R-TREES: A Dynamic Index Structure for Spatial Searching by A. Guttman, SIGMOD 1984.

1. R-TREES: A Dynamic Index Structure for Spatial Searchingby A. Guttman, SIGMOD 1984. Shahram Ghandeharizadeh Computer Science Department University of Southern California

2. Motivating Example • Type in your street address in Google

3. Example (Cont…) • Show me all the pizza places close by:

4. Terminology • Example query is termed a spatial query. • R-tree is a spatial index structure. • K-D-B trees are useful for point data only. • Exact-point lookup! • Show me the USC Salvatory Computer Science building. • R-tree represents data objects in intervals in several dimensions. • Exact-point and range lookups! • Show me all Pizza places in a 2 mile radius of USC Salvatory Computer Science building. • R-tree is: • A height-balanced tree similar to B-tree with index records in its leaf nodes containing pointers to data objects. • A node is a disk page. • Assumes each tuple has a unique identifier, RID.

5. R-Tree: Leaf Nodes • Leaf nodes contain index records: • (I, tuple-identifier) • tuple-identifier is RID, • I is an n-dimensional rectangle that bounds the indexed spatial object • I = (I0, I1, …, In-1) where n is the number of dimensions. • Ii is a closed bounded interval [a,b] describing the extent of the object along dimension i. • Values for a and b might be infinity, indicating an unbounded object along dimension i.

6. R-Tree: Non-leaf nodes • Non-leaf nodes contain entries of the form: • (I, child-pointer) • Child-pointer is the address of a lower node in the R-Tree. • I covers all rectangles in the lower node’s entries.

7. R-Tree: A 2-D (n=2) Example

8. R-Tree: Non-leaf nodes • Non-leaf nodes contain entries of the form: • (I, child-pointer) • Child-pointer is the address of a lower node in the R-Tree. • I covers all rectangles in the lower node’s entries. • Questions?

9. R-Tree: Non-leaf nodes • Non-leaf nodes contain entries of the form: • (I, child-pointer) • Child-pointer is the address of a lower node in the R-Tree. • I covers all rectangles in the lower node’s entries. • Questions? What is this?

10. R-Tree: Non-leaf nodes • Non-leaf nodes contain entries of the form: • (I, child-pointer) • Child-pointer is the address of a lower node in the R-Tree. • I covers all rectangles in the lower node’s entries. • Questions? Disk Page address!

11. R-Tree: Non-leaf nodes • Non-leaf nodes contain entries of the form: • (I, child-pointer) • Child-pointer is the address of a lower node in the R-Tree. • I covers all rectangles in the lower node’s entries. • Questions? How about this? What is it?

12. R-Tree: Non-leaf nodes • Non-leaf nodes contain entries of the form: • (I, child-pointer) • Child-pointer is the address of a lower node in the R-Tree. • I covers all rectangles in the lower node’s entries. • Questions? An n dimensional rectangle: I = (I0, I1, …, In-1)

13. R-tree: Properties • Assume: • M = Maximum number of entries in a node. • m <= M/2 • N = Number of records • R-tree has the following properties: • Every leaf node contains between m and M index records. Root node is the exception. • For each index record (I, tuple-identifier) in a leaf node, I is the smallest rectangle that spatially contains the n dimensional data object represented in the indicated tuple. • Every non-leaf node has between m and M children. Root node is the exception. • For each entry (I, child-pointer) in a non-leaf node, I is the smallest rectangle that spatially contains the rectangles in the child node. • The root node has at least two children unless it is a leaf. • All leaves appear on the same level. • Height of a tree = Ceiling(logmN)-1. • Worst case utilization for all nodes except the root is m/M.

14. Descend from root to leaf in a B+-tree manner. If multiple sub-trees contain the point of interest then follow all. Assume: EI denotes the rectangle part of an index entry E, Ep denotes the tuple-identifier or child-pointer. Search (T: Root of the R-tree, S: Search Rectangle) If T is not a leaf, check each entry E to determine whether EI overlaps S. For all overlapping entries, invoke Search(Ep, S). If T is a leaf, check all entries E to determine whether EI overlaps S. If so, E is a qualifying record. Searching

15. Insertion • Similar to B-trees, new index records are added to the leaves, nodes that overflow are split, and splits propagate up the tree. • Insert (T: Root of the R-tree, E: new index entry) • Find position for new record: Invoke ChooseLeaf to select a leaf node L in which to place E. • Add record to leaf node: If L has room for E then insert E and return. Otherwise, invoke SplitNode to obtain L and LL containing E and all the old entries of L. • Propagate changes upwards: Invoke AdjustTree on L, also passing LL if a split was performed. • Grow tree taller: If node split propagation caused the root to split, create a new root whose children are the two resulting nodes.

16. ChooseLeaf (E: new index entry) Initialize: Set N to be the root node, Leaf check: If N is a leaf, return N. Choose subtree: Let F be the entry in N whose rectangle FI needs least enlargement to include E. Resolve ties by choosing the entry with the rectangle of smallest area. Descend until a leaf is reached: Set N to be the child node pointed to by Fp and repeat from step 2. Insertion: ChooseLeaf

17. A full node contains M entries. Divide the collection of M+1 entries between 2 nodes. Objective: Make it as unlikely as possible for the resulting two new nodes to be examined on subsequent searches. Heuristic: The total area of two covering rectangles after a split should be minimized. SplitNode: Node Splitting Total area is larger!

18. A full node contains M entries. Divide the collection of M+1 entries between 2 nodes. Objective: Make it as unlikely as possible for the resulting two new nodes to be examined on subsequent searches. Heuristic: The total area of two covering rectangles after a split should be minimized. SplitNode: Node Splitting Total area is larger!

19. Node Splitting: How? • How to find the minimum area node split? • Exhaustive algorithm, • Quadratic-cost algorithm, • Linear cost algorithm.

20. Exhaustive Algorithm • Generate all possible groups and choose the best with minimum area. • Number of possibilities ~ 2 to power of M-1 • M ~ 50  Number of possibilities ~ 600 Trillion

21. Exhaustive Algorithm • Generate all possible groups and choose the best with minimum area. • Number of possibilities ~ 2 to power of M-1 • M ~ 50  Number of possibilities ~ 600 Trillion • US deficit pales!

22. A heuristic to find a small-area split. Cost is quadratic in M and linear in the number of dimensions. Pick two of the M+1 entries to be the first elements of the two new groups. Choose these in a manner to waste the most area if both were put in the same group. Assign remaining entries to groups one at a time. Quadratic-Cost algorithm

23. A heuristic to find a small-area split. Cost is quadratic in M and linear in the number of dimensions. Pick two of the M+1 entries to be the first elements of the two new groups. Choose these in a manner to waste the most area if both were put in the same group. Assign remaining entries to groups one at a time. Quadratic-Cost algorithm

24. A heuristic to find a small-area split. Cost is quadratic in M and linear in the number of dimensions. Pick two of the M+1 entries to be the first elements of the two new groups. Choose these in a manner to waste the most area if both were put in the same group. Assign remaining entries to groups one at a time. Quadratic-Cost algorithm

25. Identical to Quadratic with the following differences: Uses a different version of PickSeeds. PickNext simply chooses any of the remaining entries. Linear Cost Algorithm Linear: Choose two objects that are furthest apart. Quadratic: Choose two objects that create as much empty space as possible.

26. Comparison • Linear node-split is simple, fast, and as good as quadratic! • Quality of the splits is slightly worse!

27. Insertion • Similar to B-trees, new index records are added to the leaves, nodes that overflow are split, and splits propagate up the tree. • Insert (T: Root of the R-tree, E: new index entry) • Find position for new record: Invoke ChooseLeaf to select a leaf node L in which to place E. • Add record to leaf node: If L has room for E then insert E and return. Otherwise, invoke SplitNode to obtain L and LL containing E and all the old entries of L. • Propagate changes upwards: Invoke AdjustTree on L, also passing LL if a split was performed. • Grow tree taller: If node split propagation caused the root to split, create a new root whose children are the two resulting nodes.

28. AdjustTree • Ascend from a leaf node L to the root, adjusting covering rectangles and propagating node splits.

29. Deletes • Straightforward. The only complication is under-flows: • An under-full node can be merged with whichever sibling will have its area increased least. • Orphaned entries are inserted back into the R-Tree.

30. R-Tree

31. R-tree Variations • R+-tree enhances retrieval performance by avoiding visiting multiple paths when searching for point queries. • No overlap for minimum bounding rectangels at the same level. • Specific object’s entry might be duplicated. • Insertions might lead to a series of update operations in a chain-reaction. • Under certain circumstances, the structure may lead to a deadlock, e.g., every rectangle encloses a smaller one.

32. R*-tree [1990] • Node split is more sophisticated. • Does not obey the limitation of the number of pairs per node. • When a node overflows, p entries are extracted and reinserted in the tree (p might be 25%). • Considers minimization of: • the overlapping between minimum bounding rectangles at the same level. • the perimeter of the produced minimum bounding rectangles. • Insertion is more expensive while retrievals are faster.

33. Static R-trees • Assumes the dataset is known in advance. • Static R-trees are more efficient than dynamic ones: • Tree structure is more compact, • Contains fewer news, • Overlap between minimum bounding rectangles is reduced.

34. Summary • R-tree is a spatial index structure that provides competitive average performance. • Many different variations in the literature: • Spatio-temporal access methods, 3-d R-tree. • Historical R-trees and Time-Parameterized R-tree fo spatiotemporal applications. • Have been used to speed-up operations in OLAP applications, data warehouses and data mining.