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Table of Contents

Algorithms, Volume 9, Issue 2 (June 2016)

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Research

Jump to: Review, Other

Open AccessArticle A Direct Search Algorithm for Global Optimization
Algorithms 2016, 9(2), 40; doi:10.3390/a9020040
Received: 19 January 2016 / Revised: 25 April 2016 / Accepted: 7 June 2016 / Published: 13 June 2016
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Abstract
A direct search algorithm is proposed for minimizing an arbitrary real valued function. The algorithm uses a new function transformation and three simplex-based operations. The function transformation provides global exploration features, while the simplex-based operations guarantees the termination of the algorithm and provides
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A direct search algorithm is proposed for minimizing an arbitrary real valued function. The algorithm uses a new function transformation and three simplex-based operations. The function transformation provides global exploration features, while the simplex-based operations guarantees the termination of the algorithm and provides global convergence to a stationary point if the cost function is differentiable and its gradient is Lipschitz continuous. The algorithm’s performance has been extensively tested using benchmark functions and compared to some well-known global optimization algorithms. The results of the computational study show that the algorithm combines both simplicity and efficiency and is competitive with the heuristics-based strategies presently used for global optimization. Full article
Open AccessArticle Improved Direct Linear Transformation for Parameter Decoupling in Camera Calibration
Algorithms 2016, 9(2), 31; doi:10.3390/a9020031
Received: 11 December 2015 / Revised: 20 April 2016 / Accepted: 25 April 2016 / Published: 29 April 2016
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Abstract
For camera calibration based on direct linear transformation (DLT), the camera’s intrinsic and extrinsic parameters are simultaneously calibrated, which may cause coupling errors in the parameters and affect the calibration parameter accuracy. In this paper, we propose an improved direct linear transformation (IDLT)
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For camera calibration based on direct linear transformation (DLT), the camera’s intrinsic and extrinsic parameters are simultaneously calibrated, which may cause coupling errors in the parameters and affect the calibration parameter accuracy. In this paper, we propose an improved direct linear transformation (IDLT) algorithm for calibration parameter decoupling. This algorithm uses a linear relationship of calibration parameter errors and obtains calibration parameters by moving a three-dimensional template. Simulation experiments were conducted to compare the calibration accuracy of DLT and IDLT algorithms with image noise and distortion. The results show that the IDLT algorithm calibration parameters achieve higher accuracy because the algorithm removes the coupling errors. Full article
Open AccessArticle Visual and Textual Sentiment Analysis of a Microblog Using Deep Convolutional Neural Networks
Algorithms 2016, 9(2), 41; doi:10.3390/a9020041
Received: 16 February 2016 / Revised: 29 May 2016 / Accepted: 1 June 2016 / Published: 21 June 2016
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Abstract
Sentiment analysis of online social media has attracted significant interest recently. Many studies have been performed, but most existing methods focus on either only textual content or only visual content. In this paper, we utilize deep learning models in a convolutional neural network
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Sentiment analysis of online social media has attracted significant interest recently. Many studies have been performed, but most existing methods focus on either only textual content or only visual content. In this paper, we utilize deep learning models in a convolutional neural network (CNN) to analyze the sentiment in Chinese microblogs from both textual and visual content. We first train a CNN on top of pre-trained word vectors for textual sentiment analysis and employ a deep convolutional neural network (DNN) with generalized dropout for visual sentiment analysis. We then evaluate our sentiment prediction framework on a dataset collected from a famous Chinese social media network (Sina Weibo) that includes text and related images and demonstrate state-of-the-art results on this Chinese sentiment analysis benchmark. Full article
Open AccessArticle Joint Antenna Selection and Beamforming Algorithms for Physical Layer Multicasting with Massive Antennas
Algorithms 2016, 9(2), 42; doi:10.3390/a9020042
Received: 12 March 2016 / Revised: 17 May 2016 / Accepted: 20 June 2016 / Published: 22 June 2016
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Abstract
We investigate the problem of minimizing the total power consumption under the constraint of the signal-to-noise ratio (SNR) requirement for the physical layer multicasting system with large-scale antenna arrays. In contrast with existing work, we explicitly consider both the transmit power and the
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We investigate the problem of minimizing the total power consumption under the constraint of the signal-to-noise ratio (SNR) requirement for the physical layer multicasting system with large-scale antenna arrays. In contrast with existing work, we explicitly consider both the transmit power and the circuit power scaling with the number of antennas. The joint antenna selection and beamforming technique is proposed to minimize the total power consumption. The problem is a challenging one, which aims to minimize the linear combination of 0 -norm and 2 -norm. To our best knowledge, this minimization problem has not yet been well solved. A random decremental antenna selection algorithm is designed, which is further modified by an approximation of the minimal transmit power based on the asymptotic orthogonality of the channels. Then, a more efficient decremental antenna selection algorithm is proposed based on minimizing the 0 norm. Performance results show that the 0 norm minimization algorithm greatly outperforms the random selection algorithm in terms of the total power consumption and the average run time. Full article
Open AccessArticle Modifying Orthogonal Drawings for Label Placement
Algorithms 2016, 9(2), 22; doi:10.3390/a9020022
Received: 20 December 2015 / Revised: 10 March 2016 / Accepted: 16 March 2016 / Published: 28 March 2016
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Abstract
In this paper, we investigate how one can modify an orthogonal graph drawing to accommodate the placement of overlap-free labels with the minimum cost (i.e., minimum increase of the area and preservation of the quality of the drawing). We investigate computational
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In this paper, we investigate how one can modify an orthogonal graph drawing to accommodate the placement of overlap-free labels with the minimum cost (i.e., minimum increase of the area and preservation of the quality of the drawing). We investigate computational complexity issues of variations of that problem, and we present polynomial time algorithms that find the minimum increase of space in one direction, needed to resolve overlaps, while preserving the orthogonal representation of the orthogonal drawing when objects have a predefined partial order. Full article
(This article belongs to the Special Issue Graph Drawing and Experimental Algorithms)
Open AccessArticle Uniform vs. Nonuniform Membership for Mildly Context-Sensitive Languages: A Brief Survey
Algorithms 2016, 9(2), 32; doi:10.3390/a9020032
Received: 8 March 2016 / Revised: 19 April 2016 / Accepted: 27 April 2016 / Published: 11 May 2016
Cited by 1 | PDF Full-text (235 KB) | HTML Full-text | XML Full-text
Abstract
Parsing for mildly context-sensitive language formalisms is an important area within natural language processing. While the complexity of the parsing problem for some such formalisms is known to be polynomial, this is not the case for all of them. This article presents a
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Parsing for mildly context-sensitive language formalisms is an important area within natural language processing. While the complexity of the parsing problem for some such formalisms is known to be polynomial, this is not the case for all of them. This article presents a series of results regarding the complexity of parsing for linear context-free rewriting systems and deterministic tree-walking transducers. We discuss the difference between uniform and nonuniform complexity measures and how parameterized complexity theory can be used to investigate how different aspects of the formalisms influence how hard the parsing problem is. The main results we survey are all hardness results and indicate that parsing is hard even for relatively small values of parameters such as rank and fan-out in a rewriting system. Full article
Open AccessArticle An Improved Fireworks Algorithm Based on Grouping Strategy of the Shuffled Frog Leaping Algorithm to Solve Function Optimization Problems
Algorithms 2016, 9(2), 23; doi:10.3390/a9020023
Received: 10 October 2015 / Accepted: 21 December 2015 / Published: 1 April 2016
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Abstract
The fireworks algorithm (FA) is a new parallel diffuse optimization algorithm to simulate the fireworks explosion phenomenon, which realizes the balance between global exploration and local searching by means of adjusting the explosion mode of fireworks bombs. By introducing the grouping strategy of
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The fireworks algorithm (FA) is a new parallel diffuse optimization algorithm to simulate the fireworks explosion phenomenon, which realizes the balance between global exploration and local searching by means of adjusting the explosion mode of fireworks bombs. By introducing the grouping strategy of the shuffled frog leaping algorithm (SFLA), an improved FA-SFLA hybrid algorithm is put forward, which can effectively make the FA jump out of the local optimum and accelerate the global search ability. The simulation results show that the hybrid algorithm greatly improves the accuracy and convergence velocity for solving the function optimization problems. Full article
Open AccessArticle Mining Branching Rules from Past Survey Data with an Illustration Using a Geriatric Assessment Survey for Older Adults with Cancer
Algorithms 2016, 9(2), 33; doi:10.3390/a9020033
Received: 25 February 2016 / Revised: 25 April 2016 / Accepted: 2 May 2016 / Published: 13 May 2016
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Abstract
We construct a fast data mining algorithm that can be used to identify high-frequency response patterns in historical surveys. Identification of these patterns leads to the derivation of question branching rules that shorten the time required to complete a survey. The data mining
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We construct a fast data mining algorithm that can be used to identify high-frequency response patterns in historical surveys. Identification of these patterns leads to the derivation of question branching rules that shorten the time required to complete a survey. The data mining algorithm allows the user to control the error rate that is incurred through the use of implied answers that go along with each branching rule. The context considered is binary response questions, which can be obtained from multi-level response questions through dichotomization. The algorithm is illustrated by the analysis of four sections of a geriatric assessment survey used by oncologists. Reductions in the number of questions that need to be asked in these four sections range from 33% to 54%. Full article
Open AccessArticle Structural Damage Localization by the Principal Eigenvector of Modal Flexibility Change
Algorithms 2016, 9(2), 24; doi:10.3390/a9020024
Received: 2 February 2016 / Revised: 8 March 2016 / Accepted: 31 March 2016 / Published: 13 April 2016
Cited by 1 | PDF Full-text (3700 KB) | HTML Full-text | XML Full-text
Abstract
Using the principal eigenvector (PE) of modal flexibility change, a new vibration-based algorithm for structural defect localization was presented in this paper. From theoretical investigations, it was proven that the PE of modal flexibility variation has a turning point with a sharp peak
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Using the principal eigenvector (PE) of modal flexibility change, a new vibration-based algorithm for structural defect localization was presented in this paper. From theoretical investigations, it was proven that the PE of modal flexibility variation has a turning point with a sharp peak in its curvature at the damage location. A three-span continuous beam was used as an example to illustrate the feasibility and superiority of the proposed PE algorithm for damage localization. Furthermore, defect localization was also performed using the well-known uniform load surface approach for comparison. Numerical results demonstrated that the PE algorithm can locate structural defects with good accuracy, whereas the ULS approach occasionally missed one or two defect locations. It was found that the PE algorithm may be promising for structural defect assessment. Full article
Open AccessArticle A State Recognition Approach for Complex Equipment Based on a Fuzzy Probabilistic Neural Network
Algorithms 2016, 9(2), 34; doi:10.3390/a9020034
Received: 29 April 2016 / Revised: 16 May 2016 / Accepted: 16 May 2016 / Published: 20 May 2016
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Abstract
Due to the traditional state recognition approaches for complex electromechanical equipment having had the disadvantages of excessive reliance on complete expert knowledge and insufficient training sets, real-time state identification system was always difficult to be established. The running efficiency cannot be guaranteed and
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Due to the traditional state recognition approaches for complex electromechanical equipment having had the disadvantages of excessive reliance on complete expert knowledge and insufficient training sets, real-time state identification system was always difficult to be established. The running efficiency cannot be guaranteed and the fault rate cannot be reduced fundamentally especially in some extreme working conditions. To solve these problems, an online state recognition method for complex equipment based on a fuzzy probabilistic neural network (FPNN) was proposed in this paper. The fuzzy rule base for complex equipment was established and a multi-level state space model was constructed. Moreover, a probabilistic neural network (PNN) was applied in state recognition, and the fuzzy functions and quantification matrix were presented. The flowchart of proposed approach was designed. Finally, a simulation example of shearer state recognition and the industrial application with an accuracy of 90.91% were provided and the proposed approach was feasible and efficient. Full article
Open AccessArticle Primary User Localization Algorithm Based on Compressive Sensing in Cognitive Radio Networks
Algorithms 2016, 9(2), 25; doi:10.3390/a9020025
Received: 22 January 2016 / Revised: 3 April 2016 / Accepted: 11 April 2016 / Published: 14 April 2016
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Abstract
In order to locate source signal more accurately in authorized frequency bands, a novel primary user localization algorithm based on compressive sensing (PU-CSL) in cognitive radio networks (CRNs) is proposed in this paper. In comparison to existing centroid locating algorithms, PU-CSL shows higher
[...] Read more.
In order to locate source signal more accurately in authorized frequency bands, a novel primary user localization algorithm based on compressive sensing (PU-CSL) in cognitive radio networks (CRNs) is proposed in this paper. In comparison to existing centroid locating algorithms, PU-CSL shows higher locating accuracy for integrally exploring correlation between source signal and secondary users (SUs). Energy detection is first adopted for collecting the energy fingerprint of source signal at each SU, then degree of correlation between source signal and SUs is reconstructed based on compressive sensing (CS), which determines weights of centroid coordinates. A weighted centroid scheme is finally utilized to estimate source position. Simulation results show that PU-CSL has smaller maximum error of positioning and root-mean-square error. Moreover, the proposed PU-CSL algorithm possess excellent location accuracy and strong anti-noise performance. Full article
Open AccessArticle Application of the Energy-Conserving Integration Method to Hybrid Simulation of a Full-Scale Steel Frame
Algorithms 2016, 9(2), 35; doi:10.3390/a9020035
Received: 16 March 2016 / Revised: 11 May 2016 / Accepted: 18 May 2016 / Published: 21 May 2016
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Abstract
The nonlinear unconditionally stable energy-conserving integration method (ECM) is a new method for solving a continuous equation of motion. To our knowledge, there is still no report about its application on a hybrid test. Aiming to explore its effect on hybrid tests, the
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The nonlinear unconditionally stable energy-conserving integration method (ECM) is a new method for solving a continuous equation of motion. To our knowledge, there is still no report about its application on a hybrid test. Aiming to explore its effect on hybrid tests, the nonlinear beam-column element program is developed for computation. The program contains both the ECM and the average acceleration method (AAM). The comparison of the hybrid test results with thesetwo methods validates the effectiveness of the ECM in the hybrid simulation. We found that the energy error of hybrid test by using ECM is less than that of AAM. In addition, a new iteration strategy with reduction factor is presented to avoid the overshooting phenomena during iteration process with the finite element program. Full article
Open AccessArticle siEDM: An Efficient String Index and Search Algorithm for Edit Distance with Moves
Algorithms 2016, 9(2), 26; doi:10.3390/a9020026
Received: 25 November 2015 / Revised: 8 April 2016 / Accepted: 11 April 2016 / Published: 15 April 2016
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Abstract
Although several self-indexes for highly repetitive text collections exist, developing an index and search algorithm with editing operations remains a challenge. Edit distance with moves (EDM) is a string-to-string distance measure that includes substring moves in addition to ordinal editing operations to turn
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Although several self-indexes for highly repetitive text collections exist, developing an index and search algorithm with editing operations remains a challenge. Edit distance with moves (EDM) is a string-to-string distance measure that includes substring moves in addition to ordinal editing operations to turn one string into another. Although the problem of computing EDM is intractable, it has a wide range of potential applications, especially in approximate string retrieval. Despite the importance of computing EDM, there has been no efficient method for indexing and searching large text collections based on the EDM measure. We propose the first algorithm, named string index for edit distance with moves (siEDM), for indexing and searching strings with EDM. The siEDM algorithm builds an index structure by leveraging the idea behind the edit sensitive parsing (ESP), an efficient algorithm enabling approximately computing EDM with guarantees of upper and lower bounds for the exact EDM. siEDM efficiently prunes the space for searching query strings by the proposed method, which enables fast query searches with the same guarantee as ESP. We experimentally tested the ability of siEDM to index and search strings on benchmark datasets, and we showed siEDM’s efficiency. Full article
Open AccessArticle Robust Hessian Locally Linear Embedding Techniques for High-Dimensional Data
Algorithms 2016, 9(2), 36; doi:10.3390/a9020036
Received: 26 November 2015 / Revised: 14 May 2016 / Accepted: 16 May 2016 / Published: 26 May 2016
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Abstract
Recently manifold learning has received extensive interest in the community of pattern recognition. Despite their appealing properties, most manifold learning algorithms are not robust in practical applications. In this paper, we address this problem in the context of the Hessian locally linear embedding
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Recently manifold learning has received extensive interest in the community of pattern recognition. Despite their appealing properties, most manifold learning algorithms are not robust in practical applications. In this paper, we address this problem in the context of the Hessian locally linear embedding (HLLE) algorithm and propose a more robust method, called RHLLE, which aims to be robust against both outliers and noise in the data. Specifically, we first propose a fast outlier detection method for high-dimensional datasets. Then, we employ a local smoothing method to reduce noise. Furthermore, we reformulate the original HLLE algorithm by using the truncation function from differentiable manifolds. In the reformulated framework, we explicitly introduce a weighted global functional to further reduce the undesirable effect of outliers and noise on the embedding result. Experiments on synthetic as well as real datasets demonstrate the effectiveness of our proposed algorithm. Full article
(This article belongs to the Special Issue Manifold Learning and Dimensionality Reduction)
Open AccessArticle The Effect of Preprocessing on Arabic Document Categorization
Algorithms 2016, 9(2), 27; doi:10.3390/a9020027
Received: 21 January 2016 / Revised: 30 March 2016 / Accepted: 12 April 2016 / Published: 18 April 2016
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Abstract
Preprocessing is one of the main components in a conventional document categorization (DC) framework. This paper aims to highlight the effect of preprocessing tasks on the efficiency of the Arabic DC system. In this study, three classification techniques are used, namely, naive Bayes
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Preprocessing is one of the main components in a conventional document categorization (DC) framework. This paper aims to highlight the effect of preprocessing tasks on the efficiency of the Arabic DC system. In this study, three classification techniques are used, namely, naive Bayes (NB), k-nearest neighbor (KNN), and support vector machine (SVM). Experimental analysis on Arabic datasets reveals that preprocessing techniques have a significant impact on the classification accuracy, especially with complicated morphological structure of the Arabic language. Choosing appropriate combinations of preprocessing tasks provides significant improvement on the accuracy of document categorization depending on the feature size and classification techniques. Findings of this study show that the SVM technique has outperformed the KNN and NB techniques. The SVM technique achieved 96.74% micro-F1 value by using the combination of normalization and stemming as preprocessing tasks. Full article
Open AccessArticle A New Multi-Step Iterative Algorithm for Approximating Common Fixed Points of a Finite Family of Multi-Valued Bregman Relatively Nonexpansive Mappings
Algorithms 2016, 9(2), 37; doi:10.3390/a9020037
Received: 3 December 2015 / Revised: 11 May 2016 / Accepted: 17 May 2016 / Published: 30 May 2016
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Abstract
In this article, we introduce a new multi-step iteration for approximating a common fixed point of a finite class of multi-valued Bregman relatively nonexpansive mappings in the setting of reflexive Banach spaces. We prove a strong convergence theorem for the proposed iterative algorithm
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In this article, we introduce a new multi-step iteration for approximating a common fixed point of a finite class of multi-valued Bregman relatively nonexpansive mappings in the setting of reflexive Banach spaces. We prove a strong convergence theorem for the proposed iterative algorithm under certain hypotheses. Additionally, we also use our results for the solution of variational inequality problems and to find the zero points of maximal monotone operators. The theorems furnished in this work are new and well-established and generalize many well-known recent research works in this field. Full article
Open AccessArticle Alternating Direction Method of Multipliers for Generalized Low-Rank Tensor Recovery
Algorithms 2016, 9(2), 28; doi:10.3390/a9020028
Received: 23 November 2015 / Revised: 26 March 2016 / Accepted: 13 April 2016 / Published: 19 April 2016
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Abstract
Low-Rank Tensor Recovery (LRTR), the higher order generalization of Low-Rank Matrix Recovery (LRMR), is especially suitable for analyzing multi-linear data with gross corruptions, outliers and missing values, and it attracts broad attention in the fields of computer vision, machine learning and data mining.
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Low-Rank Tensor Recovery (LRTR), the higher order generalization of Low-Rank Matrix Recovery (LRMR), is especially suitable for analyzing multi-linear data with gross corruptions, outliers and missing values, and it attracts broad attention in the fields of computer vision, machine learning and data mining. This paper considers a generalized model of LRTR and attempts to recover simultaneously the low-rank, the sparse, and the small disturbance components from partial entries of a given data tensor. Specifically, we first describe generalized LRTR as a tensor nuclear norm optimization problem that minimizes a weighted combination of the tensor nuclear norm, the l1-norm and the Frobenius norm under linear constraints. Then, the technique of Alternating Direction Method of Multipliers (ADMM) is employed to solve the proposed minimization problem. Next, we discuss the weak convergence of the proposed iterative algorithm. Finally, experimental results on synthetic and real-world datasets validate the efficiency and effectiveness of the proposed method. Full article
(This article belongs to the Special Issue Manifold Learning and Dimensionality Reduction)
Open AccessArticle A 3/2-Approximation Algorithm for the Graph Balancing Problem with Two Weights
Algorithms 2016, 9(2), 38; doi:10.3390/a9020038
Received: 29 April 2016 / Revised: 27 May 2016 / Accepted: 2 June 2016 / Published: 8 June 2016
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Abstract
In the pursuit of finding subclasses of the makespan minimization problem on unrelated parallel machines that have approximation algorithms with approximation ratio better than 2, the graph balancing problem has been of current interest. In the graph balancing problem each job can be
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In the pursuit of finding subclasses of the makespan minimization problem on unrelated parallel machines that have approximation algorithms with approximation ratio better than 2, the graph balancing problem has been of current interest. In the graph balancing problem each job can be non-preemptively scheduled on one of at most two machines with the same processing time on either machine. Recently, Ebenlendr, Krčál, and Sgall (Algorithmica 2014, 68, 62–80.) presented a 7 / 4 -approximation algorithm for the graph balancing problem. Let r , s Z + . In this paper we consider the graph balancing problem with two weights, where a job either takes r time units or s time units. We present a 3 / 2 -approximation algorithm for this problem. This is an improvement over the previously best-known approximation algorithm for the problem with approximation ratio 1.652 and it matches the best known inapproximability bound for it. Full article
Figures

Open AccessArticle An Improved Dynamic Joint Resource Allocation Algorithm Based on SFR
Algorithms 2016, 9(2), 29; doi:10.3390/a9020029
Received: 25 January 2016 / Revised: 12 April 2016 / Accepted: 15 April 2016 / Published: 22 April 2016
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Abstract
Inter-cell interference (ICI) is the main factor affecting system capacity and spectral efficiency. Effective spectrum resource management is an important and challenging issue for the design of wireless communication systems. The soft frequency reuse (SFR) is regarded as an interesting approach to significantly
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Inter-cell interference (ICI) is the main factor affecting system capacity and spectral efficiency. Effective spectrum resource management is an important and challenging issue for the design of wireless communication systems. The soft frequency reuse (SFR) is regarded as an interesting approach to significantly eliminate ICI. However, the allocation of resource is fixed prior to system deployment in static SFR. To overcome this drawback, this paper adopts a distributed method and proposes an improved dynamic joint resource allocation algorithm (DJRA). The improved scheme adaptively adjusts resource allocation based on the real-time user distribution. DJRA first detects the edge-user distribution vector to determine the optimal scheme, which guarantees that all the users have available resources and the number of iterations is reduced. Then, the DJRA maximizes the throughput for each cell via optimizing resource and power allocation. Due to further eliminate interference, the sector partition method is used in the center region and in view of fairness among users, the novel approach adds the proportional fair algorithm at the end of DJRA. Simulation results show that the proposed algorithm outperforms previous approaches for improving the system capacity and cell edge user performance. Full article

Review

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Open AccessReview Review of Recent Type-2 Fuzzy Controller Applications
Algorithms 2016, 9(2), 39; doi:10.3390/a9020039
Received: 15 March 2016 / Revised: 29 May 2016 / Accepted: 31 May 2016 / Published: 9 June 2016
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Abstract
Type-2 fuzzy logic controllers (T2 FLC) can be viewed as an emerging class of intelligent controllers because of their abilities in handling uncertainties; in many cases, they have been shown to outperform their Type-1 counterparts. This paper presents a literature review on recent
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Type-2 fuzzy logic controllers (T2 FLC) can be viewed as an emerging class of intelligent controllers because of their abilities in handling uncertainties; in many cases, they have been shown to outperform their Type-1 counterparts. This paper presents a literature review on recent applications of T2 FLCs. To follow the developments in this field, we first review general T2 FLCs and the most well-known interval T2 FLS algorithms that have been used for control design. Certain applications of these controllers include robotic control, bandwidth control, industrial systems control, electrical control and aircraft control. The most promising applications are found in the robotics and automotive areas, where T2 FLCs have been demonstrated and proven to perform better than traditional controllers. With the development of enhanced algorithms, along with the advancement in both hardware and software, we shall witness increasing applications of these frontier controllers. Full article

Other

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Open AccessComment Comment on: On the Kung-Traub Conjecture for Iterative Methods for Solving Quadratic Equations. Algorithms 2016, 9, 1
Algorithms 2016, 9(2), 30; doi:10.3390/a9020030
Received: 21 January 2016 / Revised: 12 April 2016 / Accepted: 13 April 2016 / Published: 26 April 2016
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Abstract
Kung-Traub conjecture states that an iterative method without memory for finding the simple zero of a scalar equation could achieve convergence order 2d1, and d is the total number of function evaluations. In an article “Babajee, D.K.R. On the
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Kung-Traub conjecture states that an iterative method without memory for finding the simple zero of a scalar equation could achieve convergence order 2 d 1 , and d is the total number of function evaluations. In an article “Babajee, D.K.R. On the Kung-Traub Conjecture for Iterative Methods for Solving Quadratic Equations, Algorithms 2016, 9, 1, doi:10.3390/a9010001”, the author has shown that Kung-Traub conjecture is not valid for the quadratic equation and proposed an iterative method for the scalar and vector quadratic equations. In this comment, we have shown that we first reported the aforementioned iterative method. Full article

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