalisation of the theorems of Kolmogorov complexity is also key to proving results in the theory of Intelligent Agents, specifically the results in Universal Artificial Intelligence. In this paper, we present a mechanisation of some of these fundamental results. In particular, building on HOL4's existing model of computability, we provide a formal definition of the complexity of a binary string, and then prove (i) that Kolmogorov complexity is uncomputable; (ii) the Kolmogorov Complexity invariance theorem; (iii) the Kraft and Kolmogorov-Kraft inequalities; and (iv) key Kolmogorov Complexity inequalities.

The goal of this chapter is to provide a quick overview of the main conversion mechanism for numbers and strings that are specific to the cryptography field. Representing numbers and binary strings in different bases is one of the most critical steps in cryptography. This chapter covers the main steps and discusses how they are done properly in MATLAB.

Background Given a binary tree $\mathcal {T}$ T of n leaves, each leaf labeled by a string of length at most k, and a binary string alignment function ⊗, an implied alignment can be generated to describe the alignment of a dynamic homology for $\mathcal {T}$ T . This is done by first decorating each node of $\mathcal {T}$ T with an alignment context using ⊗, in a post-order traversal, then, during a subsequent pre-order traversal, inferring on which edges insertion and deletion events occurred using those internal node decorations. Results Previous descriptions of the implied alignment algorithm suggest a technique of “back-propagation” with time complexity $\mathcal {O}\left (k^{2} * n^{2}\right)$ O k 2 ∗ n 2 . Here we describe an implied alignment algorithm with complexity $\mathcal {O}\left (k * n^{2}\right)$ O k ∗ n 2 . For well-behaved data, such as molecular sequences, the runtime approaches the best-case complexity of Ω(k∗n). Conclusions The reduction in the time complexity of the algorithm dramatically improves both its utility in generating multiple sequence alignments and its heuristic utility.

We study the two party problem of randomly selecting a common string among all the strings of length n. We want the protocol to have the property that the output distribution has high Shannon entropy or high min entropy, even when one of the two parties is dishonest and deviates from the protocol. We develop protocols that achieve high, close to n, Shannon entropy and simultaneously min entropy close to n/2. In the literature the randomness guarantee is usually expressed in terms of “resilience”. The notion of Shannon entropy is not directly comparable to that of resilience, but we establish a connection between the two that allows us to compare our protocols with the existing ones. We construct an explicit protocol that yields Shannon entropy $$n - O(1)$$ and has $$O(\log ^* n)$$ rounds, improving over the protocol of Goldreich et al. (SIAM J Comput 27: 506–544, 1998) that also achieves this entropy but needs O(n) rounds. Both these protocols need $$O(n^2)$$ bits of communication. Next we reduce the number of rounds and the length of communication in our protocols. We show the existence, non-explicitly, of a protocol that has 6 rounds, O(n) bits of communication and yields Shannon entropy $$n- O(\log n)$$ and min entropy $$n/2 - O(\log n)$$ . Our protocol achieves the same Shannon entropy bound as, also non-explicit, protocol of Gradwohl et al. (in: Dwork (ed) Advances in Cryptology—CRYPTO ‘06, 409–426, Technical Report , 2006), however achieves much higher min entropy: $$n/2 - O(\log n)$$ versus $$O(\log n)$$ . Finally we exhibit a very simple 3-round explicit “geometric” protocol with communication length O(n). We connect the security parameter of this protocol with the well studied Kakeya problem motivated by Harmonic Analysis and Analytic Number Theory. We prove that this protocol has Shannon entropy $$n-o(n)$$ . Its relation to the Kakeya problem follows a new and different approach to the random selection problem than any of the previously known protocols.

During the decade of the 1960s several individuals independently arrived at a notion that a binary string is random if its shortest description is the string itself. Among the main protagonists in this story is the celebrated Russian mathematician Andrei Kolmogorov, whom we met earlier, and Gregory Chaitin, information theorist and computer scientist.

Structure prediction is an important and widely studied problem in network science and machine learning, finding its applications in various fields. Despite the significant progress in prediction algorithms, the fundamental predictability of structures remains unclear, as networks’ complex underlying formation dynamics are usually unobserved or difficult to describe. As such, there has been a lack of theoretical guidance on the practical development of algorithms for their absolute performances. Here, for the first time, we find that the normalized shortest compression length of a network structure can directly assess the structure predictability. Specifically, shorter binary string length from compression leads to higher structure predictability. We also analytically derive the origin of this linear relationship in artificial random networks. In addition, our finding leads to analytical results quantifying maximum prediction accuracy, and allows the estimation of the network dataset potential values through the size of the compressed network data file., The likelihood of linking within a complex network is of importance to solve real-world problems, but it is challenging to predict. Sun et al. show that the link predictability limit can be well estimated by measuring the shortest compression length of a network without a need of prediction algorithm.

Consider two sequences of ${n}$ independent and identically distributed fair coin tosses, ${X}=({X}_{1},\ldots,{X}_{n})$ and ${Y}=({Y}_{1},\ldots,{Y}_{n})$ , which are $\rho $ -correlated for each ${j}$ , i.e. $\mathbb {P}[{X}_{j}={Y}_{j}] = {\frac{1+\rho }{ 2}}$ . We study the question of how large (small) the probability $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$ can be among all sets ${A},{B}\subset \{0,1\}^{n}$ of a given cardinality. For sets $|{A}|,|{B}| = \Theta (2^{n})$ it is well known that the largest (smallest) probability is approximately attained by concentric (anti-concentric) Hamming balls, and this can be proved via the hypercontractive inequality (reverse hypercontractivity). Here we consider the case of $|{A}|,|{B}| = 2^{\Theta ({n})}$ . By applying a recent extension of the hypercontractive inequality of Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming balls of the same size approximately maximize $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$ in the regime of $\rho \to 1$ . We also prove a similar tight lower bound, i.e. show that for $\rho \to 0$ the pair of opposite Hamming balls approximately minimizes the probability $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$ .

We consider hypercubes with pairwise disjoint faulty edges. An n-dimensional hypercube Q n is an undirected graph with 2 n nodes, each labeled with a distinct binary string of length n. The parity of the vertex is 0 if the number of ones in its label is even, and is 1 if the number of ones is odd. Two vertices a and b are connected by an edge iff a and b differ in one position. If a and b differ in position i, then we say that the edge ( a , b ) goes in dimension i and we define the parity of the edge as the parity of the end with 0 on the position i. It was already known that Q n is not Hamiltonian if all edges going in one dimension and of the same parity are faulty. In this paper we show that if n ≥ 4 then all other hypercubes are Hamiltonian. In other words, every cube Q n , with n ≥ 4 and disjoint faulty edges is Hamiltonian if and only if for each dimension there are two healthy crossing edges of different parity.

The generalized Fibonacci Q d ( f ) is defined as the graph obtained from the d -cube Q d by removing all vertices that contain a given binary string f as a contiguous substring. This idea was introduced by Ilic, Klavžar and Rho. A binary string f is called isometric if Q d ( f ) is an isometric subgraph of Q d for all d ≥ 1 , otherwise it is called non-isometric. In this paper, we prove that a string f is isometric if and only if f n is isometric for any n ≥ 1 . This result can help us to construct more isometric strings and significantly increase the number of isometric strings.

Let {φ p } be an optimal Godel numbering of the family of computable functions (in Schnorr’s sense), where p ranges over binary strings. Assume that a list of strings L(p) is computable from p and for all p contains a φ-program for φ p whose length is at most e bits larger than the length of the shortest φ-programs for φ p . We show that for infinitely many p the list L(p) must have 2|p|−e−O(1) strings. Here e is an arbitrary function of p.

In a 0 - 1 sequence of Markov dependent trials we consider a statistic which counts strings of a limited length run of 0s between subsequent 1s. Its probability mass function is used to determine the chance that a stochastic process remains or not in statistical control. Illustrative numerics are presented.

Generalized Fibonacci cube Q n ( f ) is the graph obtained from the n -cube Q n by removing all vertices that contain a given binary string f as a consecutive substring. A binary string f is called bad if Q n ( f ) is not an isometric subgraph of Q n for some n , and the smallest such integer n , denoted by B ( f ) , is called the index of f . Ilic, Klavžar and Rho posed a problem that if Q n ( f ) is not an isometric subgraph of Q n , is there a dimension n ′ such that Q n ( f ) can be isometrically embedded into Q n ′ ? We give a negative answer to this problem by showing that if f is bad, then for any n ≥ B ( f ) , Q n ( f ) cannot be isometrically embedded to any hypercube.

We consider hypercube Qn with one or two vertices of degree one.In such cubes some pairs of vertices can be connected by Hamiltonian paths.We show that they can be connected even if n - 3 additional edges are faulty. The n-dimensional hypercube Qn is a graph with 2n vertices, each labeled with a distinct binary string of length n. The vertices are connected by an edge if and only if their labels differ in one bit. The hypercube is bipartite, the set of nodes is the union of two sets: nodes of parity 0 (the number of ones in their labels is even) and nodes of parity 1 (the number of ones is odd). We consider Hamiltonian paths in hypercubes with faulty edges and prove the following: (1) If Qn has one vertex u of degree 1, then u can be connected by a Hamiltonian path with every vertex v that is of a parity different than u and that is not connected with u by a healthy edge. (2) If Qn with n ? 4 has two vertices u and v of degree 1, then they can be connected by a Hamiltonian path if the distance between u and v is odd and greater than 1 or if u and v are connected by the faulty edge. (3) If Qn contains a cycle (u, v, w, x) in which all edges going away from the cycle from u and w are faulty, then u or w can be connected by a Hamiltonian path with any vertex outside the cycle that is of different parity than u and w.Moreover, in all three cases, the thesis remains true even if Qn has n - 3 additional faulty edges. Furthermore, in all three cases, no other Hamiltonian paths are possible.

In order to describe and study the communication between the cells or between cells and environment, tissue P system with symport/antiport rules and tissue-like P system with evolutional symport/antiport rules was introduced as a class of distributed parallel computing model, the object can change their location or evolve in the communication process. But the objects and evolutional rules are not specific, so in this paper, a new computing model for P system called tissue-like P system with mutational symport/antiport rules and trigger mechanism (TPMTM) is proposed, where all genetic materials are encoded by binary strings and the evolutional rules are defined by mutational rules. The TPMTM' s computational power is testified. Extraordinary, tissue-like P system with one cell what executing mutational antiport rules of length of mutational position at most 5 or executing mutational symport rules of length of mutational position at most 7 are testified Turing universal. So, this tissue P system with mutational communication rules that we proposed is computable.

With the popularity of mobile devices and the rapid development of position acquisition technology, the amount of trajectory data has soared dramatically. It is time-consuming to manage and mine massive trajectory data, because we need to access different trajectory samples or different parts of a trajectory for multiple times. Therefore, it is necessary to devise an efficient data management technology to fast retrieve the desired trajectories. In general, building indexes is a basic step for solving query problems. However, traditional spatial indexing technologies are mostly designed for moving objects, and thus are unable to achieve fast trajectory data query and efficient computing analysis. In this regard, we propose traj2bits, a bitmap-based trajectory data encoding schema, to convert trajectories into binary strings. Based on traj2bits, we also design a trajectory query method. Experiments on two real datasets have shown that traj2bits improves the spatio-temporal efficiency of trajectory query. Compared with other schemes, traj2bitsuery encoding occupies less than 1/10 of the disk space, its encoding efficiency is at least four times faster and its range query time is reduced by at least 65%.

We study the query complexity of a permutation-based variant of the guessing game Mastermind. In this variant, the secret is a pair $(z,\pi)$ which consists of a binary string $z \in \{0,1\}^n$ and a permutation $\pi$ of $[n]$. The secret must be unveiled by asking queries of the form $x \in \{0,1\}^n$. For each such query, we are returned the score \[ f_{z,\pi}(x):= \max \{ i \in [0..n]\mid \forall j \leq i: z_{\pi(j)} = x_{\pi(j)}\}\,;\] i.e., the score of $x$ is the length of the longest common prefix of $x$ and $z$ with respect to the order imposed by $\pi$. The goal is to minimize the number of queries needed to identify $(z,\pi)$. This problem originates from the study of black-box optimization heuristics, where it is known as the \textsc{LeadingOnes} problem. In this work, we prove matching upper and lower bounds for the deterministic and randomized query complexity of this game, which are $\Theta(n \log n)$ and $\Theta(n \log \log n)$, respectively., Comment: Full version of a result previously announced in 2013. Accepted subject to minor revision at Discrete Applied Mathematics

This paper proposes a novel method for fast and compact recognition of biomedical and other complex signals. It is essentially based on decomposing the signal sequences to be recognized into binary strings. Then each binary string is divided into smaller strings corresponding to Boolean functions. Based on complexities for neural implementation of Boolean functions a feature vector is computed by counting the occurrences of functions with certain complexities. Using enough labeled signal sequences, a classifier can be easily trained and then used for the automatic recognition of the signals. To decrease costs, a hardware oriented Extreme Learning Machine is used. Herein, in addition to describing the method, we consider the widely used Bonn dataset for epileptic seizure detection to demonstrate that our method offers good accuracies (more than 97%) with the advantage of very low implementation costs. Given its convenience and simplicity, the proposed method can be easily integrated in low cost portable equipment, to be operated, for instance, in the context of Internet of Things.

The self-assembly of proteins into protein quaternary structures is of fundamental importance to many biological processes, and protein misassembly is responsible for a wide range of proteopathic diseases. In recent years, abstract lattice models of protein self-assembly have been used to simulate the evolution and assembly of protein quaternary structure, and to provide a tractable way to study the genotype-phenotype map of such systems. Here we generalize these models by representing the interfaces as mutable binary strings. This simple change enables us to model the evolution of interface strengths, interface symmetry, and deterministic assembly pathways. Using the generalized model we are able to reproduce two important results established for real protein complexes: The first is that protein assembly pathways are under evolutionary selection to minimize misassembly. The second is that the assembly pathway of a complex mirrors its evolutionary history, and that both can be derived from the relative strengths of interfaces. These results demonstrate that the generalized lattice model offers a powerful new idealized framework to facilitate the study of protein self-assembly processes and their evolution., Author summary Protein complexes assemble by joining individual proteins together through interacting binding sites. Because of the long time scales of biological evolution, it can be difficult to reconstruct how these interactions change over time. We use simplified representations of proteins to simulate the evolution of these complexes on a computer. In some cases the order in which the complex assembles is crucial. We show that biological evolution increases the strength of interactions that must occur earlier, and decreases the strength of later interactions. Similar knowledge of interactions being preferred to be stronger or weaker can also help to predict the evolutionary ancestry of a complex. While these simulations are too idealized to make exact predictions, this general link between ordered pathways in assembly and evolution matches well-established observations that have been made in real protein complexes. This means that our model provides a powerful framework to help study protein complex assembly and evolution.

The Carlson-Simpson lemma is a combinatorial statement occurring in the proof of the Dual Ramsey theorem. Formulated in terms of variable words, it informally asserts that given any finite coloring of the strings, there is an infinite sequence with infinitely many variables such that for every valuation, some specific set of initial segments is homogeneous. Friedman, Simpson, and Montalban asked about its reverse mathematical strength. We study the computability-theoretic properties and the reverse mathematics of this statement, and relate it to the finite union theorem. In particular, we prove the Ordered Variable word for binary strings in ACA0., 9 pages

In this paper, we study the problem of edge exploration of an n node graph by a mobile agent. The nodes of the graph are unlabeled, and the ports at a node of degree d are arbitrarily numbered \(0,\dots , d-1\). A mobile agent, starting from some node, has to visit all the edges of the graph and stop. The time of the exploration is the number of edges the agent traverses before it stops. The task of exploration can not be performed even for a class of cycles if no additional information, called advice, is provided to the agent a priori. Following the paradigm of algorithms with advice, this priori information is provided to the agent by an Oracle in the form of a binary string. The Oracle knows the graph, but does not have the knowledge of the starting point of the agent. In this paper, we consider the following two problems of edge exploration. The first problem is: “how fast is it possible to explore an n node graph regardless of the size of advice provided to the agent?”

Colussi (2011) described the binary strings that iterate to 1 under the Collatz map.We confirm Colussi's description via elementary methods.Colussi's strings hold the repetends of 1 / 3 h , optionally rotated and replicated.It is only in binary that we see the repetends of 1 / 3 h .We see the Collatz map as multiplying up a fraction, versus reducing a large number. The Collatz function can be stated as 'for any odd positive integer x, calculate 3 x + 1 and then divide by 2 until the result is odd'. Colussi (2011) discovered and proved that if x attains 1 on the kth iteration of the Collatz function, then its binary representation can be written as the concatenation of strings s k s k - 1 ? s 1 where each s h is a finite and contiguous extract from the representation of 1 3 h . We provide an elementary confirmation of Colussi's finding, and comment on how working in binary 'protects' the repetends of 1 3 h as formed into each s h .

This paper proposes a new approach to the forensic investigation of Internet history artefacts by aggregating the history from a recovered device into sessions and comparing those sessions to other sessions to determine whether they are one-time events or form a repetitive or habitual pattern. We describe two approaches for performing the session aggregation: fixed-length sessions and variable-length sessions. We also describe an approach for identifying repetitive pattern of life behaviour and show how such patterns can be extracted and represented as binary strings. Using the Jaccard similarity coefficient, a session-to-session comparison can be performed and the sessions can be analysed to determine to what extent a particular session is similar to any other session in the Internet history, and thus is highly likely to correspond to the same user. Experiments have been conducted using two sets of test data, where multiple users have access to the same computer. By identifying patterns of Internet usage that are unique to each user, our approach exhibits a high success rate in attributing particular sessions of the Internet history to the correct user. This can provide considerable help to a forensic investigator trying to establish which user was using the computer when a web-related crime was committed.

In this work we propose a mechanism to optimize the capacity of the main corridor within a railway network with a radial-backbone or X-tree structure. The radial-backbone (or X-tree) structure is composed of two types of lines: the primary lines that travel exclusively on the common backbone (main corridor) and radial lines which, starting from the common backbone, branch out to individual locations. We define possible line configurations as binary strings and propose operators on them for their analysis, yielding an effective algorithm for generating an optimal design and train frequencies. We test our algorithm on real data for the high speed line Madrid–Seville. A frequency plan consistent with the optimal capacity is then proposed in order to eliminate the number of transfers between lines as well as to minimize the network fleet size, determining the minimum number of vehicles needed to serve all travel demand at maximum occupancy.