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Spatial Information Systems (SIS) COMP 30110 Spatial access methods: Indexing

Spatial Information Systems (SIS) COMP 30110 Spatial access methods: Indexing. In previous lectures. Oracle Spatial Object-relational model SDO_Geometry data type Spatial functionality and Query model. Today. Indexing Spatial Data Indexing mechanisms in Oracle Spatial R-trees.

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Spatial Information Systems (SIS) COMP 30110 Spatial access methods: Indexing

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  1. Spatial Information Systems (SIS) COMP 30110 Spatial access methods: Indexing

  2. In previous lectures • Oracle Spatial • Object-relational model • SDO_Geometry data type • Spatial functionality and Query model Today • Indexing Spatial Data • Indexing mechanisms in Oracle Spatial • R-trees

  3. Spatial Indexes Used to speed up spatial queries Example: Point query: return the geometric object that contains a given query point Sequentially scanning all objects of a large collection to check whether they contain the query point involves a high number of disk accesses and the repetition of the evaluation of computationally expensive geometric predicates (e.g., containment, intersection, etc.) Reducing the set of objects to be processed is highly desirable

  4. Indexes for object-based and space-based representations Indexes for raster data: based on recursive subdivision of the space Example: quadtrees Indexes for vector data: differ depending on the type of data (extensions of quadtrees are used also for vector data)

  5. Vector Data Indexing • Different indexing methods are used for point, linear and polygonal data • In the case of collections of polygons, instead of indexing the object geometries themselves, whose shapes might be complex, we consider an approximation of the geometry and index it instead • Most commonly used approximation: minimum bounding rectangle (MBR) also called minimum bounding box (MBB)

  6. MBRs • By using the MBR as the geometric key for building the spatial index, we save the cost of evaluating expensive geometric predicates during index traversal (as geometric tests againsts an MBR is constant) • Example: point-in-polygon test • In addition, the space required to store a rectangle is constant (2 points) (x,y) (x,y)

  7. MBRs (cont.d) • An operation involving a spatial predicate on a collection of objects indexed on their MBRs is performed in two steps: • Filter step: selects the objects whose MBR satisfies the spatial predicate (by traversing the spatial index and applying the predicate to the MBRs) • Refinement step: the objects that pass the filter step are a superset of the solution. An MBR might satisfy the predicate but the corresponding object might not P MBR obj

  8. Refinement step Refinement step: the objects that pass the filter step are a superset of the solution. An MBR might satisfy the predicate but the corresponding object might not Therefore, in this step the spatial predicate is applied to the actual geometry of the object P MBR obj

  9. Spatial LayerData Secondary Filter Spatial Functions Primary Filter Spatial Index Reduced Data Set Table where coordinates are stored Index retrieves area of interest Procedures that determine exact relationship Oracle Spatial Query Model Exact Result Set

  10. Oracle Spatial Indexing Methods Two types of indexes are implemented in Oracle Spatial: • R-trees • Quadtrees

  11. R-trees Based on MBRs (minimum bounding rectangles) Defined for indexing 2D objects (can be extended to higher dimensions but implemented only for 2D in Oracle Spatial) MBRs of geometric objects form the leaves of the index tree Multiple MBRs are grouped into larger rectangles (MBRs) to form intermediate nodes in the tree Repeat until one rectangle is left that contains everything

  12. a root R b R S d a b c d S c root 1 2 3 4 5 6 7 8 9 Pointers to geometries R-trees R-tree 1 2 3 4 8 6 5 9 7

  13. Remark: nodes • Intermediate nodes store: • MBRs of collections of objects • Leaf nodes store: • MBRs of individual objects • Pointers to storage location of the exact geometry

  14. Building R-trees An R-tree is a depth-balanced tree in which each node corresponds to a disk page (i.e., the number of entries in each node is limited) The structure satisfies the following properties: • For all nodes in the tree (except the root) the number of entries is between m and M • The root has at least two children (unless it is a leaf) • All leaves are at the same level

  15. a root R b R S d a b c d S c root 1 2 3 4 5 6 7 8 9 Pointers to geometries Example (1) R-tree 1 2 3 4 8 6 5 9 7 m = 2; M = 3

  16. root R2 R3 R1 ….. ….. ….. Example (2) R-tree m = 2; M = 4

  17. Searching R-trees • We consider two types of queries: • point query: “what object contains the query point” • window query: “what objects intersect the query window”

  18. O Basic spatial queries (1) P Containment Query: Given a spatial object O, find all objects in the collection that completely contain O. When O is a point, the query is called Point Query Containment Query Point Query (also Point-in-polygon, or Point Location)

  19. Basic spatial queries (2) Region Query: Given a region R, find all objects in the collection that intersect R. When R is a rectangle, the query is called Window Query R R Region Query Window Query

  20. Searching R-trees: window query • Compare search window with MBRs stored at each node • starting at root node • Stop at leaf nodes • compare contained geometries with search window

  21. a root R b R S d a b c d S c root 1 2 3 4 8 9 Pointers to geometries Searching R-trees: window query Example: R-tree 1 2 3 4 8 6 5 9 7

  22. Example: remarks If no MBRs are used: check the query window against all geometries for intersection (computationally expensive) In some cases, using R-trees to structure the set of MBRs can cause more tests (against MBRs) to be done. In general, this is not the case

  23. Searching R-trees: point query Test query point for inclusion in MBRs stored at each node • starting at root node • Stop at leaf nodes • Test query point for inclusion in exact geometries

  24. a root P R b R S d a b c d S c root 1 2 3 4 5 6 7 8 9 Pointers to geometries Exercise: point query R-tree 1 2 3 4 8 6 5 9 7

  25. a root P R b R S a b S root 3 4 Pointers to geometries Searching R-trees: point query Example: R-tree 1 2 3 4 8 6 5 9 7

  26. Summary • Indexing Vector Spatial Data • R-trees: • Based on MBRs (leaves) • Root: whole dataset • Intermediate nodes: groups of MBRs (objects) – not a partition of the underlying space!

  27. Important remarks • Note that the MBRs (at all levels) can overlap • A rectangle is stored as child of a bigger rectangle only if completely contained in it • Example: • Next lecture: Quadtrees for vector data (seen for raster)

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