Spatial Operators Research Articles

A novel transient-adaptive subcell algorithm with a hybrid application of different collision techniques in direct simulation Monte Carlo (DSMC)

A novel hybrid transient adaptive subcell (TAS) direct simulation Monte Carlo (DSMC) algorithm is proposed to simulate rarefied gas flows in a wide range of Knudsen numbers. It is derived and analyzed by using a time and spatial discrete operator approach based on the non-homogeneous, local N-particle kinetic equation, first proposed by Stefanov. The novel algorithm is considered together with the standard and hybrid collision algorithms built on uniform grids. The standard collision algorithm uses only one single scheme—the NoTime Counter (NTC), or the Generalized or Simplified Bernoulli trials (GBT, SBT). The hybrid algorithm employs NTC, GBT, or SBT depending on the instantaneous number of particles in the considered cell. The novel hybrid TAS algorithm benefits from both the hybrid collision approach and the transient adaptive subcell grid covering each collision cell to achieve a uniform accuracy of order O(Δt, Δr) independently of the number of particles in the cells. To this aim, a local time step is defined as coherent with the TAS grid covering the corresponding collision cell. The novel hybrid TAS algorithm is tested on two-dimensional benchmark problems: supersonic rarefied gas flow past of a flat plate under an angle of incidence and pressure-driven gas flow in a microchannel. The results obtained by the hybrid TAS algorithm are compared to those obtained by the standard algorithms and the available Bird's DS2V code using nearest neighbor collision and open-source OpenFOAM code. The comparison shows an excellent accuracy of the suggested algorithm in predicting the flow field.

Quantum metrology with multimode Gaussian states of multiple point sources

A unified theoretical framework is developed for measuring multimode Gaussian states of multiple point sources. The spatial and temporal mode functions are found analytically by aid of the circular symmetry of the object plane and aperture, as well as by the rectangular spectral density function of the background. They relate the modal operators and statistical moments to the locations of the point sources through the prolate spheroidal functions. An experimentally straightforward implementation is proposed, which consists of a nonabsorptive polarization- and phase-modulating spatial light modulator (SLM), three lenses, and a polarization beam splitter. Interestingly, it is found that the spatial modal annihilation operators are related to the modal field operators in the image plane by a simple complex-valued scale factor. The quantum Fisher information matrix and quantum Cram\'er-Rao bound (QCRB) are also discussed. The numerical calculation of QCRB is completed for the case where the spatially multimode aperture fields are prepared by the spatial propagation from temporally single-mode thermal states, coherent states, or quadrature squeezed states of the point sources in the object plane. The results show that the Rayleigh limit can always be surpassed by means of increasing the mean photon rate or the squeeze parameter.

A Gated Generative Adversarial Imputation Approach for Signalized Road Networks

Missing data imputation is an essential component of a robust traffic surveillance system. Despite the progress of data imputation technologies in intelligent transportation systems (ITS), only limited efforts have been devoted to tackling data missing issues on signalized roads, which present distinct missing patterns due to the effect of traffic signal timing on traffic flow. Unlike most studies that try to impute missing data at the granularity of road segment and aggregated time intervals, we impute missing data for individual traffic lanes at the granularity of signal cycle. We develop a data-driven fine-grained imputation approach based on a novel gated attentional generative adversarial network (GaGAN), which is highly responsive to the dynamic traffic environments of signalized road networks. The advantage of the network lies in that it can automatically learn inter-lane spatio-temporal correlations during each signal cycle. To model the spatial correlations, we jointly leverage spatial attention mechanisms and graph convolutional operations to quantify inter-lane influences within each signal cycle. To model the temporal correlations, we propose to integrate self-attention mechanisms into gated recurrent units (termed as SA-GRU). Two sub-discriminators are developed to model lane-level complete and missing data distributions, respectively, with the goal to improve the consistency between imputed data and overall data distribution. Experimental results demonstrate that the proposed approach is superior to other state-of-the-art methods, achieving robust performance over different data missing types.

Generalized SAV-Exponential Integrator Schemes for Allen--Cahn Type Gradient Flows

The energy dissipation law and the maximum bound principle (MBP) are two important physical features of the well-known Allen-Cahn equation. While some commonly-used first-order time stepping schemes have turned out to preserve unconditionally both energy dissipation law and MBP for the equation, restrictions on the time step size are still needed for existing second-order or even higher-order schemes in order to have such simultaneous preservation. In this paper, we develop and analyze novel first- and second-order linear numerical schemes for a class of Allen-Cahn type gradient flows. Our schemes combine the generalized scalar auxiliary variable (SAV) approach and the exponential time integrator with a stabilization term, while the standard central difference stencil is used for discretization of the spatial differential operator. We not only prove their unconditional preservation of the energy dissipation law and the MBP in the discrete setting, but also derive their optimal temporal error estimates under fixed spatial mesh. Numerical experiments are also carried out to demonstrate the properties and performance of the proposed schemes.

On the existence and uniqueness of solutions to a nonlinear variable order time-fractional reaction–diffusion equation with delay

In this article, our purpose is to study the existence and uniqueness of a solution to a damped variable order fractional subdiffusion equation with time delay. Under weak assumptions on the data, we prove the uniqueness of a weak solution to the problem under consideration. The method of semi-discretization is extended to this kind of time fractional parabolic problem with delay in the case that the time delay parameter s>0 satisfies s⩽T, where T denotes the final time. As a consequence, two a priori estimates are predicted based on a discrete variational formulation of the problem. The existence of the problem’s weak solution on the time frame 0,⌊Ts⌋s is established by the aid of these derived a priori estimates. The paper is closed by introducing a fully discrete scheme based on Galerkin Legendre spectral approximation for the spatial operator and the backward Euler difference approximation for the temporal variable order operator. Accordingly, the accuracy and efficiency of the proposed scheme are justified by giving some numerical experiments for the sake of clearness.

Realizing transmissive and reflective focusing with an on-chip metalens.

A metalens made of compact planar metastructure exhibits an excellent capability of focusing. The high-quality transmissive and reflective focusing simultaneously provides Fourier transform (FT) operation for optical information processing. Here we show a transflective on-chip metalens (TOM) made of orthogonal nano-grooves (ONGs). The TOM simultaneously converges transmitted and reflected (T&R) waves to the designed focal points. By adjusting the phase gradient profiles provided by the ONGs, the focal lengths of the T&R in-plane waves can be independently tuned. Our simulations show that the TOM possesses the advantages of broadband (>400 nm bandwidth) and high-focusing-efficiency (∼60%) dual-focusing capability. Further, we utilize the TOM to build a one-to-two 4-f optical system. Two different spatial filtering operations based on FT can be simultaneously implemented in axial transmission and off-axis reflection channels for one input signal. We expect that the dual-focusing metalens approach can realize parallel optical processing in on-chip optical computing, spatial filtering, and beyond.

Discretization Corrected Particle Strength Exchange for Steady State Linear Elasticity

Discretization corrected particle strength exchange (DC PSE) is a particle based spatial differential operator designed to solve meshless continuum mechanics problems. DC PSE is a spatial gradient operator that can discretize a computational domain with randomly distributed particles, provided that each particle has enough neighboring particles. In contrast, conventional methods such as the standard finite difference method require the computational domain to be discretized into a Cartesian grid. In linear elasticity simulations, especially steady state cases, this domain is mostly discretized using mesh-based methods such as finite element. However, while particle methods such as smoothed particle hydrodynamics (SPH) have been widely applied to solve dynamic elasticity problems, they have rarely been used in steady state simulations. In this study, a DC PSE operator was used to solve steady linear elasticity problems in a two-dimensional domain. The result of the DC PSE numerical simulation was compared to numerical results, empirical formula results, and results from conventional commercial finite element software, respectively.

Wave Equation Numerical Simulation and RTM With Mixed Staggered-Grid Finite-Difference Schemes

For the conventional staggered-grid finite-difference scheme (C-SFD), although the spatial finite-difference (FD) operator can reach 2Mth-order accuracy, the FD discrete wave equation is the only second-order accuracy, leading to low modeling accuracy and poor stability. We proposed a new mixed staggered-grid finite-difference scheme (M-SFD) by constructing the spatial FD operator using axial and off-axial grid points jointly to approximate the first-order spatial partial derivative. This scheme is suitable for modeling the stress–velocity acoustic and elastic wave equation. Then, based on the time–space domain dispersion relation and the Taylor series expansion, we derived the analytical expression of the FD coefficients. Theoretically, the FD discrete acoustic wave equation and P- or S-wave in the FD discrete elastic wave equation given by M-SFD can reach the arbitrary even-order accuracy. For acoustic wave modeling, with almost identical computational costs, M-SFD can achieve higher modeling accuracy than C-SFD. Moreover, with a larger time step used in M-SFD than that used in C-SFD, M-SFD can achieve higher computational efficiency and reach higher modeling accuracy. For elastic wave simulation, compared to C-SFD, M-SFD can obtain higher modeling accuracy with almost the same computational efficiency when the FD coefficients are calculated based on the S-wave time–space domain dispersion relation. Solving the split elastic wave equation with M-SFD can further improve the modeling accuracy but will decrease the efficiency and increase the memory usage as well. Stability analysis shows that M-SFD has better stability than C-SFD for both acoustic and elastic wave simulations. Applying M-SFD to reverse time migration (RTM), the imaging artifacts caused by the numerical dispersion are effectively eliminated, which improves the imaging accuracy and resolution of deep formation.

Video-based bird posture recognition using dual feature-rates deep fusion convolutional neural network

It is necessary to detect changes in birds’ behaviors promptly to realize their health and habitat status. Moreover, promptly provide appropriate medical treatment and environmental remediation. Automating bird behavior recognition can solve this problem and assist in breeding and protecting birds. This paper proposes the transposed non-local module based on a time pyramid network to establish a dual feature-rates deep fusion net (DF2-Net) for bird behavior recognition. The time pyramid network uses spatial alignment and time pooling operations to extract features with different rates from features of different depths and then fuse features containing the information. On this basis, the transposed non-local (TNL) module uses features with different rates to calculate the relationship matrix of each time slice and spatial position. Then TNL uses the transpose operation to make the original feature take advantage of the corresponding relationship when it multiplies the original feature by the matrix. The module further deeply integrates the relational information of behaviors at different rates to enhance features of different rates, respectively, which can improve the recognition effect of the model on dynamic behaviors. Our study has three contributions: (1) The TNL module takes features with different rates as inputs and uses a transpose mechanism to get two relationships in different directions that match their corresponding input. (2) Dual feature-rates deep fusion net (DF2-Net) transforms a single rate feature into two rate features and iteratively fuses them to obtain information and behavior relationships at various rates. (3) We collect the unique video dataset of birds’ behaviors to fill in the blank of the video dataset and support the study of birds’ behaviors. The experiments compare the DF2-Net with the well-known video-based recognition model on the self-collected birds’ behavior dataset, containing eight behaviors. DF2-Net has the best classification accuracy and achieves 80.87%, 81.35%, 80.70%, and 81.35% in precision, recall, f1-score, and OA metrics. They are 1.81%, 2.43%, 2.20%, and 2.43% higher than the second-best approach (TPN) with 8 frames, respectively. The experimental results show that DF2-Net outperforms state-of-the-art methods in various frames, in which 16 frames are the most suitable for bird behavior recognition. Moreover, we execute the various ablation experiments to demonstrate the efficiency of the TNL module, the optimal location and internal operations of TNL, and the most suitable parameters of DF2-Net. The ablation experiments demonstrate that the TNL module dramatically improves the recognition accuracy of the dynamic behavior and proves the validity and rationality of the TNL module. Therefore, the proposed model is practical and feasible for automatically recognizing bird behavior.

An Accurate and Efficient Quaternion-Based Visualization Approach to 2D/3D Vector Data for the Mobile Augmented Reality Map

Increasingly complex vector map applications and growing multi-source spatial data pose a serious challenge to the accuracy and efficiency of vector map visualization. It is true especially for real-time and dynamic scene visualization in mobile augmented reality, with the dramatic development of spatial data sensing and the emergence of AR-GIS. Such issues can be decomposed into three issues: accurate pose representation, fast and precise topological relationships computation and high-performance acceleration methods. To solve these issues, a novel quaternion-based real-time vector map visualization approach is proposed in this paper. It focuses on precise position and orientation representation, accurate and efficient spatial relationships calculation and acceleration parallel rendering in mobile AR. First, a quaternion-based pose processing method for multi-source spatial data is developed. Then, the complex processing of spatial relationships is mapped into simple and efficient quaternion-based operations. With these mapping methods, spatial relationship operations with large computational volumes can be converted into efficient quaternion calculations, and then the results are returned to respond to the interaction. Finally, an asynchronous rendering acceleration mechanism is also presented in this paper. Experiments demonstrated that the method proposed in this paper can significantly improve vector visualization of the AR map. The new approach, when compared to conventional visualization methods, provides more stable and accurate rendering results, especially when the AR map has strenuous movements and high frequency variations. The smoothness of the user interaction experience is also significantly improved.

Exploring Spatial Significance via Hybrid Pyramidal Graph Network for Vehicle Re-Identification

Existing vehicle re-identification methods commonly use spatial pooling operations to aggregate feature maps extracted via off-the-shelf backbone networks, such as visual geometry group network (VGGNet), Google network (GoogLeNet) and residual network (ResNet). They ignore exploring the spatial significance of feature maps, eventually degrading the vehicle re-identification performance. In this paper, firstly, an innovative spatial graph network (SGN) is proposed to elaborately explore the spatial significance of feature maps. The SGN stacks multiple spatial graphs (SGs). Each SG assigns feature map’s elements as nodes and utilizes spatial neighborhood relationships to determine edges among nodes. During the SGN’s propagation, each node and its spatial neighbors on an SG are aggregated to the next SG. On the next SG, each aggregated node is re-weighted with a learnable parameter to find the significance at the corresponding location. Secondly, a novel pyramidal graph network (PGN) is designed to comprehensively explore the spatial significance of feature maps at multiple scales. The PGN organizes multiple SGNs in a pyramidal manner and makes each SGN handles feature maps of a specific scale. Finally, a hybrid pyramidal graph network (HPGN) is developed by embedding the PGN behind a ResNet-50 based backbone network. Extensive experiments on three large scale vehicle databases (i.e., VeRi776, VehicleID, and VeRi-Wild) demonstrate that the proposed HPGN is superior to state-of-the-art vehicle re-identification approaches in terms of accuracy, parameter cost, and computation cost. In addition, experiments show that the proposed PGN is universal to various backbone networks.

Numerical analysis of time-fractional Sobolev equation for fluid-driven processes in impermeable rocks

This paper proposes a local meshless radial basis function (RBF) method to obtain the solution of the two-dimensional time-fractional Sobolev equation. The model is formulated with the Caputo fractional derivative. The method uses the RBF to approximate the spatial operator, and a finite-difference algorithm as the time-stepping approach for the solution in time. The stability of the technique is examined by using the matrix method. Finally, two numerical examples are given to verify the numerical performance and efficiency of the method.

Computing metasurfaces enabled chiral edge image sensing

SummaryComputing metasurfaces have shown the extraordinary ability to precisely perform optical analog operations to the input light wave, and therefore exhibit greater potentials toward sensing applications. Here, we propose a unique application of computing metasurface for chiral edge sensing by incorporating a weak-value amplification technique. The computing metasurface performs the spatial differentiation operations of phase objects and extracts the edge-enhanced images, because the phase gradient generally occurs at the edge. The chirality-induced polarization rotation acts as the preselection state and the spatial differentiation operations in the metasurface provide weak coupling. The amplified pointer shift related to the tiny polarization rotation will eventually lead to an asymmetric edge-enhanced image. Owing to the high sensitivity of the weak-value amplification, we experimentally demonstrate a high-contrast recognition of chirality by edge detection, which may have potential applications in real-time measurement and separation of chiral enantiomers.

Efficient spatial data partitioning for distributed kNN joins

Parallel processing of large spatial datasets over distributed systems has become a core part of modern data analytic systems like Apache Hadoop and Apache Spark. The general-purpose design of these systems does not natively account for the data’s spatial attributes and results in poor scalability, accuracy, or prolonged runtimes. Spatial extensions remedy the problem and introduce spatial data recognition and operations. At the core of a spatial extension, a locality-preserving spatial partitioner determines how to spatially group the dataset’s objects into smaller chunks using the distributed system’s available resources. Existing spatial extensions rely on data sampling and often mismanage non-spatial data by either overlooking their memory requirements or excluding them entirely. This work discusses the various challenges that face spatial data partitioning and proposes a novel spatial partitioner for effectively processing spatial queries over large spatial datasets. For evaluation, the proposed partitioner is integrated with the well-known k-Nearest Neighbor (kNN) spatial join query. Several experiments evaluate the proposal using real-world datasets. Our approach differs from existing proposals by (1) accounting for the dataset’s unique spatial traits without sampling, (2) considering the computational overhead required to handle non-spatial data, (3) minimizing partition shuffles, (4) computing the optimal utilization of the available resources, and (5) achieving accurate results. This contributes to the problem of spatial data partitioning through (1) providing a comprehensive discussion of the problems facing spatial data partitioning and processing, (2) the development of a novel spatial partitioning technique for in-memory distributed processing, (3) an effective, built-in, load-balancing methodology that reduces spatial query skews, and (4) a Spark-based implementation of the proposed work with an accurate kNN spatial join query. Experimental tests show up to 1.48 times improvement in runtime as well as the accuracy of results.

Implicit two-derivative deferred correction time discretization for the discontinuous Galerkin method

In this paper, we use an implicit two-derivative deferred correction time discretization approach and combine it with a spatial discretization of the discontinuous Galerkin spectral element method to solve (non-)linear PDEs. The resulting numerical method is high order accurate in space and time. As the novel scheme handles two time derivatives, the spatial operator for both derivatives has to be defined. This results in an extended system matrix of the scheme. We analyze this matrix regarding possible simplifications and an efficient way to solve the arising (non-)linear system of equations. It is shown how a carefully designed preconditioner and a matrix-free approach allow for an efficient implementation and application of the novel scheme. For both, linear advection and the compressible Euler equations, up to eighth order of accuracy in time is shown. Finally, it is illustrated how the method can be used to approximate solutions to the compressible Navier-Stokes equations.

Suitable Solid Waste Disposal Site Selection Using Geographical Information System: A Case of Debre Markos Town, Ethiopia

In this study an effort has been made to identify suitable sites for safe disposal. For proper identification and selection of appropriate sites for solid waste disposal careful and systematic procedures need to be adopted and followed. The main objective of this research was identified the suitable solid waste disposal site by using the GIS-based approaches in Debre markos town. The present study had considered various factors like road networks; rivers, soil, slope, altitude and land use/ land cover for selecting a suitable solid waste disposal site within the study area. The relative weights of the factor were estimated using AHP and factor maps were developed by using GIS spatial operations. The final weighted factor map produced an overall solid waste disposal suitability map. The solid waste disposal site suitability map was presented in four suitability index such as highly suitable, moderately suitable, low suitable and unsuitable. The result shows that around 21% area is highly suitable for solid waste disposal site, 25% is moderately suitable, 28% of study area has low suitable and 26% area is unsuitable for solid waste disposal site.

Metamaterial-enabled arbitrary on-chip spatial mode manipulation

On-chip spatial mode operation, represented as mode-division multiplexing (MDM), can support high-capacity data communications and promise superior performance in various systems and numerous applications from optical sensing to nonlinear and quantum optics. However, the scalability of state-of-the-art mode manipulation techniques is significantly hindered not only by the particular mode-order-oriented design strategy but also by the inherent limitations of possibly achievable mode orders. Recently, metamaterials capable of providing subwavelength-scale control of optical wavefronts have emerged as an attractive alternative to manipulate guided modes with compact footprints and broadband functionalities. Herein, we propose a universal yet efficient design framework based on the topological metamaterial building block (BB), enabling the excitation of arbitrary high-order spatial modes in silicon waveguides. By simply programming the layout of multiple fully etched dielectric metamaterial perturbations with predefined mathematical formulas, arbitrary high-order mode conversion and mode exchange can be simultaneously realized with uniform and competitive performance. The extraordinary scalability of the metamaterial BB frame is experimentally benchmarked by a record high-order mode operator up to the twentieth. As a proof of conceptual application, an 8-mode MDM data transmission of 28-GBaud 16-QAM optical signals is also verified with an aggregate data rate of 813 Gb/s (7% FEC). This user-friendly metamaterial BB concept marks a quintessential breakthrough for comprehensive manipulation of spatial light on-chip by breaking the long-standing shackles on the scalability, which may open up fascinating opportunities for complex photonic functionalities previously inaccessible.

A New Spatial Criteria Method to Delimit Rural Settlements towards Boundaries Equity: Land Use Optimization for Decision Making in Galicia, NW Spain

Rural settlements (RS) are a reality of rural areas. They consist of cluster of buildings and ways of life mainly associated with activities related to agriculture. As economic policies applied in rural development have evolved, the physical delimitation of rural areas has become more important because such areas are recipients of financial support, which depends on an area’s characteristics. Thus, it is necessary to formulate a new spatial approach for RS delimitation. The objective of this study is to define spatial criteria for identification and delimitation of the RS to recognize the morphological context of each RS. With respect to methodology, RS in the community of Galicia, Northwestern Spain were studied, and factors for spatial characterization were defined according to experts’ evaluations. Subsequently, spatial restrictions and conditions were identified for the delimitation of boundaries. The criteria that this research proposes reveal numerically adjustable factors that can recognize and interpret the morphological characteristics of each RS, which is also evidenced by the results of RS delimitations. It can be concluded that the numerically defined criteria associated with a spatial operation allow the adaptation to the morphological characteristics of any RS, as well as spatial equity by recognizing the differentiation of building structures and land uses of each RS, rather unlike the criteria defined by the law.

Optimized explicit Runge–Kutta schemes for high-order collocated discontinuous Galerkin methods for compressible fluid dynamics

In compressible computational fluid dynamics, the step size of explicit time integration schemes is often constrained by stability when high-order accurate spatial discretizations are used. We report a set of new optimized explicit Runge–Kutta schemes for the integration of systems of ordinary differential equations arising from the spatial discretization of wave propagation problems with high-order entropy stable collocated discontinuous Galerkin methods. The eigenvalues of the discrete spatial operator for the advection equation and the propagation of an isentropic vortex with the compressible Euler equations for various values of the problems' parameters are used to optimize the stability region of the proposed time integration schemes. To demonstrate the efficiency and the robustness of the methods, we solve the compressible turbulent flow past the Valeo controlled-diffusion airfoil and a delta wing at a Reynolds number of 8.3×105 and 106, respectively. A thorough analysis of the performance of the two families of optimized schemes revealed that methods generated using the spectra of the vortex problem are 6-to-20% faster than methods constructed using the spectra of the advection equation. Compared to widely used explicit Runge–Kutta schemes, the methods designed using the spectra of the vortex problem yield a time-to-solution saving of approximately 6-to-38%. For large-scale time-dependent partial differential equations computations, these gains mean saving hundreds of thousands if not millions of core hours. In addition, the new methods can be effectively and efficiently applied to integrate systems of ordinary differential equations arising from a wide range of spatial discretization, including discontinuous Galerkin spectral element methods, spectral difference methods, and flux reconstruction methods.

Spatial Concept Query Based on Lattice-Tree

As a basic method of spatial data operation, spatial keyword query can provide meaningful information to meet user demands by searching spatial textual datasets. How to accurately understand users’ intentions and efficiently retrieve results from spatial textual big data are always the focus of research. Spatial textual big data and their complex correlation between textual features not only enrich the connotation of spatial objects but also bring difficulties to the efficient recognition and retrieval of similar spatial objects. Because there are a lot of many-to-many relationships between massive spatial objects and textual features, most of the existing research results that employ tree-like and table-like structures to index spatial data and textual data are inefficient in retrieving similar spatial objects. In this paper, firstly, we define spatial textual concept (STC) as a group of spatial objects with the same textual keywords in a limited spatial region in order to present the many-to-many relationships between spatial objects and textual features. Then we attempt to introduce the concept lattice model to maintain a group of related STCs and propose a hybrid tree-like spatial index structure, the lattice-tree, for spatial textual big data. Lattice-tree employs R-tree to index the spatial location of objects, and it embeds a concept lattice structure into specific tree nodes to organize the STC set from a large number of textual keywords of objects and their relationships. Based on this, we also propose a novel spatial keyword query, named Top-k spatial concept query (TkSCQ), to answer STC and retrieve similar spatial objects with multiple textual features. The empirical study is carried out on two spatial textual big data sets from Yelp and Amap. Experiments on the lattice-tree verify its feasibility and demonstrate that it is efficient to embed the concept lattice structure into tree nodes of 3 to 5 levels. Experiments on TkSCQ evaluate lattice from results, keywords, data volume, and so on, and two baseline index structures based on IR-tree and Fp-tree, named the inverted-tree and Fpindex-tree, are developed to compare with the lattice-tree on data sets from Yelp and Amap. Experimental results demonstrate that the Lattice-tree has the better retrieval efficiency in most cases, especially in the case of large amounts of data queries, where the retrieval performance of the lattice-tree is much better than the inverted-tree and Fpindex-tree.