Hilbert Curve Research Articles

Toward a Dynamic Allocation Strategy for Deadline‐Oriented Resource and Job Management in HPC Systems

ABSTRACTAs high‐performance computing (HPC) becomes a tool used in many different workflows, quality of service (QoS) becomes increasingly important. In many cases, this includes the reliable execution of an HPC job and the generation of the results by a certain deadline. The resource and job management system (RJMS) or simply RMS is responsible for receiving the job requests and executing the jobs with a deadline‐oriented policy to support the workflows. In this article, we evaluate how well static resource management policies cope with deadline‐constrained HPC jobs and explore two variations of a dynamic policy in this context. As the Hilbert curve‐based approach used by the SLURM workload manager represents the state‐of‐the‐art in production environments, it was selected as one of the static allocation strategies. The Manhattan median approach as a second allocation strategy was introduced as a research work that aims to minimize the communication overhead of the parallel programs by providing compact partitions more than the Hilbert curve approach. In contrast to the static partitions provided by the Hilbert curve approach and the Manhattan median approach, the leak approach focuses on supporting dynamic runtime behavior of the jobs and assigning nodes of the HPC system on demand at runtime. Since the contiguous leak version also relies on a compact set of nodes, the noncontiguous leak can provide additional nodes at a greater distance from the nodes already used by the job. Our preliminary results clearly show that a dynamic policy is needed to meet the requirements of a modern deadline‐oriented RMS scenario.

Development of a novel computational technique to create DNA and cell geometrical models for Geant4-DNA

BackgroundThis study aimed to develop a novel human cell geometry for the Geant4-DNA simulation toolkit that explicitly incorporates all 23 chromosome pairs of the human cell. This approach contrasts with the existing, default human cell, geometrical model, which utilizes a continuous Hilbert curve. MethodsA Python-based tool named “complexDNA” was developed to facilitate the design of both simple and complex DNA geometries. This tool was employed to construct a human cell geometry with individual pairs of chromosomes. Subsequently, the performance of this chromosomal model was compared to the standard human cell model provided in the “molecularDNA” Geant4-DNA example. ResultsSimulations using the new chromosomal model revealed minimal discrepancies in DNA damage yield and fragment size distribution compared to the default human cell model. Notably, the chromosomal model demonstrated significant computational efficiency, requiring approximately three times less simulation time to achieve equivalent results. ConclusionsThis work highlights the importance of incorporating chromosomal structure into human cell models for radiation biology research. The “complexDNA” tool offers a valuable resource for creating intricate DNA structures for future studies. Further refinements, such as implementing smaller voxels for euchromatin regions, are proposed to enhance the model’s accuracy.

GLMI: An Efficient Spatiotemporal Index Leveraging Geohash and Piecewise Linear Models for Optimized Query Performance

Spatiotemporal big data contain information in multiple dimensions such as space and time. Spatiotemporal data have the characteristics of large volume, intricate spatiotemporal relationship, and uneven spatiotemporal distribution. Index structure is one of the most important technologies used to improve system data analysis and workload. However, it is difficult to dynamically adjust with data density, resulting in increased maintenance costs and retrieval complexity. At the same time, maintaining the proximity of spatiotemporal data in spatial or temporal dimensions is crucial for efficient spatiotemporal analysis. To address these challenges, this paper proposes a learned index method, GLMI (Geohash and piecewise linear model-based index for spatiotemporal data). GLMI uses dynamic space partitioning based on the Hilbert curve to reduce the impact of data skew on index performance. In the time dimension, a piecewise linear model was constructed using the ShrinkingCone algorithm, and a buffer was designed to support the fast writing of spatiotemporal data. Compared with the current mainstream traditional high-dimensional indexes and the ZM index, GLMI has a smaller space consumption and shorter construction time compared to high-dimensional learned indexes on real traffic itinerary and trajectory record datasets. Meanwhile, GLMI also has an advantage in query efficiency.

Ncorpi[formula omitted]N : A [formula omitted] software for N-body integration in collisional and fragmenting systems

NcorpiON is a general purpose N-body software initially developed for the time-efficient integration of collisional and fragmenting systems of planetesimals or moonlets orbiting a central mass. It features a fragmentation model, based on crater scaling and ejecta models, able to realistically simulate a violent impact.The user of NcorpiON can choose between four different built-in modules to compute self-gravity and detect collisions. One of these makes use of a mesh-based algorithm to treat mutual interactions in O(N) time. Another module, much more efficient than the standard Barnes–Hut tree code, is a O(N) tree-based algorithm called FalcON. It relies on fast multipole expansion for gravity computation and we adapted it to collision detection as well. Computational time is reduced by building the tree structure using a three-dimensional Hilbert curve. For the same precision in mutual gravity computation, NcorpiON is found to be up to 25 times faster than the famous software REBOUND.NcorpiON is written entirely in the C language and only needs a C compiler to run. A python add-on, that requires only basic python libraries, produces animations of the simulations from the output files. NcorpiON can communicate with REBOUND’s webGL viewer via MPI for 3D visualization. The name NcorpiON, reminding of a scorpion, comes from the French N-corps, meaning N-body, and from the mathematical notation O(N), due to the running time of the software being almost linear in the total number N of bodies. NcorpiON detects collisions and computes mutual gravity faster than REBOUND, and unlike other N-body integrators, it can resolve a collision by fragmentation. The fast multipole expansions are implemented up to order eight to allow for a high precision in mutual gravity computation.

Space-filling designs on Riemannian manifolds

This paper proposes a new approach to generating space-filling designs over Riemannian manifolds by using a Hilbert curve. Different from ordinary Euclidean spaces, a novel transformation is constructed to link the uniform distribution over a Riemannian manifold and that over its parameter space. Using this transformation, the uniformity of the design points in the sense of Riemannian volume measure can be guaranteed by the intrinsic measure preserving property of the Hilbert curve. It is proved that these generated designs are not only asymptotically optimal under minimax and maximin distance criteria, but also perform well in minimizing the Wasserstein distance from the target distribution and controlling the estimation error in numerical integration. Furthermore, an efficient algorithm is developed for numerical generation of these space-filling designs. The advantages of the new approach are verified through numerical simulations.

AMP: Total Variation Reduction for Lossless Compression via Approximate Median-based Preconditioning

With the increasing scale of cloud computing applications of next-generation embedded systems, a major challenge that domain scientists are facing is how to efficiently store and analyze the vast volume of output data. Compression can reduce the amount of data that needs to be transferred and stored. However, most of the large datasets are in floating-point format, which exhibits high entropy. As a result, existing lossless compressors cannot provide enough performance for such applications. To address this problem, we propose a total variation reduction method for improving the compression ratio of lossless compressors (namely, FPC + and FPZIP + ), which employs a median-based hyperplane to precondition the data. In particular, we first try to exploit the space-filling curve (SFC), a well-known technique to preserve data locality for a multi-dimensional dataset. We show and explain why a raw SFC, such as Hilbert and Z-order curves, cannot improve the compression ratio. Then, we explore the opportunity and theoretical feasibility of the proposed total variation reduction-based algorithm. The experiment results show the effectiveness of the proposed method. The compression ratios are improved up to 48.2% (20.6% on average) for FPZIP and 42.4% (18.4% on average) for FPC. Moreover, through observing the time composition of the proposed method, it is found that the median finding holds a high percentage of the execution time. Hence, we further introduce an approximate median finding algorithm, providing a linear-time overhead reduction scheme. The experiment results clearly demonstrate that this algorithm reduces execution time by an average of 56.7% and 40.7% compared to FPC + and FPZIP + , respectively.

Metamaterial-based Nano-Antenna Design of Enhanced Plasmonic Electromagnetic Properties

This work, presents a design of a nano-antenna based plasmonic materials at the frequency range from 192THz to 195THz for modern applications, including THz communication systems. The proposed antenna is constructed from the 3rd iteration of the Hilbert curve patch fed with a modified coplanar 50Ω port. The back layer of the antenna was covered with a partial ground plane defective with a metamaterial-based electromagnetic band gap (EBG) layer. The antenna shows high directivity up to 68dBi at 193.4THz with an excellent matching over the entire frequency band of interest. The proposed antenna is considered of gold conductive layer mounted on a silicon dioxide (SiO2) layer. The proposed EBG was designed as the second order of the Minkowski fractal geometry. The obtained results in terms of S11 were below -10dB for the frequency band of 3THz. These results were validated using numerical techniques based on CST MWS by invoking the time domain (TD) and frequency domain (FD) solvers. The results achieved from the considered techniques agreed very well with each other.

On Recursive Tiling of some Mathematical Objects.

This essay, in a novel way, uses various tiles, such as dominoes, square-, interlocking-, and flow-tiles, to visualize approximations to recursively defined, potentially infinite mathematical objects, such as binary code tables, and variants of the space-filling Hilbert curve. The power of L-systems is shown by defining these tiled objects as L-systems.

LDMNet: Enhancing the Segmentation Capabilities of Unmanned Surface Vehicles in Complex Waterway Scenarios

Semantic segmentation-based Complex Waterway Scene Understanding has shown great promise in the environmental perception of Unmanned Surface Vehicles. Existing methods struggle with estimating the edges of obstacles under conditions of blurred water surfaces. To address this, we propose the Lightweight Dual-branch Mamba Network (LDMNet), which includes a CNN-based Deep Dual-branch Network for extracting image features and a Mamba-based fusion module for aggregating and integrating global information. Specifically, we improve the Deep Dual-branch Network structure by incorporating multiple Atrous branches for local fusion; we design a Convolution-based Recombine Attention Module, which serves as the gate activation condition for Mamba-2 to enhance feature interaction and global information fusion from both spatial and channel dimensions. Moreover, to tackle the directional sensitivity of image serialization and the impact of the State Space Model’s forgetting strategy on non-causal data modeling, we introduce a Hilbert curve scanning mechanism to achieve multi-scale feature serialization. By stacking feature sequences, we alleviate the local bias of Mamba-2 towards image sequence data. LDMNet integrates the Deep Dual-branch Network, Recombine Attention, and Mamba-2 blocks, effectively capturing the long-range dependencies and multi-scale global context information of Complex Waterway Scene images. The experimental results on four benchmarks show that the proposed LDMNet significantly improves obstacle edge segmentation performance and outperforms existing methods across various performance metrics.

Multilayer multiband hybrid fractal antenna for public safety and 5G Sub-6 GHz bands

In this work, a multiband hybrid fractal antenna is developed for public safety and 5G Sub-6 GHz bands. Proposed antenna design is designed, analyzed, and optimized using HFSS simulator. Radiating element of the proposed antenna consists of Hybrid Fractal shape which is designed using Koch and Hilbert curve fractal geometry. Hybrid Fractal Antenna (HFA) is designed on a Multilayer FR4 substrate which has an overall size of 0.22 × 0.47 × 0.044 ƛ3. Parametric analysis of the proposed design is done to get the optimum results. Proposed HFA resonates at 2.64 GHz (2.34–2.80 GHz), 3.87, 4.54, 5.02, 5.35 GHz (3.73–5.55 GHz), and 6.42 GHz (5.70–6.72 GHz) with a gain of 4.2, 0.9, 2.9, 3.3, 4.6, and 3.7 dBi respectively. The proposed HFA is useful for 5G bands such as: N7 band, N38 band, N40 band, N41 band, N47 band, N53 band, N79 band, N90 band and N93 band, and also useful for public safety band (4.9 GHz). Prototype of the proposed HFA is fabricated and tested in lab for the verification of simulated results. The results show good performance in terms of reflection coefficient, gain, and bandwidth. Measured results are in good agreement with the simulated results.

Improvement of 3D printing and subsequent microwave freeze-drying solidification accuracy of pineapple gel-based inks: Infill percentage control and internal models design

The purpose of this study was to improve the 3D printing accuracy of pineapple gel and its subsequent microwave freeze drying (MFD) solidification precision based on infill percentage control (30%, 50%, 70%, and 90%) and internal models design (Hilbert curve, honeycomb, and rectilinear). Through comparing the designed models with the physical dimensions of the printed and dehydrated products, the optimal regulation strategy was obtained. Results showed that the printing deviation of samples decreased initially and then increased with higher infill percentages, with the 70% infill percentage having the lowest deviation. Among all internal models, the rectilinear infill pattern showed the best printing accuracy. Porosity and the shape deformation rate of MFD solidification products initially decreased and then increased with higher infill percentages. The 70% infill percentage products had the lowest shape deformation rate. The honeycomb infill pattern performed best in reducing solidification sample shrinkage and deformation. Additionally, micro-CT scans revealed that the honeycomb infill pattern helped maintain the integrity of each layer's lines of MFD solidification products, closely matching the model's line distribution. In the 50% and 70% infill percentage, honeycomb and rectilinear samples exhibited better crispness. A 70% honeycomb infill pattern was recommended for optimal printing and post-solidification accuracy.

Plasma Jet Mediated Reduction of Metal Salt

We demonstrate electrodeposition where the plasma acts as both an electrolyte and electrode, to deposit metallic conductive coatings. In conventional electroplating metal ions are reduced at a working electrode with an applied electrochemical potential. Here an RF atmospheric pressure plasma jet (APPJ) is able to induce a electrochemical reduction due to the presence of free electrons. The advantage of this method is demonstrated by the ability to deposit conducting metal tracks on delicate surfaces including insulators and polymers. The small size and gaseous nature of the plasma allows deposition in difficult geometries with precise spatial control.The reduction process occurs within the plasma. As the aerosolized metal salt crystals interact with the plasma, the free electrons reduce the metal and the hydrogen combines with the anion producing a water soluble acid. The metal reduction takes place within the plasma jet, as show by varying the plasma power and determining the amount of zerovalent metal deposited. Even at low powers, <6 W, the plasma was able to fully reduce the silver, however the copper requires plasma powers of 12 W and above in order to be fully reduced. Once reduced the deposited tracks were shown to be conductive with around 20 % and around 5 % of bulk conductivity for silver and copper respectively. A sintering process occurs also within the plasma, leaving a contiguous track of fused particles of zerovalent metal producing the conducting tracks. The technique is demonstrated mainly for copper and silver, however multiple other metals were capable of being deposited. In addition to the conductivity, the track adherence was tested both using a Scotch Tape© pull test and an in-house constructed scratch test. The track was completely unaffected by the tape pull and the stylus type scratch test showed that the films were scratch resistant up to a pressure up to of 420 kg cm-2, with no measurable drop in conductivity.The APPJ is an accessible, low power, sustainable, precise electroplating device, capable of depositing pure metallic films onto a variety of surfaces. Furthermore, we demonstrate deposition on rough surfaces such as damaged fiberglass, as well as sensitive surfaces including plant tissue and pigskin, with limited damage to the surface.Figure 1. a) showing the deposition setup including nebuliser, salt solution injection, gas flow and power supply. An optical image of the APPJ during deposition operation is included, as well as Schlieren imagine showing the laminar gas flow. b) A silver Hilbert curve printed onto Kapton 0.05 mm thickness via the plasma deposition technique, in a single step process, with no sample prep or post processing. c) Copper deposited on porcine skin showing conductance as part of a LED circuit. Figure 1

Hilbert-curve assisted structure embedding method

MotivationChemical space embedding methods are widely utilized in various research settings for dimensional reduction, clustering and effective visualization. The maps generated by the embedding process can provide valuable insight to medicinal chemists in terms of the relationships between structural, physicochemical and biological properties of compounds. However, these maps are known to be difficult to interpret, and the ‘‘landscape’’ on the map is prone to ‘‘rearrangement’’ when embedding different sets of compounds.ResultsIn this study we present the Hilbert-Curve Assisted Space Embedding (HCASE) method which was designed to create maps by organizing structures according to a logic familiar to medicinal chemists. First, a chemical space is created with the help of a set of ‘‘reference scaffolds’’. These scaffolds are sorted according to the medicinal chemistry inspired Scaffold-Key algorithm found in prior art. Next, the ordered scaffolds are mapped to a line which is folded into a higher dimensional (here: 2D) space. The intricately folded line is referred to as a pseudo-Hilbert-Curve. The embedding of a compound happens by locating its most similar reference scaffold in the pseudo-Hilbert-Curve and assuming the respective position. Through a series of experiments, we demonstrate the properties of the maps generated by the HCASE method. Subjects of embeddings were compounds of the DrugBank and CANVASS libraries, and the chemical spaces were defined by scaffolds extracted from the ChEMBL database.Scientific contributionThe novelty of HCASE method lies in generating robust and intuitive chemical space embeddings that are reflective of a medicinal chemist’s reasoning, and the precedential use of space filling (Hilbert) curve in the process.Availabilityhttps://github.com/ncats/hcaseGraphical

Integrating NoSQL, Hilbert Curve, and R*-Tree to Efficiently Manage Mobile LiDAR Point Cloud Data

The widespread use of Light Detection and Ranging (LiDAR) technology has led to a surge in three-dimensional point cloud data; although, it also poses challenges in terms of data storage and indexing. Efficient storage and management of LiDAR data are prerequisites for data processing and analysis for various LiDAR-based scientific applications. Traditional relational database management systems and centralized file storage struggle to meet the storage, scaling, and specific query requirements of massive point cloud data. However, NoSQL databases, known for their scalability, speed, and cost-effectiveness, provide a viable solution. In this study, a 3D point cloud indexing strategy for mobile LiDAR point cloud data that integrates Hilbert curves, R*-trees, and B+-trees was proposed to support MongoDB-based point cloud storage and querying from the following aspects: (1) partitioning the point cloud using an adaptive space partitioning strategy to improve the I/O efficiency and ensure data locality; (2) encoding partitions using Hilbert curves to construct global indices; (3) constructing local indexes (R*-trees) for each point cloud partition so that MongoDB can natively support indexing of point cloud data; and (4) a MongoDB-oriented storage structure design based on a hierarchical indexing structure. We evaluated the efficacy of chunked point cloud data storage with MongoDB for spatial querying and found that the proposed storage strategy provides higher data encoding, index construction and retrieval speeds, and more scalable storage structures to support efficient point cloud spatial query processing compared to many mainstream point cloud indexing strategies and database systems.

Analysis of Dynamic Behavior of Gravity Model Using the Techniques of Road Saturation and Hilbert Curve Dimensionality Reduction

Analysis of Dynamic Behavior of Gravity Model Using the Techniques of Road Saturation and Hilbert Curve Dimensionality Reduction

Hilbert Curve Projection Distance for Distribution Comparison.

Distribution comparison plays a central role in many machine learning tasks like data classification and generative modeling. In this study, we propose a novel metric, called Hilbert curve projection (HCP) distance, to measure the distance between two probability distributions with low complexity. In particular, we first project two high-dimensional probability distributions using Hilbert curve to obtain a coupling between them, and then calculate the transport distance between these two distributions in the original space, according to the coupling. We show that HCP distance is a proper metric and is well-defined for probability measures with bounded supports. Furthermore, we demonstrate that the modified empirical HCP distance with the Lp cost in the d-dimensional space converges to its population counterpart at a rate of no more than O(n-1/2max{d,p}). To suppress the curse-of-dimensionality, we also develop two variants of the HCP distance using (learnable) subspace projections. Experiments on both synthetic and real-world data show that our HCP distance works as an effective surrogate of the Wasserstein distance with low complexity and overcomes the drawbacks of the sliced Wasserstein distance.

Fractal Circuit Architectures for Spatially Controlled Heating and Multi-Modal Shape Programming of Electro-Active Shape Memory Polymers

Shape memory polymers (SMPs) are commonly activated through external heating or rigid embedded heaters, which lack precise temperature control and restrict programmable shape transformations. This study demonstrates integrating thin films of hard electronic materials patterned in fractal designs into SMPs yields novel thermomechanical responses, enabling flexible implementation of stretchable electronic functionality. Space-filling Hilbert, Moore and Peano fractal curves generate distributed electronic circuitry patterns within the SMP matrices. To investigate the coupled electro-thermal-mechanical behavior, a multi-physics modeling approach is adopted. A nonlinear thermo-visco-hyperelastic constitutive model captures the SMP’s shape memory characteristics. Governing equations for the multi-physics problem are formulated. Experimental data validates the model’s predictions of the SMP’s nonlinear material response under uniaxial loading. This validated framework analyzes SMP composites with integrated fractal circuits. Findings reveal composites with higher-order fractal circuits’ exhibit faster, more uniform resistive heating and smaller electrical resistance changes during stretching, enabling consistent and predictable shape recovery under various deformations. Notably, these advantageous heating and resistance characteristics enhance the force recovery rate during the shape memory cycle. Furthermore, the composites demonstrate excellent shape recovery under bending. Remarkably, uniaxial stretching can induce controlled out-of-plane shape transformations, enabling novel actuation modalities. This work provides insights into leveraging fractal circuit integration to enhance SMP performance and expand capabilities, enabling multi-functional actuators, sensors and programmable soft grippers.

Research on Additive Manufacturing Path Planning of a Six-Degree-of-Freedom Manipulator

The research on additive manufacturing (AM) path planning mainly focuses on the traditional three-axis AM path planning and five-degree-of-freedom (DOF) AM path planning, while there is less research on six-DOF AM path planning. In the traditional AM path planning algorithm, the filling path is discontinuous and there is long straight-line printing in a certain direction, which can easily lead to warpage deformation. Therefore, in this work, the six-DOF manipulator is taken as the main object to build an AM platform, and the mechanism of AM path planning of the manipulator is studied. The path planning algorithm combining the contour offset filling method and Hilbert curve filling is optimized by using a cubic uniform B-spline curve, and an AM path planning algorithm suitable for a six-DOF manipulator is obtained. A continuous printing path can be generated by this algorithm. It reduces the existence of long straight-line printing in a certain direction, thereby reducing the warpage deformation of the model and improving the molding quality of the model. The traditional three-axis AM device and the six-DOF AM platform were used to print two kinds of models. By comparing the printing time, the six-DOF AM platform was 43.70% and 37.94% shorter than the traditional three-axis AM device. The same model was printed on a six-DOF AM platform by using the parallel scanning filling method, the path planning algorithm combining contour offset and Hilbert curve, and the method proposed in this paper. Through experimental verification, the average warpage deformation of the model printed by the method proposed in this paper was reduced by 37.81% and 13.79%, respectively, compared with the other two methods.

Analysis of Reader Orientation on Detection Performance of Hilbert Curve-based Fractal Chipless RFID Tags

The role of the orientation of the radio frequency identification (RFID) reader is vital in the RFID-based communication system. This study presents the design and analysis of the impact of the orientation of the RFID reader (angle of incidence of the plane wave) on the detection and sensitivity characteristics of fractal chipless RFID tags. Four fractal (irregular) shaped tags are developed using four iterations of the Hilbert curve filling algorithm. The full-wave EM analysis of the designed tags in Matlab is performed by exporting them in Computer Simulation Technologies Micro-Wave Studio (CST MWS) in the frequency range of 2 to 20 GHz. Firstly, the performance of the tags is analyzed by observing the radar cross-section (RCS) of the tag for the fixed orientation of the incident plane wave in three different polarizations (horizontal, vertical, and oblique). Later, the variations in the EM spectrum (RCS results) are analyzed for oblique polarization by varying the incidence plane wave in both elevations (for two cases of 0∘ and 90∘) and azimuth planes (sweeping from 0∘ to 180∘ with a step size of 10∘). The analysis of the proposed aggregated RCS response for all cases in oblique polarization produces higher coding capacity (169 bits), coding spatial capacity (16.504 bits/cm2), coding spectral capacity (9.826 bits/GHz), and coding density (0.960 bits/GHz/cm2) for the realized highly irregular tag using fourth-iteration (4T) of Hilbert curve filling algorithm. The proposed procedure of detection based on aggregated response makes the developed RFID communication system more secure and reliable.

Improving the gas sensing performance of halide perovskite MAPbI3 film via fractal geometry electrode structure

The first successful fabrication of metal halide Perovskite MAPbI3 Film for NH3 gas sensing using fractal electrode structure is reported. High performance NH3 gas sensor was fabricated on ITO glass substrate by laser-patterning method. Gas sensors were fabricated using Hilbert, Peano, and Gosper curve structure electrodes, and their performance is compared to conventional interdigitated electrode. Gas sensing performance on Hilbert electrode design showed the highest response of 2.82 at 10 ppm NH3 gas, which is 145 % times higher compared to conventional interdigitated structures. Simulation study demonstrates that the electric field intensity and number of nodes of the fractal electrode are higher than those of the traditional interdigitated electrode. This study demonstrates that fractal electrode is a feasible strategy for optimizing the potential performance of MAPbI3-based Perovskite gas sensor with low-cost fabrication with no extra processing.