Parallel time complexity of a heuristic algorithm for the k-center problem with usage weights

Big data is technology that has the ability to manage very large amounts of data, in very fast time to allow real-time analysis and reactions. Several clustering methods which are used to group data are Fuzzy C-Means (FCM) and K-Means Clustering. K-Means Clustering algorithm is a method of partitioning existing data into two or more group. This research goal was to compare the performance of K-Means and Fuzzy C-Means algorithms in clustering data using big data technology. In this research, Hadoop and Hive were chosen the big data technology. The knowledge of Shia history on student and lecturer of Syarif Hidayatullah State Islamic University Jakarta were the data which used in this research, The testing was done by constructing application K-Means Fuzzy C-Means using Java language, Hadoop and Hive and then test the performance of K-Means and Fuzzy C-Means algorithms in data clustering. It compares both algorithms in terms of accuracy, execution time, and time complexity of the algorithms. In the application K-Means Fuzzy C-Means, evaluation were performed with data filter and the average accuracy difference result of K-Means and Fuzzy C-Means is 8.03% with the better accuracy owned by K-Means. The average execution time difference is 718.58 ms, which K-Means was faster than is Fuzzy C-Means. The time complexities of both algorithms have the same value O(n2) and the Big O equation resulted in an average difference of 93,568 with the smallest value on K-Means. Thus, K-Means algorithm is better than the Fuzzy C-Means in terms of accuracy, execution time, and the time complexity.

We consider a network facility location problem with unbounded production levels. This problem is NP-hard in the general case and is known to have an optimal solution with quadratic complexity on a tree network. We study the case of a network representable by an outerplanar graph, i.e., by a graph whose vertices belong to one (outer) face. This problem is known to have an optimal algorithm with time complexity O(nm3), where n is the number of vertices and m is the number of possible facility locations. Using some properties of outerplanar graphs (binary 2-trees) and the existence of an optimal solution with a family of centrally connected service areas, we obtain recurrence relations for the construction of an optimal algorithm with time complexity that is smaller by a factor of \(\sqrt m \) than the time complexity of the earlier algorithm.

The sample average approximation (SAA) approach is a widely used technique, based on Monte‐Carlo simulation, often applied to large‐scale stochastic optimization problems. In this approach, a set of sample average problems with multiple copies of sampled scenarios are generated and solved exactly. In other words, there is an implicit assumption that the sample average problems are solvable to optimality. In some instances, however, the sample average problems might be NP‐hard problems, often difficult or impractical to solve to optimality. In this article, we broaden the scope of the SAA approach and show that even without solving the sample problems to optimality, by combining a heuristic and a lower bounding approach, high‐quality solutions with tight confidence bounds on the optimal solution value can be obtained. We demonstrate this “inexact SAA approach” on two problems. First, we apply it to the Stochastic Connected Facility Location (SConFL) problem, the motivating application for this article, that arises in the design of telecommunications networks. As an additional application, we also use it for the Stochastic Uncapacitated Facility Location (SUFL) problem. Our computational results demonstrate the effectiveness of the inexact SAA approach. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(1), 19–33 2017

Advancements in technology make the application domains to collect large amounts of data that is used for future decision making by applying various techniques such as classification, regression, etc. These techniques comprise machine learning algorithms. The running time complexity of these machine learning algorithms is determined by the number of instances and the total of all features or dimensions in the dataset. All the instances are generally required. But all the features in the dataset may not be related or useful in the machine learning task, for example, in classification. In such a case, the dimensionality of the dataset is reduced to optimize the running time complexity of the algorithms without compromising the accuracy. Reducing the dimensionality is achieved by extracting only those features which are more relevant. Hence feature extraction or dimensionality reduction plays a significant role in obtaining high accuracy of machine learning algorithms within the acceptable time complexity. The existing techniques for feature extraction are principal component analysis and by using correlation coefficients. But these techniques required more time complexity and some sort of manual decisions for selecting optimal features. To extract the most optimal features, a novel algorithm is proposed using an evolutionary optimization technique called particle swarm optimization (PSO). As a fitness function in PSO, an artificial neural network classifier pertaining to its efficient performance in classification tasks. The performance of the proposed algorithm is verified by five benchmark datasets from the UCI repository. The model is evaluated with different performance measures:accuracy, precision, recall, and the F-1 score. Research results have proven that algorithm proposed here reduces the number of features by improving the performance of the machine learning sub-system when compared to conventional algorithms.

Most of works on the time complexity analysis of evolutionary algorithms have always focused on some artificial binary problems. The time complexity of the algorithms for combinatorial optimisation has not been well understood. This paper considers the time complexity of an evolutionary algorithm for a classical combinatorial optimisation problem, to find the maximum cardinality matching in a graph. It is shown that the evolutionary algorithm can produce a matching with nearly maximum cardinality in average polynomial time.

The computational complexity study of algorithms is highly germane to the design and development of high-speed computing devices. The whole essence of computation is principally influenced by efficiency of algorithms; this is more so the case with the algorithms whose solution space explodes exponentially. Cryptographic algorithms are good examples of such algorithms. The goal of this study is to compare the computational speeds of RSA and ElGamal cryptographic algorithms by carrying out a survey of works done so far by researchers. This study has therefore examined some of the results of the studies already done and highlighted which of the RSA and ElGamal algorithms performed better under given parameters. It is expected that this study would spur further investigation of the behaviour of cryptographic structures in order to ascertain their complexity and impact on the field of theoretical computer science. The experimental results of many of the papers reviewed showed that RSA cryptographic algorithm performs better as regards to energy usage, time complexity and space complexity of text, image and audio data during encryption process while some studies showed that ElGamal performs better in terms of time complexity during decryption process.

A directed graph D is semicomplete if for every pair x, y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph D=(V,A) and a pair of natural numbers k and ell , we are to decide whether there is a subset X of V of size k such that the largest strongly connected component in D-X has at most ell vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for ell =1. We study the parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: k, ell ,ell +k and n-ell . In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time O^*(2^{16k}) but not in time O^*(2^{o(k)}) unless the Exponential Time Hypothesis (ETH) fails. The upper bound O^*(2^{16k}) implies the upper bound O^*(2^{16(n-ell )}) for the parameter n-ell . We complement the latter by showing that there is no algorithm of time complexity O^*(2^{o({n-ell })}) unless ETH fails. Finally, we improve (in dependency on ell ) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter ell +k on general digraphs from O^*(2^{O(kell log (kell ))}) to O^*(2^{O(klog (kell ))}). Note that Drange, Dregi and van ’t Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time O^*(2^{o(klog ell )}) unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity O^*(2^{o(klog k)}).

The traditional fault isolation strategy based on the rollout algorithm will increase the algorithm time complexity and use too many test points. The optimization strategy of tests for fault isolation based on rollout algorithm can solve these problems, and make the final optimization result is still better than the normal rollout algorithm. This optimization strategy not only makes the rollout algorithm effective, maximizes the system fault isolation rate, but also reduces the time complexity of the algorithm, reduces the use of unnecessary tests, simplifies the complexity of the system, and improves the compatibility of the algorithm for complex systems. The principle of this optimization strategy is to perform fault isolation operations on all tests before using the rollout algorithm, and then use the rollout algorithm to sort the tests after selecting the necessary tests. Finally, the theoretical and case analysis results prove that the optimization strategy can use the least tests under the premise of ensuring the maximum fault isolation rate, reduce the time complexity of the rollout algorithm, and is suitable for optimizing the fault diagnosis of complex systems.

Improved shortest path algorithms for nearly acyclic graphs

We present an asymptotically fast algorithm for the numerical evaluation of modular functions such as the elliptic modular function j j . Our algorithm makes use of the natural connection between the arithmetic-geometric mean (AGM) of complex numbers and modular functions. Through a detailed complexity analysis, we prove that for a given τ \tau , evaluating N N significative bits of j ( τ ) j(\tau ) can be done in time O ( M ( N ) log ⁡ N ) O(\mathcal {M}(N)\log N) , where M ( N ) \mathcal {M}(N) is the time complexity for the multiplication of two N N -bit integers. However, this is only true for a fixed τ \tau and the time complexity of this first algorithm greatly increases as I m ( τ ) \mathrm {Im}(\tau ) does. We then describe a second algorithm that achieves the same time complexity independently of the value of τ \tau in the classical fundamental domain F \mathcal {F} . We also show how our method can be used to evaluate other modular forms, such as the Dedekind η \eta function, with the same time complexity.

New algorithms for fair k-center problem with outliers and capacity constraints

Transportation problems are one of the particular forms that often appear in linear programs, one of which is the distribution of goods. A transportation method is needed to determine the optimal result, namely, the minimum cost from source to destination with all demand and supply fulfilled. There are several methods, one of which is the Lowest Supply Lowest Cost Method (LSLC) and the Exponential Approach Method (PE). Both methods are made in a MATLAB program, generating a script that calculates the algorithm's time complexity. Using the function notation, the Big-O Algorithm complexity of the Lowest Supply Lowest Cost method is more efficient than the Exponential Approach Method algorithm. At the same time, the optimal result for the minimum cost between the two methods is obtained by using the Exponential Approach Method.

The emerging of mobile social networks opens opportunities for viral marketing. However, before fully utilizing mobile social networks as a platform for viral marketing, many challenges have to be addressed. In this paper, we address the problem of identifying a small number of individuals through whom the information can be diffused to the network as soon as possible, referred to as the diffusion minimization problem. Diffusion minimization under the probabilistic diffusion model can be formulated as an asymmetric k-center problem which is NP-hard, and the best known approximation algorithm for the asymmetric k-center problem has approximation ratio of log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> n and time complexity O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> ). Clearly, the performance and the time complexity of the approximation algorithm are not satisfiable in large-scale mobile social networks. To deal with this problem, we propose a community based algorithm and a distributed set-cover algorithm. The performance of the proposed algorithms is evaluated by extensive experiments on both synthetic networks and a real trace. The results show that the community based algorithm has the best performance in both synthetic networks and the real trace, and the distributed setcover algorithm outperforms the approximation algorithm in the real trace in terms of diffusion time.

Parallel shape recognition and its implementation on a fixed-size VLSI architecture

The traditional algorithm of L. Stockmeyer (1983) for area minimization of slicing floorplans has time (and space) complexity O(n/sup 2/) in the worst case, or O(n log n) for balanced slicing. For more than a decade, it is considered the best possible. In this paper, we present a new algorithm of worst-case time (and space) complexity O(n log n), where n is the total number of realizations for the basic blocks, regardless whether the slicing is balanced or not. We also prove /spl Omega/(n log n) is the lower bound and the time complexity of any area minimization algorithm. Therefore, the new algorithm not only finds the optimal realization, but also has an optimal running time.