Snehal Thakkar Spatial Data Structures Hanan Samet Computer Science Department

1. Snehal Thakkar Spatial Data Structures Hanan Samet Computer Science Department University of Maryland Snehal Thakkar

2. Spatial Data Structures • Introduction • Spatial Indexing • Region Data • Point Data • Rectangle Data • Line Data • Conclusion Snehal Thakkar

3. Introduction • Spatial Objects Points, Lines, Regions, Rectangles ….. • Spatial Indexing Unlike conventional data sort has to be on space occupied by data • Hierarchical Data Structures Based on recursive decomposition, similar to divide and conquer method Snehal Thakkar

4. Spatial Indexing • Mapping Spatial Data into Point - Same, Higher or Lower Dimension - Good storage purposes, queries like intersect - Problems with queries like nearest • Bucketing Methods - Grid file, BANG file, LSD trees, Buddy trees…. - Buckets based on not the representative point, but based on actual space. Snehal Thakkar

5. R1 R2 R3 R4 R5 R6 a b d g h c i e f R-tree • Based on Minimum Bounding Rectangle Snehal Thakkar

6. R-Trees (Continued) • Organize spatial objects into d-dimensional rectangles. • Each node in the tree corresponds to smallest d-dimensional rectangle that encloses child nodes. • If an object is spatially contained in several nodes, it is only stored in one node. • Tree parameters are adjusted so that small number of pages are visited during a spatial query • All leaf nodes appear at same level • Each leaf node is (R,O) where R is smallest rectangle containing O, e.g. R3,R4…… Snehal Thakkar

7. R-trees (Continued) • Each non-leaf node is (R,P) where R is smallest rectangle containing all child rectangles, e.g. R1,R2 • R-tree of order (m,M) means that each node in the tree has between floor M/2 and M nodes, with exception of root node. Root node has two entries unless it is a leaf node. • R-tree is not unique, rectangles depend on how objects are inserted and deleted from the tree. • Problem is that to find some object you might have to go through several rectangles or whole database. Snehal Thakkar

8. R1 R2 R3 R4 R5 R6 d g h c h i a b e i c f i R+ - Trees • Decomposition of Space into Disjoint Cells Snehal Thakkar

9. R+ Trees (Continued) • R+-tree and Cell Trees used approach of discomposing space into cells • R+-trees deals with collection of objects bounded by rectangles • Cell tree deals with collection of objects bounded by convex polyhedra • R+-trees is extension of k-d-B-tree. • Try not to overlap the rectangles. • If object is in multiple rectangles, it will appear multiple times. Snehal Thakkar

10. R+Trees(Continued) • Multiple paths to object from the root • Height of the tree is increased • Retrieval times are smaller • When summing the objects, needs eliminate duplicates • It is not possible to guarantee that all properties of B-trees is fulfilled without going through difficult insert and deletion routines. • It is data-dependent, so depending on how you insert or delete records R+-tree will be different. Snehal Thakkar

11. More Spatial Indexing • Uniform Grid - Ideal for uniformly distributed data - More data-independence then R+-trees - Space decomposed on blocks on uniform size - Higher overhead • Quadtree - Space is decomposed based on data points - Sensitive to positioning of the object - Width of the blocks is restricted to power of two - Good for Set-theory type operations, like composition of data. Snehal Thakkar

12. Region Data • Focus on Interior Representation • Represented as Image array of pixels • Runlength Code - Break array into 1*m blocks, row representation • Metal Axis Transformation (MAT) - Union of Maximal Square blocks - Blocks may overlap - Block are specified by center and radius Snehal Thakkar

13. More Region Data • Region Quadtree - Is Metal Axis Transformation - Whose blocks are required to be disjoint - To have standard sizes(squares whose sides are power of two) - To be at standard locations - Based on successive subdivision of image array into four equal size quadrants. Snehal Thakkar

14. A 1 2 3 4 5 NW SW NE SE 6 7 8 13 14 9 10 1 B C F 11 12 15 16 19 17 18 E 2 3 4 5 6 D 11 12 13 14 19 15 16 17 18 7 8 9 10 Region Quadtree Snehal Thakkar

15. Region Quadtree (Continued) • Each leaf node is either Black or White • All non-leaf nodes are Gray(Circle is previous example • You can also use it for non-binary images • Resolution of the decomposition may be governed by data or predetermined • Can be used for several object representations. Snehal Thakkar

16. Variations of Quadtree • Point Quadtree - Quadtree with rectangular quadrants - Adoption of Binary Search Tree to two dimensions or more - Useful for location based queries like where is nearest theatre from the location. - Descending the tree till you find the node for location based queries. - For nearest neighbor, search is continued in the neighborhood of the node containing object. - Feature based queries tough because index is based on spatial occupancy not on features. Snehal Thakkar

17. Variations of Quadtree • Pyramid - Exponentially tapering stack of arrays, each one quarter size of previous - Useful for feature based queries like where does wheat grow in California. - Nodes that are not at maximum level of resolution contain summary information • Octree - Three dimensional analog of quadtree - Recursively subdivide into eight octants Snehal Thakkar

18. More Variations of Quadtree • Locational Code Based Quadtree - Treats image as a collection of leaf nodes, each encoded by pair of numbers - First is base 4 number, sequence of directional codes that locates leaf from the root - Second depth at which node is found or size • DF-expression - Represents the image in form of traversal of nodes of its quadtree - Very Compact storage, each node type can be encoded with two bits. - Not easy to use when random access to nodes is required. Snehal Thakkar

19. Searching with Quadtree • Useful for performing set operations • When performing intersection, it only returns black node when both quadtrees have black nodes. • Operation is performed using three quadtrees. • Worst case scenario is sum of nodes in two quadtrees Snehal Thakkar

20. Algorithms with Quadtree • Most algorithms are preorder traversals • Execution time is linear function of number of nodes • Quadtree Complexity Theorem - Number of nodes in quadtree representation is O(p+q) for 2q*2q image with perimeter p measured in pixel width. - It also holds for more dimensions. Snehal Thakkar

21. Point Data • PR Quadtree - Regular decomposition of space into quadrants - Organized same way as the region quadtree - Leaf nodes are either empty or contain data point and its co-ordinates - A quadrant contains at most one data point - Shape of the tree is independent of the order in which points are inserted - If points are close together then decomposition can be deep - Can use quadrants with capacity c - Good for search within specified distance of given record Snehal Thakkar

22. PR-tree (Continued) (50,50) (75,75) (25,25) (75,25) (20,88) (0,100) (100,100) (88,65) (52,15) (92,1) (0,0) (100,0) Snehal Thakkar

23. Rectangle Data • Used to approximate other objects in the image and in VLSI design rule checking • If environment is static, solution is based on use of plane sweep paradigm • Any addition to database forces re-execution of algorithm on whole database Snehal Thakkar

24. Rectangle Data (Continued) • Grid File Based Approach - Each rectangle reduced to a point in higher dimension - Made up of Cartesian product of two one dimensional intervals - Each interval is represented by center and extent - Set of intervals is represented by Grid File - Grid File uses two dimensional array of grid blocks called Grid Directory Snehal Thakkar

25. Rectangle Data (Continued) • Grid File Based Approach (Continued) - Grid Directory has address of the bucket - Set of linear scales is kept in the core to access grid block in the grid directory - Guarantees access to record in two operations - First operation to access the grid block - Second operation to access the grid bucket Snehal Thakkar

26. Rectangle Data (Continued) • MX-CIF Quadtree - Based on Quadtree - Decomposition of space into rectangles - Each rectangle is associated with a quadtree node corresponding to the smallest block which contains it in its entirety - Subdivision stops when nodes block contains no rectangles or at predetermined size - Rectangles can be associated with terminal and non-terminal nodes Snehal Thakkar

27. {A,E} B {G} A C {B,C,D} D F {F} G E MX-CIF Quadtree Snehal Thakkar

28. Line Data • PM1 quadtree - Based on regular decomposition of space - Partitioning occurs as long as a block contains more than one line segment unless the line segments are incident at a vertex in the block - Vertex-based implementation - Useful because space requirements for polyhedral objects are smaller then conventional octree Snehal Thakkar

29. PM1 Quadtree(Continued) Snehal Thakkar

30. Line Data (Continued) • PMR Quadtree - Edge-based variant of PM quatree - Uses probabilistic splitting rule - Block contains variable number of line segments - Each line segment is inserted into all blocks that it intersects or occupies - If block has more line segments than permitted, it is divided into four blocks once and only once - During deletion line segment is removed from all blocks and blocks are checked for merging Snehal Thakkar

31. PMR Quadtree Snehal Thakkar

32. PMR Quadtree Snehal Thakkar

33. PMR Quadtree Snehal Thakkar

34. Conclusion • Questions ? • Comments ? • Email me at snehalth@usc.edu Snehal Thakkar