Average Time Complexity Research Articles

Advanced In-Place Merge Sort Approach for Enhanced Sorting Performance

This paper introduces a new sorting algorithm designed to sort array elements directly within the array itself. The algorithm features a best-case time complexity of O(n) and an average and worst-case time complexity of O (n log n). It achieves this performance through a method that combines recursive breakdown with in-place merging techniques. We compare this new approach with existing popular sorting algorithms to assess its relative effectiveness. The paper concludes with insights into the algorithm's strengths and limitations, and proposes potential areas for further development and refinement. Keywords - Complexity Analysis, In-Place Sorting, Recursive Breakdown

Interpolation Once Binary Search over a Sorted List

Searching over a sorted list is a classical problem in computer science. Binary Search takes at most log2⁡n+1 tries to find an item in a sorted list of size n. Interpolation Search achieves an average time complexity of O(log⁡log⁡n) for uniformly distributed data. Hybrids of Binary Search and Interpolation Search are also available to handle data with unknown distributions. This paper analyzes the computation cost of these methods and shows that interpolation can significantly affect their performance—accordingly, a new method, Interpolation Once Binary Search (IOBS), is proposed. The experimental results show that IOBS outperforms the hybrids of Binary Search and Interpolation Search for nonuniformly distributed data.

CSIM: A Fast Community Detection Algorithm Based on Structure Information Maximization

Community detection has been a subject of extensive research due to its broad applications across social media, computer science, biology, and complex systems. Modularity stands out as a predominant metric guiding community detection, with numerous algorithms aimed at maximizing modularity. However, modularity encounters a resolution limit problem when identifying small community structures. To tackle this challenge, this paper presents a novel approach by defining community structure information from the perspective of encoding edge information. This pioneering definition lays the foundation for the proposed fast community detection algorithm CSIM, boasting an average time complexity of only O(nlogn). Experimental results showcase that communities identified via the CSIM algorithm across various graph data types closely resemble ground truth community structures compared to those revealed via modularity-based algorithms. Furthermore, CSIM not only boasts lower time complexity than greedy algorithms optimizing community structure information but also achieves superior optimization results. Notably, in cyclic network graphs, CSIM surpasses modularity-based algorithms in effectively addressing the resolution limit problem.

On the average time complexity of computation with random partition

On the average time complexity of computation with random partition

A machine‐learning approach to biomarker evaluation for AD precision medicine

AbstractBackgroundLate‐onset Alzheimer disease (AD) is a heterogeneous disease, as demonstrated by its wide range of genetic and environmental risk factors, as well as diversity of clinical manifestations. Cognizant of the futility of a one‐size‐fits‐all approach for heterogeneous diseases, precision medicine aims to treat each subtype appropriately, but methods for successful diagnosis of AD subtypes has been limited. Real‐valued biomarkers, such as omics measurements in plasma or CSF, frequently are evaluated based on fold change (FC) and/or the area under the receiver operating characteristic curve (AUC) metric. We previously demonstrated the inability for FC and AUC to capture signals for subtypes that comprise less than 50% of the diseased individuals due to inherently low true positive rates and proposed an alternative approach based on the bimodality of the data [Smith and Climer, https://doi.org/10.1101/2022.02.14.22270972]. This approach is based on statistical characteristics of the data, including skewness, cardinality, and kurtosis. k‐medians clustering, a machine learning approach to identify natural clusters in data, removes these statistical assumptions, but optimally solving k‐medians is generally NP‐hard, yielding computational intractability for datasets of interest. Lloyd’s approximation algorithm is commonly used in practice, however, its accuracy is highly sensitive to the random start state, thus is not suitable when accuracy is paramount.MethodOwing to the nature of the problem, we present an evaluation metric based on the optimal k‐medians objective with k = 2. Additionally, due to the single dimensionality of the data, we introduce a dynamic programming algorithm to improve computational performance.ResultAlthough optimally solving k‐medians is NP‐hard, our 2‐medians objective over 1‐dimensional data using dynamic programming has an average time complexity of O(n log n), and worst‐case complexity of O(n 2) where n is the number of data values. This objective has no reliance upon false positive rate, skewness, or kurtosis, making it robust for real‐valued data drawn from heterogeneous AD subtypes.ConclusionBiomarker data drawn from heterogeneous AD subtypes will have inherently low true positive rates, resulting with low FC and AUC values. Our machine‐learning based approach for evaluating potential biomarkers efficiently produces optimal clustering of the data and scales well to large datasets.

Enhanced Physical Layer Security and PAPR Performance Based on Disturbance of Data Cluster Under Chaotic Sequence Color Seeking Mechanism in CO- OFDM System

This study proposes a physical layer encryption scheme based on disturbance of data cluster under color seeking mechanism of chaotic sequence which is generated by Brownian motion determined by keys aiming to improve security of coherent optical orthogonal frequency division multiplexing (CO-OFDM) system. Chaotic sequences with good randomness and unpredictability are endowed with a unique color composed of two primary colors and go through order transformation by sorting and seeking, which forms the chaotic sequence color seeking mechanism. The disturbance of transmitted data is realized according to chaotic sequence color seeking mechanism and PAPR performance is optimized simultaneously. The security of system is enhanced due to a key space of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">90</sup> and PAPR can be reduced by 0.5 dB. An experiment conducted in a 40 GHz 16QAM CO-OFDM system over an 80 km standard single mode fiber (SSMF) shows that the authorized user can successfully decrypt the received signal, while the eavesdroppers cannot derive valid information with bit error rate (BER) at approximately 0.5. Analyses of peak to average power ratio (PAPR) performance and time complexity of the scheme are further carried out under different cluster forming methods to derive the optimized configuration which is critical in encryption.

Correction to: Minimum average case time complexity for sorting algorithms

Correction to: Minimum average case time complexity for sorting algorithms

Minimum average case time complexity for sorting algorithms

Minimum average case time complexity for sorting algorithms

A novel hierarchical aggregation algorithm for optimal repartitioning of statistical regions

Data regionalisation allows spatial inference over a population. The statistical regions must be updated to account for population changes, but this update process is more restrictive and iterative than ab initio regionalisation. This creates a need for an algorithmic solution that minimises human-in-the-loop involvement in population-driven regionalisation. The new method must address the basic regionalisation criteria – contiguity, compactness, homogeneity, equinumeriosity, and temporal consistency. We present a novel validation metric to assess the quality of partition based on these criteria. We have developed a novel hybrid aggregation algorithm (HeLP), combining elements of hierarchical and graph-theoretic approaches, for the primary purpose of repartitioning. This algorithm operates in average computational time complexity. HeLP was tested on simulated data and the Australian Statistical Geography Standard. The method can emulate the human operator successfully, providing statistically significant results in repartitioning parcel-based systems, such as the Cadastre.

Unconstrained model and merge sorted serpentine arrangement method for dynamic reconfiguration of large PV arrays

• An unconstrained optimization model, and a heuristic method namely, Merge Sorted Serpentine Arrangement, are proposed for modelling and solving the problem of finding the optimal reconfiguration of a photovoltaic array to minimize partial shading losses. • The proposed unconstrained optimization model formulation enables the application of different meta-heuristic and heuristic optimization algorithms for solving the problem of finding the optimal reconfiguration of a photovoltaic array to minimize partial shading losses. • Extensive case studies show that the proposed method reduces average computation time by up to 80 % when compared with other methods. • The proposed method consistently reduces partial shading losses by up to 11 % under dynamic and static partial shading conditions due to its lower computation time as demonstrated by extensive case studies. • The main drawback of the proposed method is that it increases memory requirements by up 100 % when compared with other methods. The problem of finding the optimal interconnection between photovoltaic modules in a dynamic photovoltaic array to minimize partial shading losses is a Non-deterministic Polynomial time-hard (i.e., NP-hard) problem. Therefore, no known deterministic method can solve the problem in polynomial time. However, meta-heuristic and heuristic methods can provide solutions to this problem. In this paper, an unconstrained optimization model and a heuristic method, namely, Merge Sorted Serpentine Arrangement (MSA), are proposed for the first time for modelling and solving the problem. In the proposed method, the irradiance levels of the PV modules are accurately estimated using the double diode model and then sorted according to the merge sorting algorithm. After that, the PV modules are arranged in the PV array according to their sorted irradiance levels in a serpentine way using the proposed switching matrix. The proposed method has consistently resulted in lower computational time and partial shading losses when compared with the other methods in the literature, due to the application of the Merge Sorting Algorithm, which has an average time complexity of O (n log(n)). However, the proposed method has resulted in increased memory, switches, and sensor requirements when compared with some of the other methods in the literature.

A novel adaptive optimization framework for SVM hyper-parameters tuning in non-stationary environment: A case study on intrusion detection system

Building an Intrusion Detection System (IDS) in non-stationary environment is challenging because, in such an environment, intrusion-related data grow every day. A machine learning model trained in a stationary environment where training data does not change, often fails to retain its performance in real world environment. This is because dynamism in data makes the hyper-parametric values of the underlying classifier shift in the search space. For making such a model work for intrusion detection in non-stationary environment, one must have to run hyper-parametric optimization algorithm again and again at various time instances. But the expansion of the existing data in non-stationary environment, makes such a way of tunning the hyper-parameters computationally expensive. So, there is a requirement of more adaptive and computationally efficient optimization frameworks for hyper-parameters to build IDS in non-stationary environment. This paperwork proposes a novel framework to train a Support Vector Machine (SVM) for intrusion detection by optimizing its hyper-parameters C and γ dynamically. For designing this framework, we have used Moth-Flame Optimization (MFO) as the base optimization algorithm which can be run with random initialization. Further, for utilizing the knowledge generated by running the base optimization algorithm, we have introduced two algorithms- a Lightweight MFO and a simple Knowledge-Based Search. The Lightweight MFO uses the knowledge for initializing the starting solutions and the Knowledge-Based Search uses the knowledge as search space. Based on the result of a drift detection module, the proposed framework identifies the most appropriate algorithm to be used at a particular time instance when re-training of the model is required due to the change in the data. Results have shown a significant reduction in the average time complexity of the hyper-parametric optimization process. We have evaluated our proposed framework on benchmark NSL-KDD dataset and got significantly encouraging convergence rate and detection performance. The obtained average accuracy for IDS built using our proposed framework is 97.5%. Further, we have also compared our framework by considering other metaheuristic algorithms as base optimization algorithms and found that our proposed framework, which uses MFO as a base optimization algorithm outperforms the others.

Efficient Adaptive Online Learning via Frequent Directions.

By employing time-varying proximal functions, adaptive subgradient methods (ADAGRAD) have improved the regret bound and been widely used in online learning and optimization. However, ADAGRAD with full matrix proximal functions (ADA-FULL) cannot handle large-scale problems due to the impractical O(d³) time and O(d²) space complexities, though it has better performance when gradients are correlated. In this paper, we propose two efficient variants of ADA-FULL via a matrix sketching technique called frequent directions (FD). The first variant named as ADA-FD directly utilizes FD to maintain and manipulate low-rank matrices, which reduces the space and time complexities to O(τd) and O(τ²d) respectively, where d is the dimensionality and τ &lt;&lt; d is the sketching size. The second variant named as ADA-FFD further adopts a doubling trick to accelerate FD used in ADA-FD, which reduces the average time complexity to O(τd) while only doubles the space complexity of ADA-FD. Theoretical analysis reveals that the regret of ADA-FD and ADA-FFD is close to that of ADA-FULL as long as the outer product matrix of gradients is approximately low-rank. Experimental results demonstrate the efficiency and effectiveness of our algorithms.

Optimizing Time Complexity of Searching Operation: A Noble Algorithm

By the passage of time , there are many searching algorithm and data structure evolved to make basic operations more efficient with respect to time and space complexity like storing, searching, deleting the data and they play very important role in real-world applications. Based on mechanism of searching, Searching algorithm can be classified into three types of algorithm: linear, binary, hashing, but excluding these, there are many searching techniques proposed by many researchers. The aim of all these proposed searching technique is to optimize the time complexity of searching operation generally compared to Binary Search, which have average and worst case time complexity of log(n), where n is number of elements. This paper proposes a noble searching algorithm that works for sorted array data structure and performs better in most of the cases than that of binary search. The proposed searching Technique will either find index of target element or find a small space within array space by using statistical operation, and then binary search will be applied to that small space, thus it will reduce the overall time complexity. This paper briefly discusses, proposed algorithm and it’s time complexity and comparing the efficiency of the algorithm with some of the well-known other algorithms, using graphs and tables and its practical applications. We will analyze the behavior of algorithm in different parameters like:-size of input data, Nature of elements in list, Different Case Scenario (worst, best, and avg. case) etc.

An assistive interface protocol for communication between visually and hearing-speech impaired persons in internet platform

Purpose The article presents a design and development of a generic assistive system to establish an independent conversation-platform for hearing-speech impaired and visually impaired persons. Materials The developed software system is accomplished through programming using python and html. Methods Considering the constraints associated to the above mentioned impairments, the system implements both speech-to-text/gesture and text/gesture-to-speech conversion in its operation. In real-time hand-gesture to speech generation process is implemented using static image tracking, CNN based deep learning technique and MediaPipe hand-tracking solution. The software-prototype-terminals can be accessed through internet using MQTT protocol to accomplish the communicative conversation between visually impaired and hearing-speech impaired persons. Results The software system exhibits an average prediction time of less than approximately 1 s and 2 s for a four-letter based audio-word and a single hand-gesture, respectively, which are commensurate to the average time-complexity during human-to-human conversation. The average accuracy and loss for the hand-gestures through the CNN based deep learning are 0.9996 and 0.0008, respectively. The confusion matrix related to the prediction of alphabet-specific hand-gestures shows its satisfactory performance in gesture recognition. Conclusions The software-prototype of the generic assistive device shows its potential to establish an exclusive communication between a visually impaired and a hearing-speech impaired person through the internet. The same software-interface can also be used to accomplish a communicative conversation between either only visually-impaired persons or only hearing-speech impaired persons. IMPLICATIONS FOR REHABILITATION The article presents a design and development of a generic assistive interface to establish an independent conversation-platform for hearing-speech impaired and visually impaired people via internet network. The same software-interface can also be used to accomplish a communicative conversation between either only visually-impaired persons or only hearing-speech impaired persons. The design can be further extended by incorporating multi-modal impairments to make a universal assistive device for all-in-one communication.

Automatic Street Lighting System with Vehicle Detection using Deep-Learning Based Remote Sensing

Automated street lightning is advantageous to society as it decreases the rate of accidents, vandalism and street crimes. The ability to detect vehicles and smartly manage the street light system is among the major duties of electrical distribution companies. To recognize an objects of interest in an image by a classical technique called object detection. In order to recover the enactment and reduce the complexity of object detection, numerous computer vision methods have been proposed over the past decade years. Object detection has a wide variety of applications including vehicle detection especially in remote sensing applications. Vigorous and accurate detection of vehicles for such solicitations is a moderately stimulating problem because of discrepancy of color, size, aspect ratio and alignment of vehicles and complex backgrounds of satellite images. Nevertheless, the modern deep learning-based object detection frameworks and convolutional neural networks have great potential for improving the performance of vehicle detection methods in terms of exactness, sturdiness and detection time. This paper will review both conventional and modern object detection systems. Assortment of the supreme appropriate set of object sensor, mainstay story abstraction system, train-test proportion and hyper-parameters for recognition of automobiles for nifty street light system are studied in detail and compared by deep learning based object detectors Single shot multibox detector (SSD) and You Only Look Once (YOLO) and feature abstraction systems counting Alexnet, VGG-16, Resnet-18, Resnet-50. As between accuracy and detection time these methods offer several trade-off options. On mean average precision (mAP), precision vs, recall curves, computational complexity and time complexity is established by relative investigation.

Super Edge 4-Points Congruent Sets-Based Point Cloud Global Registration

With the acceleration in three-dimensional (3D) high-frame-rate sensing technologies, dense point clouds collected from multiple standpoints pose a great challenge for the accuracy and efficiency of registration. The combination of coarse registration and fine registration has been extensively promoted. Unlike the requirement of small movements between scan pairs in fine registration, coarse registration can match scans with arbitrary initial poses. The state-of-the-art coarse methods, Super 4-Points Congruent Sets algorithm based on the 4-Points Congruent Sets, improves the speed of registration to a linear order via smart indexing. However, the lack of reduction in the scale of original point clouds limits the application. Besides, the coplanarity of registration bases prevents further reduction of search space. This paper proposes a novel registration method called the Super Edge 4-Points Congruent Sets to address the above problems. The proposed algorithm follows a three-step procedure, including boundary segmentation, overlapping regions extraction, and bases selection. Firstly, an improved method based on vector angle is used to segment the original point clouds aiming to thin out the scale of the initial point clouds. Furthermore, overlapping regions extraction is executed to find out the overlapping regions on the contour. Finally, the proposed method selects registration bases conforming to the distance constraints from the candidate set without consideration about coplanarity. Experiments on various datasets with different characteristics have demonstrated that the average time complexity of the proposed algorithm is improved by 89.76%, and the accuracy is improved by 5 mm on average than the Super 4-Points Congruent Sets algorithm. More encouragingly, the experimental results show that the proposed algorithm can be applied to various restrictive cases, such as few overlapping regions and massive noise. Therefore, the algorithm proposed in this paper is a faster and more robust method than Super 4-Points Congruent Sets under the guarantee of the promised quality.

Voronoi Diagram and Delaunay Triangulation with Independent and Dependent Geometric Uncertainties

We address the problems of constructing the Voronoi diagram (VD) and Delaunay triangulation (DT) of points in the plane with mutually dependent location uncertainties, testing their stability, and computing their components. Point coordinate uncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), a deterministic, expressive and computationally efficient first order linear approximation of geometric uncertainty that supports parametric dependencies between point locations. We define an uncertain three-point circle and its properties and present in-circle-test algorithms for the dependent and the independent cases whose respective time complexity is [Formula: see text] and [Formula: see text], where [Formula: see text] is the number of parameters that describe the points location uncertainty and [Formula: see text] is the complexity of quartic [Formula: see text]-variable optimization. We define the uncertain VD and DT of [Formula: see text] LPGUM points and show that an unstable VD may have an exponential number of topologically different instances. We describe algorithms for testing VD and DT stability whose time complexity is [Formula: see text] and [Formula: see text] for the dependent and independent cases when the nominal (exact) VD is given. We present algorithms to compute the vertices, edges, and faces of uncertain VD and DT for the independent case whose complexity is [Formula: see text] time and [Formula: see text] space. Finally, we describe algorithms for answering exact and uncertain point location queries in a stable uncertain VD and for dynamically updating it in [Formula: see text] and [Formula: see text] average case time complexity for the independent and dependent cases.

D-PBWT: dynamic positional Burrows-Wheeler transform.

Durbin's positional Burrows-Wheeler transform (PBWT) is a scalable data structure for haplotype matching. It has been successfully applied to identical by descent (IBD) segment identification and genotype imputation. Once the PBWT of a haplotype panel is constructed, it supports efficient retrieval of all shared long segments among all individuals (long matches) and efficient query between an external haplotype and the panel. However, the standard PBWT is an array-based static data structure and does not support dynamic updates of the panel. Here, we generalize the static PBWT to a dynamic data structure, d-PBWT, where the reverse prefix sorting at each position is stored with linked lists. We also developed efficient algorithms for insertion and deletion of individual haplotypes. In addition, we verified that d-PBWT can support all algorithms of PBWT. In doing so, we systematically investigated variations of set maximal match and long match query algorithms: while they all have average case time complexity independent of database size, they have different worst case complexities and dependencies on additional data structures. The benchmarking code is available at genome.ucf.edu/d-PBWT. Supplementary data are available at Bioinformatics online.

Polarization-Adjusted Convolutional (PAC) Codes: Sequential Decoding vs List Decoding

In the Shannon lecture at the 2019 International Symposium on Information Theory (ISIT), Arikan proposed to employ a one-to-one convolutional transform as a pre-coding step before the polar transform. The resulting codes of this concatenation are called polarization-adjusted convolutional (PAC) codes . In this scheme, a pair of polar mapper and demapper as pre- and post- processing devices are deployed around a memoryless channel, which provides polarized information to an outer decoder leading to improved error correction performance of the outer code. In this paper, the list decoding and sequential decoding (including Fano decoding and stack decoding) are first adapted for use to decode PAC codes. Then, to reduce the complexity of sequential decoding of PAC/polar codes, we propose (i) an adaptive heuristic metric, (ii) tree search constraints for backtracking to avoid exploration of unlikely sub-paths, and (iii) tree search strategies consistent with the pattern of error occurrence in polar codes. These contribute to the reduction of the average decoding time complexity from 50% to 80%, trading with 0.05 to 0.3 dB degradation in error correction performance within FER = $10^{-3}$ range, respectively, relative to not applying the corresponding search strategies. Additionally, as an important ingredient in Fano decoding of PAC/polar codes, an efficient computation method for the intermediate LLRs and partial sums is provided. This method is effective in backtracking and avoids storing the intermediate information or restarting the decoding process. Eventually, all three decoding algorithms are compared in terms of performance, complexity, and resource requirements.

IDEC

Approximate Nearest Neighbor (ANN) search is a fundamental algorithmic problem, with numerous applications in many areas of computer science. In this work, we propose indexable distance estimating codes (iDEC) , a new solution framework to ANN that extends and improves the locality sensitive hashing (LSH) framework in a fundamental and systematic way. Empirically, an iDEC-based solution has a low index space complexity of O ( n ) and can achieve a low average query time complexity of approximately O (log n ). We show that our iDEC-based solutions for ANN in Hamming and edit distances outperform the respective state-of-the-art LSH-based solutions for both in-memory and external-memory processing. We also show that our iDEC-based in-memory ANN-H solution is more scalable than all existing solutions. We also discover deep connections between Error-Estimating Codes (EEC), LSH, and iDEC.