Runtime Analysis Research Articles

An algorithm and computation to verify Legendre’s conjecture up to 7·1013

We state a general purpose algorithm for quickly finding primes in evenly divided sub-intervals. Legendre’s conjecture claims that for every positive integer n, there exists a prime between n2 and (n+1)2. Oppermann’s conjecture subsumes Legendre’s conjecture by claiming there are primes between n2 and n(n+1) and also between n(n+1) and (n+1)2. Using Cramér’s conjecture as the basis for a heuristic run-time analysis, we show that our algorithm can verify Oppermann’s conjecture, and hence also Legendre’s conjecture, for all n≤N in time O(NlogNloglogN) and space NO(1/loglogN). We implemented a parallel version of our algorithm and improved the empirical verification of Oppermann’s conjecture from the previous N=2·109 up to N=7.05·1013>246 2^{46}$$\\end{document}]]>, so we were finding 27 digit primes. The computation ran for about half a year on each of two platforms: four Intel Xeon Phi 7210 processors using a total of 256 cores, and a 192-core cluster of Intel Xeon E5-2630 2.3GHz processors.

A Genetic Particle Swarm Optimization with Policy Gradient for Hydraulic Fracturing Optimization

Summary Particle swarm optimization (PSO) has been widely developed and applied as one of the important algorithms for optimization problems because of its few control parameters and low complexity. However, conventional PSO has the problem of premature convergence. Most of the control parameters of optimized PSO are adjusted manually, and the technology of adaptive adjustment based on reinforcement learning (RL) is still immature. Therefore, this study proposes a new genetic PSO (GAPSO) with policy gradient (PGGAPSO). The algorithm constructs a strategy neural network as an agent to dynamically output the action probability distribution and passes the action with the highest selection probability to the GAPSO section through the designed action selection function, thereby updating the parameters. The GAPSO section solves the optimization problem based on the new parameters obtained and uses the individual and global optimums obtained from the iterations as inputs to the policy neural network. In addition, an elite strategy is introduced to enhance the exploration capability of the PGGAPSO. Then, the performance of PGGAPSO was comprehensively evaluated using convergence analysis, statistical tests, Friedman rank, and runtime analysis. Finally, PGGAPSO is applied to the construction parameter optimization of hydraulic fracturing. The results show that due to the influence of the number of samples, the optimization performance of PGGAPSO is general in the early stage of iteration but shows excellent with continuous iteration. For unimodal and most of multimodal functions, PGGAPSO shows significant difference and excellence. Additional processes, such as elite strategy and policy gradient (PG), resulted in a 7.3% increase in runtime for PGGAPSO compared with GAPSO, but this additional computational loss is acceptable relative to the improved accuracy. The algorithm does not require an objective function and can be combined with the complex agent model of hydraulic fracturing to optimize the fracturing parameters. In the case of parameter optimization for hydraulic fracturing in the Jimsar sag, the optimized production was increased by more than 10% on average.

Runtime Analysis for the NSGA-II: Proving, Quantifying, and Explaining the Inefficiency for Many Objectives

Runtime Analysis for the NSGA-II: Proving, Quantifying, and Explaining the Inefficiency for Many Objectives

Distance distributions and runtime analysis of perceptual hashing algorithms

Perceptual image hashing refers to a class of algorithms that produce content-based image hashes. These systems use specialized perceptual hash algorithms like Phash, Microsoft’s PhotoDNA, or Facebook’s PDQ to generate a compact digest of an image file that can be roughly compared to a database of known illicit-content digests. Time taken by perceptual hashing algorithms to generate hash code has been computed. Then, we evaluated perceptual hashing algorithms on two million dataset of images. The produced nine variants of the original images were computed and then several distances were calculated. There have been several studies in the past, but in the existing literature size of the data is small and there are very few studies with hash code computation time and robustness tradeoff. This work shows that existing perceptual hashing algorithms are robust for most of the content-preserving operations and there is a tradeoff between computation time and robustness.

Runtime Analysis of Quality Diversity Algorithms

AbstractQuality diversity (QD) is a branch of evolutionary computation that gained increasing interest in recent years. The Map-Elites QD approach defines a feature space, i.e., a partition of the search space, and stores the best solution for each cell of this space. We study a simple QD algorithm in the context of pseudo-Boolean optimisation on the “number of ones” feature space, where the ith cell stores the best solution amongst those with a number of ones in $$[(i-1)k, ik-1]$$ [ ( i - 1 ) k , i k - 1 ] . Here k is a granularity parameter $$1 \le k \le n+1$$ 1 ≤ k ≤ n + 1 . We give a tight bound on the expected time until all cells are covered for arbitrary fitness functions and for all k and analyse the expected optimisation time of QD on OneMax and other problems whose structure aligns favourably with the feature space. On combinatorial problems we show that QD finds a $${(1-1/e)}$$ ( 1 - 1 / e ) -approximation when maximising any monotone sub-modular function with a single uniform cardinality constraint efficiently. Defining the feature space as the number of connected components of an edge-weighted graph, we show that QD finds a minimum spanning forest in expected polynomial time. We further consider QD’s performance on classes of transformed functions in which the feature space is not well aligned with the problem. The asymptotic performance is unaffected by transformations on easy functions like OneMax. Applying a worst-case transformation to a deceptive problem increases the expected optimisation time from $$O(n^2 \log n)$$ O ( n 2 log n ) to an exponential time. However, QD is still faster than a (1+1) EA by an exponential factor.

Runtime Analysis of Single- and Multi-Objective Evolutionary Algorithms for Chance Constrained Optimization Problems with Normally Distributed Random Variables.

Chance constrained optimization problems allow to model problems where constraints involving stochastic components should only be violated with a small probability. Evolutionary algorithms have been applied to this scenario and shown to achieve high quality results. With this paper, we contribute to the theoretical understanding of evolutionary algorithms for chance constrained optimization. We study the scenario of stochastic components that are independent and normally distributed. Considering the simple single-objective (1+1) EA, we show that imposing an additional uniform constraint already leads to local optima for very restricted scenarios and an exponential optimization time. We therefore introduce a multi-objective formulation of the problem which trades off the expected cost and its variance. We show that multi-objective evolutionary algorithms are highly effective when using this formulation and obtain a set of solutions that contains an optimal solution for any possible confidence level imposed on the constraint. Furthermore, we prove that this approach can also be used to compute a set of optimal solutions for the chance constrained minimum spanning tree problem. In order to deal with potentially exponentially many trade-offs in the multi-objective formulation, we propose and analyze improved convex multi-objective approaches. Experimental investigations on instances of the NP-hard stochastic minimum weight dominating set problem confirm the benefit of the multi-objective and the improved convex multi-objective approach in practice.

First Steps Towards a Runtime Analysis When Starting With a Good Solution

The mathematical runtime analysis of evolutionary algorithms traditionally regards the time an algorithm needs to find a solution of a certain quality when initialized with a random population. In practical applications it may be possible to guess solutions that are better than random ones. We start a mathematical runtime analysis for such situations. We observe that different algorithms profit to a very different degree from a better initialization. We also show that the optimal parameterization of an algorithm can depend strongly on the quality of the initial solutions. To overcome this difficulty, self-adjusting and randomized heavy-tailed parameter choices can be profitable. Finally, we observe a larger gap between the performance of the best evolutionary algorithm we found and the corresponding black-box complexity. This could suggest that evolutionary algorithms better exploiting good initial solutions are still to be found. These first findings stem from analyzing the performance of the \((1+1)\) evolutionary algorithm and the static, self-adjusting, and heavy-tailed \((1+(\lambda,\lambda))\) genetic algorithms on the OneMax benchmark. We are optimistic that the question of how to profit from good initial solutions is interesting beyond these first examples.

PACMAN: A Robust Partial Atomic Charge Predicter for Nanoporous Materials Based on Crystal Graph Convolution Networks.

We report a fast and easy method (PACMAN) to assign partial atomic charges on metal-organic framework (MOF) and covalent-organic framework (COF) crystal structures based on graph convolution networks (GCNs) trained on >1.8 million high-fidelity partial atomic charge data obtained from the Quantum Metal-Organic Framework (QMOF) database. The developed model shows outstanding performance, achieving a mean absolute error (MAE) of 0.0055 e (test set performance) while maintaining consistency with DDEC6, Bader, and CM5 charges across diverse chemistry and topologies of MOFs and COFs. We find that the new method accurately assigns partial atomic charges for ion-containing nanoporous materials, which has not been possible in previous machine learning (ML) models. Grand canonical Monte Carlo (GCMC) simulation results for CO2 and N2 uptakes and the Widom particle insertion calculation for Henry's law constant of water results based on PACMAN and the original DDEC6 charges show excellent agreements compared to other ML models reported in the literature. The runtime analysis of the new method demonstrates that the partial atomic charges of MOF and COF structures with up to 500 atoms can be obtained in less than 10 s. An easy-to-use web interface has been developed to facilitate the adoption of the developed model.

Deep learning to assess microsatellite instability directly from histopathological whole slide images in endometrial cancer

Molecular classification, particularly microsatellite instability-high (MSI-H), has gained attention for immunotherapy in endometrial cancer (EC). MSI-H is associated with DNA mismatch repair defects and is a crucial treatment predictor. The NCCN guidelines recommend pembrolizumab and nivolumab for advanced or recurrent MSI-H/mismatch repair deficient (dMMR) EC. However, evaluating MSI in all cases is impractical due to time and cost constraints. To overcome this challenge, we present an effective and efficient deep learning-based model designed to accurately and rapidly assess MSI status of EC using H&E-stained whole slide images. Our framework was evaluated on a comprehensive dataset of gigapixel histopathology images of 529 patients from the Cancer Genome Atlas (TCGA). The experimental results have shown that the proposed method achieved excellent performances in assessing MSI status, obtaining remarkably high results with 96%, 94%, 93% and 100% for endometrioid carcinoma G1G2, respectively, and 87%, 84%, 81% and 94% for endometrioid carcinoma G3, in terms of F-measure, accuracy, precision and sensitivity, respectively. Furthermore, the proposed deep learning framework outperforms four state-of-the-art benchmarked methods by a significant margin (p < 0.001) in terms of accuracy, precision, sensitivity and F-measure, respectively. Additionally, a run time analysis demonstrates that the proposed method achieves excellent quantitative results with high efficiency in AI inference time (1.03 seconds per slide), making the proposed framework viable for practical clinical usage. These results highlight the efficacy and efficiency of the proposed model to assess MSI status of EC directly from histopathological slides.

Estimation-of-distribution algorithms for multi-valued decision variables

The majority of research on estimation-of-distribution algorithms (EDAs) concentrates on pseudo-Boolean optimization and permutation problems, leaving the domain of EDAs for problems in which the decision variables can take more than two values, but which are not permutation problems, mostly unexplored. To render this domain more accessible, we propose a natural way to extend the known univariate EDAs to this setting. Different from a naïve reduction to the binary case, our approach avoids additional constraints.Since understanding genetic drift is crucial for an optimal parameter choice, we extend the known quantitative analysis of genetic drift to EDAs for multi-valued, categorical variables. Roughly speaking, when the variables take r different values, the time for genetic drift to become significant is r times shorter than in the binary case. Consequently, the update strength of the probabilistic model has to be chosen r times lower now.To investigate how desired model updates take place in this framework, we undertake a mathematical runtime analysis on the r-valued LeadingOnes problem. We prove that with the right parameters, the multi-valued UMDA solves this problem efficiently in O(rln⁡(r)2n2ln⁡(n)) function evaluations. This bound is nearly tight as our lower bound Ω(rln⁡(r)n2ln⁡(n)) shows.Overall, our work shows that our good understanding of binary EDAs naturally extends to the multi-valued setting, and it gives advice on how to set the main parameters of multi-values EDAs.

CAREx: context-aware read extension of paired-end sequencing data

BackgroundCommonly used next generation sequencing machines typically produce large amounts of short reads of a few hundred base-pairs in length. However, many downstream applications would generally benefit from longer reads.ResultsWe present CAREx—an algorithm for the generation of pseudo-long reads from paired-end short-read Illumina data based on the concept of repeatedly computing multiple-sequence-alignments to extend a read until its partner is found. Our performance evaluation on both simulated data and real data shows that CAREx is able to connect significantly more read pairs (up to 99%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$99\\%$$\\end{document} for simulated data) and to produce more error-free pseudo-long reads than previous approaches. When used prior to assembly it can achieve superior de novo assembly results. Furthermore, the GPU-accelerated version of CAREx exhibits the fastest execution times among all tested tools.ConclusionCAREx is a new MSA-based algorithm and software for producing pseudo-long reads from paired-end short read data. It outperforms other state-of-the-art programs in terms of (i) percentage of connected read pairs, (ii) reduction of error rates of filled gaps, (iii) runtime, and (iv) downstream analysis using de novo assembly. CAREx is open-source software written in C++ (CPU version) and in CUDA/C++ (GPU version). It is licensed under GPLv3 and can be downloaded at (https://github.com/fkallen/CAREx).

Automatic creation of personalised virtual construction safety training in digital twins

The construction industry faces persistent safety challenges resulting in far too many occupational injuries and fatalities. Even advanced measures like virtual safety training methods still fall short in terms of realism. This paper introduces a pioneering approach that exploits digital twins to create virtual training environments (VTEs) tailored for construction safety. The digital twin for construction safety (DTCS) framework seamlessly integrates building information modelling (BIM) models, schedules, resources and hazard data into a prototype using the game engine Unity. It enables the generation of VTEs with minimal user interaction, empowering safety trainers to define personalised scenarios based on specific project parameters and training requirements tailored to the trainees’ needs. The evaluation of this approach through three distinct real-world projects and a case study with trainees demonstrates its efficacy. The results indicate the ability of the DTCS framework to efficiently create VTEs independently of the project once digital project information is available. Moreover, the framework provides meaningful feedback to the trainees by evaluating the performance using runtime data collection and analysis. The collected data promise use for several applications, but future research must validate their effectiveness. Beyond training, the study highlights the utility of VTEs in improving safety design and hazard mitigation strategies through data-driven insights.

Development of a Machine Learning (ML)-Based Computational Model to Estimate the Engineering Properties of Portland Cement Concrete (PCC)

Portland cement concrete (PCC) is the construction material most used worldwide. Hence, its proper characterization is fundamental for the daily-basis engineering practice. Nonetheless, the experimental measurements of the PCC’s engineering properties (i.e., Poisson’s Ratio -v-, Elastic Modulus -E-, Compressive Strength -ComS-, and Tensile Strength -TenS-) consume considerable amounts of time and financial resources. Therefore, the development of high-precision indirect methods is fundamental. Accordingly, this research proposes a computational model based on deep neural networks (DNNs) to simultaneously predict the v, E, ComS, and TenS. For this purpose, the Long-Term Pavement Performance database was employed as the data source. In this regard, the mix design parameters of the PCC are adopted as input variables. The performance of the DNN model was evaluated with 1:1 lines, goodness-of-fit parameters, Shapley additive explanations assessments, and running time analysis. The results demonstrated that the proposed DNN model exhibited an exactitude higher than 99.8%, with forecasting errors close to zero (0). Consequently, the machine learning-based computational model designed in this investigation is a helpful tool for estimating the PCC’s engineering properties when laboratory tests are not attainable. Thus, the main novelty of this study is creating a robust model to determine the v, E, ComS, and TenS by solely considering the mix design parameters. Likewise, the central contribution to the state-of-the-art achieved by the present research effort is the public launch of the developed computational tool through an open-access GitHub repository, which can be utilized by engineers, designers, agencies, and other stakeholders.

Quantitative Bounds on Resource Usage of Probabilistic Programs

Cost analysis, also known as resource usage analysis, is the task of finding bounds on the total cost of a program and is a well-studied problem in static analysis. In this work, we consider two classical quantitative problems in cost analysis for probabilistic programs. The first problem is to find a bound on the expected total cost of the program. This is a natural measure for the resource usage of the program and can also be directly applied to average-case runtime analysis. The second problem asks for a tail bound, i.e. ‍given a threshold t the goal is to find a probability bound p such that ℙ[total cost ≥ t ] ≤ p . Intuitively, given a threshold t on the resource, the problem is to find the likelihood that the total cost exceeds this threshold. First, for expectation bounds, a major obstacle in previous works on cost analysis is that they can handle only non-negative costs or bounded variable updates. In contrast, we provide a new variant of the standard notion of cost martingales, that allows us to find expectation bounds for a class of programs with general positive or negative costs and no restriction on the variable updates. More specifically, our approach is applicable as long as there is a lower bound on the total cost incurred along every path. Second, for tail bounds, all previous methods are limited to programs in which the expected total cost is finite. In contrast, we present a novel approach, based on a combination of our martingale-based method for expectation bounds with a quantitative safety analysis, to obtain a solution to the tail bound problem that is applicable even to programs with infinite expected cost. Specifically, this allows us to obtain runtime tail bounds for programs that do not terminate almost-surely. In summary, we provide a novel combination of martingale-based cost analysis and quantitative safety analysis that is able to find expectation and tail cost bounds for probabilistic programs, without the restrictions of non-negative costs, bounded updates, or finiteness of the expected total cost. Finally, we provide experimental results showcasing that our approach can solve instances that were beyond the reach of previous methods.

Runtime Analysis of Competitive Co-evolutionary Algorithms for Maximin Optimisation of a Bilinear Function

Co-evolutionary algorithms have a wide range of applications, such as in hardware design, evolution of strategies for board games, and patching software bugs. However, these algorithms are poorly understood and applications are often limited by pathological behaviour, such as loss of gradient, relative over-generalisation, and mediocre objective stasis. It is an open challenge to develop a theory that can predict when co-evolutionary algorithms find solutions efficiently and reliable. This paper provides a first step in developing runtime analysis for population-based competitive co-evolutionary algorithms. We provide a mathematical framework for describing and reasoning about the performance of co-evolutionary processes. To illustrate the framework, we introduce a population-based co-evolutionary algorithm called PDCoEA, and prove that it obtains a solution to a bilinear maximin optimisation problem in expected polynomial time. Finally, we describe settings where PDCoEA needs exponential time with overwhelmingly high probability to obtain a solution.

Analysing Equilibrium States for Population Diversity

Population diversity is crucial in evolutionary algorithms as it helps with global exploration and facilitates the use of crossover. Despite many runtime analyses showing advantages of population diversity, we have no clear picture of how diversity evolves over time. We study how the population diversity of (μ+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\mu +1)$$\\end{document} algorithms, measured by the sum of pairwise Hamming distances, evolves in a fitness-neutral environment. We give an exact formula for the drift of population diversity and show that it is driven towards an equilibrium state. Moreover, we bound the expected time for getting close to the equilibrium state. We find that these dynamics, including the location of the equilibrium, are unaffected by surprisingly many algorithmic choices. All unbiased mutation operators with the same expected number of bit flips have the same effect on the expected diversity. Many crossover operators have no effect at all, including all binary unbiased, respectful operators. We review crossover operators from the literature and identify crossovers that are neutral towards the evolution of diversity and crossovers that are not.

Towards Running Time Analysis of Interactive Multi-Objective Evolutionary Algorithms

Evolutionary algorithms (EAs) are widely used for multi-objective optimization due to their population-based nature. Traditional multi-objective EAs (MOEAs) generate a large set of solutions to approximate the Pareto front, leaving a decision maker (DM) with the task of selecting a preferred solution. However, this process can be inefficient and time-consuming, especially when there are many objectives or the DM has subjective preferences. To address this issue, interactive MOEAs (iMOEAs) combine decision making into the optimization process, i.e., update the population with the help of the DM. In contrast to their wide applications, there has existed only two pieces of theoretical works on iMOEAs, which only considered interactive variants of the two simple single-objective algorithms, RLS and (1+1)-EA. This paper provides the first running time analysis (the essential theoretical aspect of EAs) for practical iMOEAs. Specifically, we prove that the expected running time of the well-developed interactive NSGA-II (called R-NSGA-II) for solving the OneMinMax, OneJumpZeroJump problems are all asymptotically faster than the traditional NSGA-II. Meanwhile, we present a variant of OneMinMax, and prove that R-NSGA-II can be exponentially slower than NSGA-II. These results provide theoretical justification for the effectiveness of iMOEAs while identifying situations where they may fail. Experiments are also conducted to validate the theoretical results.

Runtime Analysis of the SMS-EMOA for Many-Objective Optimization

The widely used multiobjective optimizer NSGA-II was recently proven to have considerable difficulties in many-objective optimization. In contrast, experimental results in the literature show a good performance of the SMS-EMOA, which can be seen as a steady-state NSGA-II that uses the hypervolume contribution instead of the crowding distance as the second selection criterion. This paper conducts the first rigorous runtime analysis of the SMS-EMOA for many-objective optimization. To this aim, we first propose a many-objective counterpart, the m-objective mOJZJ problem, of the bi-objective OJZJ benchmark, which is the first many-objective multimodal benchmark used in a mathematical runtime analysis. We prove that SMS-EMOA computes the full Pareto front of this benchmark in an expected number of O(M^2 n^k) iterations, where n denotes the problem size (length of the bit-string representation), k the gap size (a difficulty parameter of the problem), and M=(2n/m-2k+3)^(m/2) the size of the Pareto front. This result together with the existing negative result on the original NSGA-II shows that in principle, the general approach of the NSGA-II is suitable for many-objective optimization, but the crowding distance as tie-breaker has deficiencies. We obtain three additional insights on the SMS-EMOA. Different from a recent result for the bi-objective OJZJ benchmark, the stochastic population update often does not help for mOJZJ. It results in a 1/Θ(min(Mk^(1/2)/2^(k/2),1)) speed-up, which is Θ(1) for large m such as m&gt;k. On the positive side, we prove that heavy-tailed mutation still results in a speed-up of order k^(0.5+k-β). Finally, we conduct the first runtime analyses of the SMS-EMOA on the bi-objective OneMinMax and LOTZ benchmarks and show that it has a performance comparable to the GSEMO and the NSGA-II.

How to Use the Metropolis Algorithm for Multi-Objective Optimization?

The Metropolis algorithm can cope with local optima by accepting inferior solutions with suitably small probability. That this can work well was not only observed in empirical research, but also via mathematical runtime analyses on single-objective benchmarks. This paper takes several steps towards understanding, again via theoretical means, whether such advantages can also be obtained in multi-objective optimization. The original Metropolis algorithm has two components, one-bit mutation and the acceptance strategy, which allows accepting inferior solutions. When adjusting the acceptance strategy to multi-objective optimization in the way that an inferior solution that is accepted replaces its parent, then the Metropolis algorithm is not very efficient on our multi-objective version of the multimodal DLB benchmark called DLTB. With one-bit mutation, this multi-objective Metropolis algorithm cannot optimize the DLTB problem, with standard bit-wise mutation it needs at least Ω(n^5) time to cover the full Pareto front. In contrast, we show that many other multi-objective optimizers, namely the GSEMO, SMS-EMOA, and NSGA-II, only need time O(n^4). When keeping the parent when an inferior point is accepted, the multi-objective Metropolis algorithm both with one-bit or standard bit-wise mutation solves the DLTB problem efficiently, with one-bit mutation experimentally leading to better results than several other algorithms. Overall, our work suggests that the general mechanism of the Metropolis algorithm can be interesting in multi-objective optimization, but that the implementation details can have a huge impact on the performance.

Runtime Analysis of the (μ + 1) GA: Provable Speed-Ups from Strong Drift towards Diverse Populations

Most evolutionary algorithms used in practice heavily employ crossover. In contrast, the rigorous understanding of how crossover is beneficial is largely lagging behind. In this work, we make a considerable step forward by analyzing the population dynamics of the (µ+1) genetic algorithm when optimizing the Jump benchmark. We observe (and prove via mathematical means) that once the population contains two different individuals on the local optimum, the diversity in the population increases in expectation. From this drift towards more diverse states, we show that a diversity suitable for crossover to be effective is reached quickly and, more importantly, then persists for a time that is at least exponential in the population size µ. This drastically improves over the previously best known guarantee, which is only quadratic in µ. Our new understanding of the population dynamics easily gives stronger performance guarantees. In particular, we derive that population sizes logarithmic in the problem size n suffice to gain an Ω(n)-factor runtime improvement from crossover (previous works achieved comparable bounds only with µ = Θ(n) or a non-standard mutation rate).