Complexity Analysis Of Algorithm Research Articles

Complexity analysis and numerical implementation of a new interior-point algorithm for semidefinite optimization

We generalize Zhang and Xu's (2011) [22] interior point algorithm for linear optimization to semidefinite optimization problems in order to define a new search direction. The symmetrization of the search direction is based on the full Nesterov-Todd scaling scheme. Moreover, we show that the obtained algorithm solves the studied problem in polynomial time and that the short-step algorithm has the best-known iteration bound, namely O(nlog⁡nε)-iterations. Finally, we report a comparative numerical study to show the efficiency of our proposed algorithm.

Sample complexity analysis for adaptive optimization algorithms with stochastic oracles

Sample complexity analysis for adaptive optimization algorithms with stochastic oracles

Scaling private proximity testing protocols for geofenced information exchange: A metropolitan-wide case study

Private Proximity Testing (PPT) protocols allow two entities to exchange location-and-time-specific information without any tangible breach of privacy between the two or any third-party intermediary such as a server or a service provider. Numerous PPT protocols have been proposed, however, to our knowledge, experimentation relies on simulations and algorithmic complexity analysis, without tests on large-scale scenarios and actual information exchange over wide metropolitan areas. In this paper, we implement and evaluate a novel PPT geofencing algorithm based on RSA factorization. The prototype used in this work was built in Android and Kotlin, initially funded by Google Digital News Initiative Innovation Fund (digitalnewsinitiative.com). We utilize WoM to implement and test a geofenced PPT protocol using prime factorization over a wide area while users move around different municipalities and (i) present findings concerning efficiency and feasibility of prime factorization PPT implementations over large geographical distributions in terms of energy and data needs, (ii) evaluate complementary technologies needed for such private distribution of location-and-time-specific information like geospatial fences and fragmentation and (iii) consider numerous legal considerations under the EU legal context.

A survey of dynamic programming algorithms

Dynamic programming is an important algorithmic idea with widespread applications in computer science and other disciplines. With the continuous improvement of computing power and the increasing complexity of practical problems, dynamic programming algorithms have also received more and more attention. However, due to the limitations of the dynamic programming algorithm itself, it also brings challenges to the optimization of the algorithm and its application in different fields. This paper uses the methods of literature review and case analysis to systematically summarize and sort out the basic principles, complexity analysis, classic problems, and applications of dynamic programming algorithms. First of all, this paper discusses the basic concepts of splitting and optimal substructure characteristics of related problems in dynamic programming; secondly, it discusses the time complexity and space complexity of dynamic programming algorithms in detail; then, in the classic dynamic programming The case part focuses on the knapsack problem, including the 0-1 knapsack problem and multiple knapsack problems; finally, this paper discusses the wide application and optimization of dynamic programming algorithms in practice, covering natural language processing, bioinformatics and How to optimize the dynamic programming algorithm to improve the efficiency of problem solving and reduce resource consumption. To sum up, this paper fully demonstrates the basic principles and applications of dynamic programming algorithms, as well as optimization methods and development trends, and provides guidance and references for their academic and practical applications.

Memory backtracking strategy: An evolutionary updating mechanism for meta-heuristic algorithms

The search domain of meta-heuristic algorithms is always constantly changing, which make it difficult to adapt the diverse optimization issues. To overcome above issue, an evolutionary updating mechanism called Memory Backtracking Strategy (MBS) is proposed, which contains thinking stage, recall stage, and memory stage. Overall, the adoption of the MBS enhances the efficiency of MHSs by incorporating group memory, clue recall, and memory forgetting mechanisms. These strategies improve the algorithm's ability to explore the search space, optimize the search process, and escape local optima. MBS will be applied to three different types of MHS algorithms: evolutionary based (LSHADE_SPACMA), physical based (Stochastic Fractal Search, SFS), and biological based (Marine Predators Algorithmnm, MPA) to demonstrate the universality of MBS. In the experimental section including 57 engineering problems, algorithm complexity analysis, CEC2020 Friedman ranking, convergence curve, Wilcoxon statistical, and box plot. Among them, 21 algorithms participated in the Friedman experiment, including MBS_LSHADE_SPACMA ranked first, LSHADE_SPACMA ranked second, MBS_MPA ranked 6th, MPA ranked 8th, MBS_SFS ranked 9th and SFS ranked 12th. Combined with the analysis of "MBS testing analysis" and the experimental results of engineering problems, it has proven that MBS has universality and good ability to improve optimization algorithm performance. The source codes of the proposed MBS (MBS_MPA) can be accessed by https://github.com/luchenghao2022/Memory-Backtracking-Strategy

METHODS OF NAVIGATING ALGORITHMIC COMPLEXITY: BIG-OH AND SMALL-OH NOTATIONS

This article provides an in-depth exploration of Big-Oh and small-oh notations, shedding light on their practical implications in the analysis of algorithm complexity. Big-Oh notation offers a valuable tool for estimating an upper bound on the growth rate of an algorithm's running time, whereas small-oh notation delineates a lower limit on this growth rate. The piece delves into a comprehensive examination of various complexity classes that emerge through the application of Big-Oh notation, underscoring the significance of small-oh notation as it complements and enriches complexity analysis. In the realm of programming and computer science, the employment of these notations holds paramount importance. They empower developers and researchers to make informed decisions regarding algorithm selection and optimization. It is crucial to recognize that while complexity analysis is a vital facet of effective programming, ongoing research endeavors may yield more refined methodologies and approaches within this domain. By understanding and harnessing the power of Big-Oh and small-oh notations, professionals can effectively evaluate algorithm efficiency and scalability. This knowledge equips them with the ability to design and implement algorithms that meet specific performance criteria, which is pivotal in the ever-evolving landscape of technology and computation. As pushing the boundaries of what is possible in the field of algorithm design is being continued, these notations remain invaluable tools for navigating the complex terrain of algorithmic analysis and optimization. By embracing Big-Oh and small-oh notations, professionals can finely assess algorithmic efficiency, ensuring they meet performance criteria in the evolving technological landscape. These notations remain indispensable for algorithmic analysis.

Тізбекті кодтардың көп шекті декодтау параметрлерін оңтайландыру алгоритмін әзірлеу

The most known and practically used coding/decoding schemes are investigated. The results of the analysis of algorithmic complexity and simulation of the operation of coding schemes for Gaussian channels show that only multi-threshold decoders of convolutional codes outperform all others for all considered block sizes and coding rates. The multithreshold decoder (MTD) is a development of the simplest Messian threshold decoder with an additional difference register added. This decoder allows decoding block and convolutional self-orthogonal codes. The operating principles of convolutional encoders are considered in detail, the characteristics of multithreshold decoders and the Viterbi decoder for convolutional codes are compared. The best self-orthogonal codes for MTD are chosen, the efficiency of MTD in channels with additive white Gaussian noise is studied. An algorithm for increasing the performance of multithreshold decoding (MTD) based on the optimization of the MTD parameters of symbolic self-orthogonal codes is proposed.

User grouping and resource management in efficient NOMA‐OMA transmission scheme for infotainment message dissemination in VANETs

AbstractThe data distribution for infotainment services at vehicular ad‐hoc networks (VANETs) has become a research hotspot in recent years. Non‐orthogonal multiple access (NOMA) is accepted as an enabling technology for 5G and beyond due to its improved spectral efficiency. In this work, we consider a VANET communication scenario where users have common data demands such as high‐density maps, local entertainment/shopping options or other popular media content. For such a scenario, we propose an efficient clustering scheme together with a hybrid NOMA‐OMA scheme for improved spectral and power efficiency. For this setup, we also develop an optimal bandwidth allocation strategy to improve the power efficiency further. We formulate a one dimensional nonlinear power minimization problem such that the bandwidth allocation for the system can be optimized. The complexity and optimality analysis of optimization algorithms used for power minimization are performed. The improved performance of the proposed scheme in terms of power efficiency, spectral efficiency and latency performance is showcased in comparison to conventional NOMA and OMA approaches.

Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations

We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear representations or exact mappings between nonlinear ODEs/HJE and linear partial differential equations (the Liouville equation and the Koopman-von Neumann equation). The connection between the linear representations and the original nonlinear system is established through the Dirac delta function or the level set mechanism. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations, including the finite difference discretisations and the Fourier spectral discretisations for the two different linear representations, with the result showing that the quantum simulation methods usually give the best performance in time complexity. We also propose the Schrödinger framework to solve the Liouville equation for the HJE with the Hamiltonian formulation of classical mechanics, since it can be recast as the semiclassical limit of the Wigner transform of the Schrödinger equation. Comparison between the Schrödinger and the Liouville framework will also be made.

Distributed Variable Sample-Size Stochastic Optimization With Fixed Step-Sizes

In this article, we consider distributed stochastic optimization over randomly switching networks, where agents collaboratively minimize the average of all agents’ local expectation-valued convex cost functions. Due to the stochasticity in gradient observations, distributedness of local functions, and randomness of communication topologies, distributed algorithms with an exact convergence guarantee under fixed step-sizes have not been achieved yet. This work incorporates variance reduction scheme into the distributed stochastic gradient tracking algorithm, where local gradients are estimated by averaging across a variable number of sampled gradients. With an identically and independently distributed random network, we show that all agents’ iterates converge almost surely to the same optimal solution under fixed step-sizes. When the global cost function is strongly convex and the sample size increases at a geometric rate, we prove that the iterates geometrically converge to the unique optimal solution, and establish the iteration, oracle, and communication complexity. The algorithm performance, including rate and complexity analysis, are further investigated with constant step-sizes and a polynomially increasing sample size. Finally, the empirical algorithm performance are illustrated with numerical examples.

Quantum random access stored-program machines

Random access machines (RAMs) and random access stored-program machines (RASPs) are models of computing that are closer to the architecture of real-world computers than Turing machines (TMs). They are also convenient in complexity analysis of algorithms. The relationships between RAMs, RASPs and TMs are well-studied. However, clear relationships between their quantum counterparts are still missing in the literature. We fill in this gap by formally defining the models of quantum random access machines (QRAMs) and quantum random access stored-program machines (QRASPs) and clarifying the relationships between QRAMs, QRASPs and quantum Turing machines (QTMs). In particular, we show that P⊆EQRAMP⊆EQP⊆BQP=BQRAMP, where EQRAMP and BQRAMP stand for the sets of problems that can be solved by polynomial-time QRAMs with certainty and bounded-error, respectively. At the heart of our proof, is a standardisation of QTM with an extended halting scheme, which is of independent interest.

On the Complexity of the Plantinga–Vegter Algorithm

We introduce tools from numerical analysis and high dimensional probability for precision control and complexity analysis of subdivision-based algorithms in computational geometry. We combine these tools with the continuous amortization framework from exact computation. We use these tools on a well-known example from the subdivision family: the adaptive subdivision algorithm due to Plantinga and Vegter. The only existing complexity estimate on this rather fast algorithm was an exponential worst-case upper bound for its interval arithmetic version. We go beyond the worst-case by considering both average and smoothed analysis, and prove polynomial time complexity estimates for both interval arithmetic and finite-precision versions of the Plantinga-Vegter algorithm.

Mini-Workshop: Analysis of Data-driven Optimal Control

This hybrid mini-workshop discussed recent mathematical methods for analyzing the opportunities and limitations of data-driven and machine- learning approaches to optimal feedback control. The analysis concerned all aspects of such approaches, ranging from approximation theory particularly for high-dimensional problems via complexity analysis of algorithms to robustness issues.

Inexact Newton-CG algorithms with complexity guarantees

Abstract We consider variants of a recently developed Newton-CG algorithm for nonconvex problems (Royer, C. W. & Wright, S. J. (2018) Complexity analysis of second-order line-search algorithms for smooth nonconvex optimization. SIAM J. Optim., 28, 1448–1477) in which inexact estimates of the gradient and the Hessian information are used for various steps. Under certain conditions on the inexactness measures, we derive iteration complexity bounds for achieving $\epsilon $-approximate second-order optimality that match best-known lower bounds. Our inexactness condition on the gradient is adaptive, allowing for crude accuracy in regions with large gradients. We describe two variants of our approach, one in which the step size along the computed search direction is chosen adaptively, and another in which the step size is pre-defined. To obtain second-order optimality, our algorithms will make use of a negative curvature direction on some steps. These directions can be obtained, with high probability, using the randomized Lanczos algorithm. In this sense, all of our results hold with high probability over the run of the algorithm. We evaluate the performance of our proposed algorithms empirically on several machine learning models. Our approach is a first attempt to introduce inexact Hessian and/or gradient information into the Newton-CG algorithm of Royer & Wright (2018, Complexity analysis of second-order line-search algorithms for smooth nonconvex optimization. SIAM J. Optim., 28, 1448–1477).

Monte Carlo Methods for the Neutron Transport Equation

This paper continues our treatment of the Neutron Transport Equation (NTE) building on the work in [arXiv:1809.00827v2], [arXiv:1810.01779v4] and [arXiv:1901.00220v3], which describes the flux of neutrons through inhomogeneous fissile medium. Our aim is to analyse existing and novel Monte Carlo (MC) algorithms, aimed at simulating the lead eigenvalue associated with the underlying model. This quantity is of principal importance in the nuclear regulatory industry for which the NTE must be solved on complicated inhomogenous domains corresponding to nuclear reactor cores, irradiative hospital equipment, food irradiation equipment and so on. We include a complexity analysis of such MC algorithms, noting that no such undertaking has previously appeared in the literature. The new MC algorithms offer a variety of advantages and disadvantages of accuracy vs cost, as well as the possibility of more convenient computational parallelisation.

A theoretical approach to the computational complexity measure of abstract DEVS simulators

DEVS is a sound Modeling and Simulation (M&S) framework that describes a model in a modular and hierarchical way. It comes along with an abstract simulation algorithm which defines its operational semantics. Many variants of such an algorithm have been proposed by DEVS researchers. Yet, the proper interpretation and analysis of the computational complexity of such approaches have not been systematically addressed and defined. As systems become larger and more complex, the efficiency of the DEVS simulation algorithms in terms of time complexity measure becomes a major issue. Therefore, it is necessary to devise a method for computing this complexity. This paper proposes a generic method to address such an issue, taking advantage of the recursion embedded in the triggered-by-message principle of the DEVS simulation protocol. The applicability of the method is shown through the complexity analysis of various DEVS simulation algorithms.

Leakage detection method of natural gas pipeline combining improved variational mode decomposition and Lempel–Ziv complexity analysis

With the continuous development of pipeline transportation industry, pipeline leakage often occurs, posing a great threat to people’s lives and property safety. In order to improve the detection accuracy of natural gas pipeline leakage, a pipeline leakage detection method based on improved variational mode decomposition algorithm and Lempel–Ziv complexity analysis is proposed. In this work, the normalized mutual information is used to determine the decomposition level K of variational mode decomposition, and the Lempel–Ziv complexity analysis algorithm is used to extract pipeline signal feature. The results show that the proposed leakage detection method has higher classification accuracy than other methods, which verifies the effectiveness of this method in the process of pipeline leakage detection.

Efficient multi-partition topology optimization

In topology optimization, the state of structures is typically obtained by numerically evaluating a discretized PDE-based model. The degrees of freedom of such a model can be partitioned in free and prescribed sets to define the boundary conditions. A multi-partition problem involves multiple partitions of the same discretization, typically corresponding to different loading scenarios. As a result, solving multi-partition problems involves multiple factorization/preconditionings of the system matrix, requiring a high computational effort. In this paper, a novel method is proposed to efficiently calculate the responses and accompanying design sensitivities in such multi-partition problems using static condensation for use in gradient-based topology optimization. A main problem class that benefits from the proposed method is the topology optimization of small-displacement multi-input–multi-output compliant mechanisms. However, the method is applicable to any linear problem. We present its formulation and an algorithmic complexity analysis to estimate computational advantages for both direct and iterative solution methods to solve the system of equations, verified by numerical experiments. It is demonstrated that substantial gains are achievable for large-scale multi-partition problems. This is especially true for problems with both a small set of number of degrees of freedom that fully describes the performance of the structure and with large similarities between the different partitions. A major contribution to the gain is the lack of large adjoint analyses required to obtain the sensitivities of the performance measure.

Development of near real-time wireless image sequence streaming cloud using Apache Kafka for road traffic monitoring application.

In this paper, the authors have designed and implemented the prototype for a near real-time wireless image sequence streaming cloud with two-layered restoration for a road traffic monitoring application of a small-scale network. Since the proposed design is targeted to implement outdoors where the link or node failure could occur, the fault-tolerant capability must be considered. Having only one layer restoration may not provide a good quality of service. Therefore, a two-layer restoration framework is designed in the proposed system by restoring the network layer with the underlying software-defined wireless mesh network capability and at the local broker selection over the Apache Kafka framework. The monitoring application performance has been investigated for the end-to-end average latency and image loss percentage by outdoor testing for 13 hours from 5:40 P.M. 17th November 2020 to 6:40 A.M. 18th November 2020. The end-to-end average latency and image loss percentage have been found to be within the acceptable condition i.e. less than 5 seconds on average with approximated 10% image losses. The proposed system has also been compared with the traditional ad-hoc network, running the OLSR-based network layer, in terms of the rerouting time, restoration time and end-to-end average latency. Based on the emulated wireless network in controllable laboratory environments, the proposed SDWMN-based system outperforms the conventional OLSR-based system with potentially faster rerouting/restoration time due to SDN central controllability and with only marginally increased end-to-end average latency after re-routing/restoration completion. Algorithm complexity analysis has also been given for both the systems. Both the experimental and complexity analysis results thus suggest the practical applicability of the proposed system. Given this promising result, it is therefore recommended as the future research in further developing from the prototype design into the actual deployment for daily traffic monitoring operations.

Построение сети маршрутизации точек пешеходной панорамной съемки городских территорий

The author describes technical and algorithmic solution of the task of building a spatial routing network for pedestrian panoramic survey points at monitoring urban infrastructure objects. Its urgency is caused by the necessity of making a panoramic measuring the yard, intra-block territories, as well as park areas and other objects which have no reliable data on the road network there. The matter of spatial routing network providing between panoramic images without reference to the road graph is considered and formulated. Based on the presented conditional variant of solving the task, as well as considering the problem in the view of graph theory, geometric conditions are compiled, the necessary sets of Geodata are determined, and the method of spatial routing network building is developed. Algorithm complexity analysis is performed. An intelligent GIS agent was developed providing filling of the DBMS table with spatial features, which are the object implementation of the spatial routing network in the GIS. Geoportal that helps the final user analyze and adjust the mentioned network with the introduction of changes to the DBMS on the server side in an interactive mode is presented.