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Greedy algorithms are a fundamental class of mathematics and computer science algorithms, defined by their iterative approach of making locally optimal decisions to approximate global optima. In this review, we focus on two greedy algorithms. First, we examine the relaxed greedy algorithm in the context of dictionaries in Hilbert spaces, analyzing the optimality of the definition of this algorithm. Next, we provide a general overview of the thresholding greedy algorithm and the Chebyshev thresholding greedy algorithm, with particular attention to their applications to bases in p -Banach spaces with 0 < p ≤ 1 . In both cases, we conclude by posing several questions for future research.

We investigate the prediction capability of the orthogonal greedy algorithm (OGA) in high-dimensional regression models with dependent observations. The rates of convergence of the prediction error of OGA are obtained under a variety of sparsity conditions. To prevent OGA from overfitting, we introduce a high-dimensional Akaike’s information criterion (HDAIC) to determine the number of OGA iterations. A key contribution of this work is to show that OGA, used in conjunction with HDAIC, can achieve the optimal convergence rate without knowledge of how sparse the underlying high-dimensional model is.

The combination of unmanned aerial vehicles (UAVs) and mobile edge computing (MEC) technology breaks through the limitations of traditional terrestrial communications. The effective line-of-sight channel provided by UAVs can greatly improve the communication quality between edge servers and mobile devices (MDs). To further enhance the Quality-of-Service (QoS) of MEC systems, a multi-UAV-enabled MEC system model is designed. In the proposed model, UAVs are regarded as edge servers to offer computing services for MDs, aiming to minimize the average task response time by jointly optimizing UAV deployment and computation offloading. Based on the problem definition, a two-layer joint optimization method (PSO-GA-G) is proposed. First, the outer layer utilizes a Particle Swarm Optimization algorithm combined with Genetic Algorithm operators (PSO-GA) to optimize UAV deployment. Next, the inner layer adopts a greedy algorithm to optimize computation offloading. The extensive simulation experiments verify the feasibility and effectiveness of the proposed PSO-GA-G. The results show that the PSO-GA-G can achieve a lower average task response time than the other three baselines.

This study introduces a greedy algorithm for deriving decision rules from decision tree ensembles, targeting enhanced interpretability and generalization in distributed data environments. Decision rules, known for their transparency, provide an accessible method for knowledge extraction from data, facilitating decision-making processes across diverse fields. Traditional decision tree algorithms, such as CART and ID3, are employed to induce decision trees from bootstrapped datasets, which represent distributed data sources. Subsequently, a greedy algorithm is applied to derive decision rules that are true across multiple decision trees. Experiments are performed, taking into account knowledge representation and discovery perspectives. They show that, as the value of α, 0≤α<1, increases, shorter rules are obtained, and also it is possible to improve the classification accuracy of rule-based models.

Herein, a combined multipoint picking scheme was proposed, and the sizes of the end of the bud picker were selectively designed. Firstly, the end of the bud picker was abstracted as a fixed-size picking box, and it was assumed that the tea buds in the picking box have a certain probability of being picked. Then, the picking box coverage and the greedy algorithm were designed to make as few numbers of picking box set as possible to cover all buds to reduce the numbers of picking. Furthermore, the Graham algorithm and the minimum bounding box were applied to fine-tune the footholds of each picking box in the optimal coverage picking box set, so that the buds were concentrated in the middle of the picking boxes as much as possible. Moreover, the geometric center of each picking box was taken as a picking point, and the ant colony algorithm was used to optimize the picking path of the end of the bud picker. Finally, by analyzing the influence of several parameters on the picking performance of the end of the bud picker, the optimal sizes of the picking box were calculated successfully under different conditions. The experimental results showed that the average picking numbers of the combined multipoint picking scheme were reduced by 31.44%, the shortest picking path was decreased by 11.10%, and the average consumed time was reduced by 50.92% compared to the single-point picking scheme. We believe that the proposed scheme can provide key technical support for the subsequent design of intelligent bud-picking robots.

The autonomous task success of an unmanned aerial vehiclel (UAV) or its military specialization called the unmanned combat aerial vehicle (UCAV) has a direct relationship with the planned path. However, planning a path for a UAV or UCAV system requires solving a challenging problem optimally by considering the different objectives about the enemy threats protecting the battlefield, fuel consumption or battery usage and kinematic constraints on the turning maneuvers. Because of the increasing demands to the UAV systems and game-changing roles played by them, developing new and versatile path planning algorithms become more critical and urgent. In this study, a greedy algorithm named as the Back-and-Forth (BaF) was designed and introduced for solving the path planning problem. The BaF algorithm gets its name from the main strategy where a heuristic approach is responsible to generate two predecessor paths, one of which is calculated from the start point to the target point, while the other is calculated in the reverse direction, and combines the generated paths for utilizing their advantageous line segments when obtaining more safe, short and maneuverable path candidates. The performance of the BaF was investigated over three battlefield scenarios and twelve test cases belonging to them. Moreover, the BaF was integrated into the workflow of a well-known meta-heuristic, artificial bee colony (ABC) algorithm, and detailed experiments were also carried out for evaluating the possible contribution of the BaF on the path planning capabilities of another technique. The results of the experiments showed that the BaF algorithm is able to plan at least promising or generally better paths with the exact consistency than other tested meta-heuristic techniques and runs nine or more times faster as validated through the comparison between the BaF and ABC algorithms. The results of the experiments further proved that the integration of the BaF boosts the performance of the ABC and helps it to outperform all of fifteen competitors for nine of twelve test cases.

Linear kinetic transport equations play a critical role in optical tomography, radiative transfer and neutron transport. The fundamental difficulty hampering their efficient and accurate numerical resolution lies in the high dimensionality of the physical and velocity/angular variables and the fact that the problem is multiscale in nature. Leveraging the existence of a hidden low-rank structure hinted by the diffusive limit, in this work, we design and test the angular-space reduced order model for the linear radiative transfer equation, the first such effort based on the celebrated reduced basis method (RBM). Our method is built upon a high-fidelity solver employing the discrete ordinates method in the angular space, an asymptotic preserving upwind discontinuous Galerkin method for the physical space, and an efficient synthetic accelerated source iteration for the resulting linear system. Addressing the challenge of the parameter values (or angular directions) being coupled through an integration operator, the first novel ingredient of our method is an iterative procedure where the macroscopic density is constructed from the RBM snapshots, treated explicitly and allowing a transport sweep, and then updated afterwards. A greedy algorithm can then proceed to adaptively select the representative samples in the angular space and form a surrogate solution space. The second novelty is a least squares density reconstruction strategy, at each of the relevant physical locations, enabling the robust and accurate integration over an arbitrarily unstructured set of angular samples toward the macroscopic density. Numerical experiments indicate that our method is effective for computational cost reduction in a variety of regimes.

Cloud RAN, a novel architecture for modern mobile networks, relocates processing units from antenna to distant data centers. This shift introduces the challenge of ensuring low latency for the periodic messages exchanged between antennas and their respective processing units. In this study, we tackle the problem of devising an efficient periodic message assignment scheme under the constraints of fixed message size and period without contention nor buffering. We address this problem by modeling it on a common network topology, wherein contention arises from a single shared link servicing multiple antennas. While reminiscent of coupled task scheduling, the introduction of periodicity adds a unique dimension to the problem. We study how the problem behaves with regard to the load of the shared link, and we focus on proving that, for load as high as possible, a solution always exists and it can be found in polynomial time. The main contributions of this article are two polynomial time algorithms, which find a solution for messages of any size and load at most 2/5 or for messages of size one and load at most ϕ-1, the golden ratio conjugate. We also prove that a randomized greedy algorithm finds a solution on almost all instances with high probability, shedding light on the effectiveness of greedy algorithms in practical applications.

We address the solution of large-scale Bayesian optimal experimental design (OED) problems governed by partial differential equations (PDEs) with infinite-dimensional parameter fields. The OED problem seeks to find sensor locations that maximize the expected information gain (EIG) in the solution of the underlying Bayesian inverse problem. Computation of the EIG is usually prohibitive for PDE-based OED problems. To make the evaluation of the EIG tractable, we approximate the (PDE-based) parameter-to-observable map with a derivative-informed projected neural network (DIPNet) surrogate, which exploits the geometry, smoothness, and intrinsic low-dimensionality of the map using a small and dimension-independent number of PDE solves. The surrogate is then deployed within a greedy algorithm-based solution of the OED problem such that no further PDE solves are required. We analyze the EIG approximation error in terms of the generalization error of the DIPNet and show they are of the same order. Finally, the efficiency and accuracy of the method are demonstrated via numerical experiments on OED problems governed by inverse scattering and inverse reactive transport with up to 16,641 uncertain parameters and 100 experimental design variables, where we observe up to three orders of magnitude speedup relative to a reference double loop Monte Carlo method.

We address the sensor selection problem where linear measurements under correlated noise are gathered at the selected nodes to estimate the unknown parameter. Since finding the best subset of sensor nodes that minimizes the estimation error requires a prohibitive computational cost especially for a large number of nodes, we propose a greedy selection algorithm that uses the log-determinant of the inverse estimation error covariance matrix as the metric to be maximized. We further manipulate the metric by employing the QR and LU factorizations to derive a simple analytic rule which enables an efficient selection of one node at each iteration in a greedy manner. We also make a complexity analysis of the proposed algorithm and compare with different selection methods, leading to a competitive complexity of the proposed algorithm. For performance evaluation, we conduct numerical experiments using randomly generated measurements under correlated noise and demonstrate that the proposed algorithm achieves a good estimation accuracy with a reasonable selection complexity as compared with the previous novel selection methods.