Spatial Indexing of Global Geographical Data with HTM

Transcription

1 Spatial Indexing of Global Geographical Data ith HTM Zhenhua Lv Yingjie Hu Haidong Zhong Bailang Yu Jianping Wu* Key Laboratory of Geographic Information Science Ministry of Education East China Normal University Shanghai China *Corresponding author: Bo Li Hui Zhao Institute of Softare Engineering East China Normal University Shanghai China Abstract Spatial indexing is one of the most important techniques in the field of spatial data management. Many kinds of techniques of spatial indexing have been successfully developed and each of them has advantages toards special applications. As a type of spatial data structure Hierarchical Triangular Mesh (HTM) has excellent features of global continuity stability hierarchy and uniformity hich has attracted much interest of researchers for many years. This paper investigates the method that using HTM as indexing for global geographical data (only point-like objects no). The HTM is defined by subdividing a unit sphere recursively and the basic elements in it are spherical triangles that are coded as integers called HTM codes in the computer system. At the global scale all the regions on the sphere are spherical hich can be intersected ith HTM elements obeying some equations. The spatial position of each input object can also be represented by a HTM code. HTM codes thus become the bridge beteen query regions and input objects. Our system is based on the combination of database management system (DBMS) and distributed file system. The major information of input files is extracted as metadata that are stored on tables of DBMS hile the original files are stored on the distributed file system (called HDFS) hich has potential abilities to support parallel processing. Millions of point-like objects on the global ere examined and the experiments indicated the system ere acceptable. Keyords-spatial indexing; hierarchical triangular mesh; spherical triangle; HDFS I. INTRODUCTION With the development of spatial information science more and more spatial data need to be stored and processed immediately. Managing massive spatial data on global scale requires efficient schemes of spatial indexing. Hierarchical Triangular Mesh (HTM) model [] is a kind of data structure hich has excellent features of global continuity stability hierarchy and uniformity. The elements of HTM are spherical triangles on the sphere and each of them can be uniquely assigned an integral code that has the information of both geographical position and spatial resolution. Goodchild [ 3] and Song et al [4] created the Discrete Global Grid ith precisely equal areas. Szalay et al [] and Gray [5] made use of HTM as spatial indexing by spherical partitioning mapping onto B-Tree index in SQL Server. In this paper e folloed the theory of HTM from them but took a different approach to implement the system architecture. The major differences lie in that our system is based on the combination of database management system (DBMS) and distributed file system. And metadata are used to describe the dataset and are stored on tables of DBMS. We adopt this scheme on the follo factors. First most types of input data in our system are nonstructural indicating that they are not greatly suitable for DBMS especially relational database [6]. Metadata that extracted from input datasets contain the primary information hich is structural and lighteight; hence they can be easily managed in tables of DBMS. This ill lighten the burden of DBMS and elevate the system efficiency if ra dataset and metadata store in separate. Second extracting metadata from large amounts of dataset is time-consuming because it depends on a great deal of IO operations that slo don the processing. If the IO operations execute in parallel at different machines the efficiency raises dramatically. Besides the distributed file system provides the considerable capacity in storage and high stability in fault tolerance. Moreover fast storage and access to data are just basic requirements for spatial data hile distributed processing is another advanced topic. Hence it is important to devise a spatial data system considering the easy support of distributed processing. Metadata in our study are managed by MySQL hile ra dataset are stored on Hadoop Distributed File System (HDFS) that is a part of Hadoop frameork. Hadoop is an open source system that hosted by Apache Softare Foundation [7]. It is a kind of implementation of MapReduce programming model that is firstly proposed by Google Inc and has been successfully used for many different applications [8]. This model is easy to use even for programmers ithout experience ith parallel and distributed systems and it hides the details of parallelization fault-tolerance locality optimization and load balancing thus making users focus on applications themselves. The rest of this paper is organized as follo. We briefly introduce the theories of HTM model in section to including ho to create the model the schemes of coding and the transformations of coordinates. Spherical areas are defined at the beginning of the section three. And the algorithms of intersecting ith spherical areas are then elaborated folloed

2 by the discussions of optimization and limitation. In section four e talk about the implementation of the system and analyze the experiment s results. Conclusions and further ork are listed in the end. II. HIERARCHICAL TRIANGULAR MESH MODEL A discrete global grid system (DGGS) is a hierarchical structure in hich the earth surface is subdivided into a series of discrete grids and the grids represents spatial regions or points [9]. Generally a DGGS can be designed and developed in five explicit steps [] including a base polyhedron a fixed inscribed orientation in the coordinate system a transformation relationship beteen the polyhedron and the earth surface a hierarchical partition method and an efficient indexing method for the hierarchical facets []. A. Definition HTM is originated from an octahedron that inscribed in a unit sphere. Both the octahedron and the unit sphere are placed in the Cartesian coordinate system. The six initial vertices of the octahedron have the follo coordinates represented by vectors as in (). ( ( ( ( ( ( ) : = v ) : = v ) : = v ) : = v ) : = v 3 ) : = v The globe is represented by a unit sphere located in the Cartesian coordinate system like the octahedron; the relations beteen them are shon in the Figure. 4 5 () The XZ plane represents the Prime (Greenich) Meridian having the value of longitude zero. The north and south poles (9 and -9 ) are v = ( ) and v = ( -). Consequently the relations beteen geographical positions and Cartesian coordinates are established by (): x = cos( lat) cos( lon) y = cos( lat)sin( lon) z = sin( lat) The hierarchical subdivisions of the sphere start ith eight spherical triangles that are defined by projecting octahedron onto the sphere. A spherical triangle is given by three points on the unit sphere connected by great circle segments hich means each edge of a spherical triangle is a segment of great circle and its midpoint is also located on the sphere hich can be expressed by (3). v + v = v + v v + v = v + v v + v = v + v () (3) here (v v v ) are three vertices of the spherical triangle and ( ) are the midpoints of their edges. The vertices of the initial octahedron are connected to their midpoints by the great circle to get the next level of subdivision and then each spherical triangle generates four small spherical triangles. The ne children are given by (4): Triangle : = ( v Triangle: = ( v Triangle : = ( v Triangle3 : = ( ) ) ) ) (4) This process is then repeated as many times as needed until an appropriate spatial resolution reached. The number of triangles multiplies by four each time. The octahedron s edges all span 9 and the angle that a spherical triangle spans ill be recursively cut in half in the further subdivisions. Finally the hierarchical triangle meshes are formed. Figure. The globe s orientations in the three-dimensional cartesian coordinate [5]. After placing the sphere into the three-dimensional Cartesian coordinate system every point on the sphere ith lon-lat values can be represented by a unit vector v = (x y z). B. HTM Coding and Transformation As mentioned previously each spherical triangle ill generate four children in subdivisions. There are many schemes coding the children but here they are labeled and 3 by keeping order counter-clockise. The initial eight triangles also have fixed codes. Each triangle has a unique code

3 called HTM ID hich is given by concatenating its order to parent ID. The HTM ID uniquely defines its depth (spatial resolution) and its location on the sphere thus it can be used identifying the spatial objects on the earth. Most geospatial data have geographical positions represented by longitude and latitude pairs or they can easily be transformed into that forms. Thus relationships beteen geo-positions and HTM IDs are necessary before associating HTM IDs ith geo-objects. Since the geographical coordinates can be directly transformed into Cartesian coordinates by the equations () calculating HTM IDs from geographical coordinates is equal to that calculating from Cartesian coordinates. TABLE I. TRANSFORMING CARTESIAN COORDINATES TO HTM IDS of HTM IDs it becomes the simple selections of integers in the database. A. Geometry Pimitives Halfspace A halfspace defines a cap on the unit sphere that is sliced by a plane and it has the form (5): h : = { v; d} v = d (5) here v is the normal vector of the cutting plane from the origin pointing into the halfspace. Scalar d is the distance from the origin to the plane along the normal vector v. Figure. illustrates a halfspace. Algorithm: xyztohtmid Input: (x y z depth) here x y z are the 3-d Cartesian coordinates of input point p and depth indicates the depth of subdivision. Output: (htmid). Decide hich face of the initial octahedron that the input point p locates at and then get initial id and vertices of selected face. They are marked as topid and v v and v. The result htmid is initialized by topid.. Decrease the depth. If the value of depth is still greater than zero then continue the process; otherise the process is finished. 3. Generate four ne children of the triangle that contains p. They are marked as t t t and t If one of the four triangles t i contains p the number i is appended to topid. 5. Repeat the steps to through four until it is finished. At a fixed level of subdivision a lon-lat point can be represented by a spherical triangle that identified by HTM ID. Since the point is zero in area but the spherical triangle is not zero the deviations are qualified by the areas of this spherical triangle depending on the depths of subdivisions. The resolutions of HTM ill approximate to one meter if the value of depth is tenty one [9]. Up to here all the point-like objects on the sphere can be associated ith HTM IDs that specify there geographical positions. Hoever the HTM ID must support intersections ith arbitrary spherical areas to perform the functions of query. III. SPHERICAL AREAS AND QUERY There are three types of geometry areas. They are halfspace convex and region. Halfspace is the basic building block of convexes and regions. A convex is defined by the intersection of some halfspaces and a region is in turn defined by the union of some convexes. Every shape on the sphere can be represented by regions. Querying by regions is that selecting all objects of hich geographical positions are inside in the regions. In other ords e need select all HTM IDs that are inside regions. Since the query regions are composed of a list Figure. A typical example of halfspace[5]. The center of Shanghai in China is located at the point ( ) hich can be expressed as (6). v = ( ) (6) The query circle around Shanghai of hich angle is one degree can be expressed as (7). h = ( ) (7) The radius of this circle is sixty nautical miles (one degree is equal to sixty arc minutes) hich approximates to. kilometers. Convex A convex is the intersection of some halfspace i.e. the intersection of caps on the unit sphere. It can be defined as (8). c = h & h & & h n n N : (8) Most simple shapes can be directly represented by convexes like spherical rectangles and polygons. Region A region is a union of a number of convexes hich can represents any area on the sphere. It can be defined as (9). +

4 r = c c c n n N : (9) B. Intersecting ith HTM Triangles Given a region on the unit sphere e compute a list of triangles (HTM IDs) that cover the region. Considering the fact that a region is composed of some halfspaces e should first find out the relationships beteen halfspaces and spherical triangles of HTM. Halfspace defines a cap by {v h d} hile a spherical triangle is identified by three corner points i.e. three unit vector v v v 3. If all three corner points are inside the halfspace then the spherical triangle is fully inside it. Otherise the spherical triangle is outside or partially inside the halfspace. Any corner point hich is inside a halfspace of the triangle satisfies (). v + v d () h i > Obviously if one or to corner points of the spherical triangle are contained by the halfspace this is the case of partial intersection. If the spherical triangle has not corner point intersecting ith the halfspace it may be outside the halfspace or contain the halfspace depending on the further decisions. If one of the spherical sides intersects ith the halfspace this is the case of partial intersection. Test the value of (). ( v v ) v < ( i j) { ();(3);(3) } i j h () If none of (i j) has the value true then the triangle (v v v 3 ) embraces the halfspace otherise it is outside the halfspace. Intersecting of a convex ith HTM triangles is not as easy as the halfspace because the convex is composed of some halfspaces and may have holes hich complicates the process of decisions. Nevertheless intersections of halfspaces are still basic computations. Intersecting ith convexes or regions is inevitably based on the computations of halfspaces. If e need a list of HTM triangles to cover a halfspace at a fixed depth the process starts intersecting from the initial eight spherical faces of the octahedron to their descendants until the specified depth is reached. The HTM IDs of triangles that are fully inside the halfspace are accepted by the list. Triangles that are partially intersected ith the halfspace ill be further examined by subdivisions if the depth is alloed. Thus given query region a list of HTM IDs that cover it can be computed. C. Optimization There are several simplifications of computing triangles list for regions including removing the duplicates of halfspaces identifying the complements and nulls dropping the halfspaces that cover the hole sphere. In addition to reduce the length of the list each item in the list specifies a range of triangles' value instead of listing each one. For more details please reference the paper []. D. Limitation The algorithms of HTM model have by no only supported point-like objects of input data hich means that the areas of objects on the sphere are seemed as zeros. The types of query triggered by users are thus limited as hich objects are nearby a given point or hich objects are contained in a region. If the objects of input data have the polygon-like shapes e choose any points that are inside the objects to represent them hich ill inevitably incur deviations. IV. SYSTEM IMPLEMENTATION Gray [5] has implemented a database system that supports spatial queries based on SQL Server 5. Different ith them in this paper e have re-implemented HTM library by Java and deployed it on the distributed system. In general there are three types of architectures managing massive data by using file system DBMS and the combination of the both. We have adopted the latter on to factors. For one thing it is not easy for users to manage massive data only ith file system because there is much ork needs to be done by the users. These include many techniques such as searching indexing scheduling etc hich are difficult but have existed in the database management system for many years. For another the DBMS usually demands strict schemas of data and it is sloer to insert raster data to tables (especially ith large size) than that to store them on the file system directly. Instead the metadata of input files ith features of strict structure light data size and poerful description are more suitable to be stored on tables. Additionally extracting metadata from large amounts of input files ill cost much time. This is caused by lots of IO operations. But it ill be greatly alleviated if the timeconsuming operations are distributed on several parallel machines. The file system used here is Hadoop Distributed File System (HDFS) hich is a basic block of Hadoop frameork. Hadoop is an open source system that supports the MapReduce model hich as introduced by Google Inc as a method of solving huge scale problems ith large clusters. In this model applications are based on to distinct steps that are map and reduce operations. Input files are automatically split into logical chunks and each chunk ill be processed independently []. HDFS provides high reliability and fault tolerance. Input data are mirrored to multiple storage nodes and this technique is called replication. As long as one replica of data chunk is available the user ill not kno of storage nodes failures. A. Logical Architecture The system is composed of ra files metadata DBMS Hadoop HTM Library and Query Application. The logical vie of our system is shon in Figure 3. Metadata contain the primary and frequently-used information. They may be defined at least by keys that identify the objects geographical positions HTM IDs that transformed from geographical positions file paths in the form of HDFS s schema. The file paths tell the system here the original files are. This is necessary hen the users ant to access after getting metadata from the database.

5 The HTM Library as re-implemented by Java because the old one as ritten by CSharp language hich as not compatible ith Hadoop. Although long integers ith sixty four bits in the computer can hold HTM IDs of hich depths are thirty e only used the depth tenty one by considering the complexities of computation and the requirements in the real orld. The spatial resolutions on the sphere at this depth approximate to one meter. Tables in MySQL should at least have five columns. They are key longitude latitude path and htmid hich are the same as the structure of metadata. B-Tree index created for the column of htmid is necessary. TABLE II. HARDWARE CONFIGURATION Name Number Details ControlNode CPU 8.GHz 4GB memory ComputingNode 8 CPU 8.GHz 8GB memory Netork -- All nodes are connected by a gigabit sitch StorageDisk 3 75G SATA 3 TABLE III. SOFTWARE CONFIGURATION Name Version Details Hadoop.9 Installed on each the computer. The control node of the cluster as specified as NameNode of Hadoop hile other computing nodes ere used as DataNodes. MySQL 5. Installed on the control node. HTMLibrary -- Implemented in Java. QueryApp -- Implemented in Java. RedHat Linux Enterprise 4 AS Installed on each computer. Figure 3. A logical vie of the system The processes are divided into the folloing five steps. Step. Copy the original files to HDFS. This can be done by shell commands or by the use of its interface functions. This step may cost much time but this runs only once. Step 3. Extract metadata from the input files that stored on HDFS. There are several map tasks in a Hadoop program. A map task includes three functions such as configure map and close. In our system the configure function as responsible for initializing the database connections hile the map function inserted the metadata to the database. The close function cleared up the connections. Generating HTM IDs as also finished in the map function. Step 4 5. Users applications requested objects from the database. When the records returned the applications accessed the ra files thro the paths. B. Experiments The environments of our experiments are composed of hardare and softare. They are listed by TABLE II. and TABLE III. separately. The datasets e used ere point-like objects that uniformly distributed on the sphere hich ere generated by a computer automatically. Each object as contained in a grid hich as. in longitude and. in latitude. The number of the objects approximated to sixty five millions but they ere small in size. All the objects ere stored by tenty hundreds text files in hich each line represented an object. Each text file thus contained about fifty thousands of objects. It cost about seven hours to put all the objects into the database if there as only one map task used. In this case the insertions ran sequentially and only one database connection orked hich as the best efficiency of the database. With the increase in the number of map tasks the time spent on IO operations as distributed on different nodes. This decreased the cost of IO operations but brought the concurrencies to the database. Hence there must be a tradeoff beteen the number of maps and the burden of database. The best result in the experiments as four hours to put all the objects into the database by tuning the number of maps. The number as sixteen then. But this may vary a lot according to the different datasets especially the differences in file types. The HTM IDs ere organized by the B-Tree indexes through hich the searching as very fast. Moreover MySQL provided some techniques of caching and it thus greatly enhanced the efficiencies of the similar searches. To avoid the effects of caching the experiments ere performed on the initial state of the database. This can be done by restarting the instance of MySQL before each exercise to ensure there is no cached data available. The orst result of searching by a circular region of hich radius as ten nautical miles as only about seven hundreds milliseconds and about three hundreds objects returned. The result increased to five seconds hen the radius of the circular query region as one hundred nautical miles and more than tenty thousand objects returned. These to results became to zero and less than a half of one second hen the cache as used. V. CONCLUSION Hierarchical Triangular Mesh (HTM) has excellent features of global continuity stability hierarchy and uniformity hich has abilities to index the spatial objects on the sphere. Each element of HTM has a unique and poerful ID that contains

6 the information of both position and resolution. By defining the spherical regions on the surface the relationships beteen the regions and HTM s elements are established. The most important characteristics of the HTM ID are that it is onedimensional and very suitable for B-Tree index. The amount of datasets in the experiment is small as the types of the objects are texts. But it may multiply by at least one hundred thousand if the input objects contain the raster images because it is common for the remote sensing images that are more than one hundred thousand in size. To be a useful system there are many things to do in the future. Firstly the system should support polygon-like objects. Associated algorithms for intersecting ith spherical polygons are developing no. Secondly there are fe file types that are supported ell on the HDFS. Many kinds of files are incompatible ith this file system. It is urgent for us to extend the functions of HDFS to support some common types such as TIFF JPG Hierarchical Data Format (HDF) [3] etc. Although the HDF Group provided some tools they are not suitable for accessing in parallel. Additionally a nice and friendly user interface is necessary for the system. REFERENCES [] A. S. Szalay J. Gray et al Indexing the sphere ith the Hierarchical Triangular Mesh Microsoft Technical Report MSR-TR [] M. F. Goodchild Y. Shiren and G. Dutton Spatial data representation and basic operations on triangular hierarchical data structure National Center for Geographic Information and Analysis Santa Barbara Technical Report [3] M. F. Goodchild and Y. Shiren A hierarchical data structure for global geographic information systems CVGIP: Graphical Models and Image Processing vol. 54 pp [4] L. Song A. J. Kimerling and K. Sahr Developing an equal area global grid by small circle subdivision Proc. International Conference on Discrete Global Grids Santa BarbaraCA March 6-8. [5] J. Gray A. S. Szalay and G. Fekete Using table valued functions in SQL Server 5 to implement a spatial data library Microsoft Technical Report MSR-TR-5-5. [6] A. Pavlo E. Paulson et al A comparison of approaches to large-scale data analysis Proceedings of the 35th SIGMOD international conference on Management of data. Providence Rhode Island USA ACM: [7] Hadoop Website [8] J. Dean and S. Ghemaat Mapreduce: simplified data processing on large clusters Communications of the Acm vol.5 pp [9] G. Dutton Planetary modelling via hierarchical tessellation Proceedings of Ninth International Symposium on Automated Cartography (AutoCarto 9) Bethesda MDUSA pp [] K. Sahr D. White and A. J. Kimerling Geodesic discrete global grid Systems Cartography and Geographic Information Science vol. 3 () pp [] W. Yuan C. Q. Cheng A. N. MA and X. J. Guan L curve for spherical triangle region quadtrees Science in China Series E- Engineering and Materials Science vol 47 pp [] J. Venner Pro Hadoop. USA: Apress 9. [3] Hierarchical Data Format Website