R-Trees: A Dynamic Index Structure for Spatial Data Antonin Guttman

R-Tree: Why, What … ? Why do we need R-Trees? What are R-Trees? How do I perform operations? Alternatives? Why not a B+ tree?

Properties of R-Trees Height Balanced 2 types of nodes Leaves point to disk pages Records in the leaves point to actual data objects For a max capacity of M, min occupancy should be M/2 Completely dynamic Guaranteed Fan-out of M/2 Every leaf record is a smallest bounding box. Root has at least two children

R-Trees: The Structure. Internal nodes : ( rectangle, child pointer) – N dimensional rectangle. – Pointer to all rectangles that are cointained. Leaf Nodes : (MBR, tuple-identifier) – MBR is minimum bounding rectangle – Tuple-identifier is a pointer to the data object.

R-tree of order 4

Example a b cde fghij kl mnop

a b c d m a b cde fghij kl mnop

a b c d m e f n a b cde fghij kl mnop

a b c d m e f n h g i o p a b cde fghij kl mnop

R-Trees: Operations Inserts Deletes Updates ( delete and re-insert) Queries/Searches – Names of all the roads in 1 sq km area? – Which buildings would be encountered between Roger’s Hall and Reitz Union? – Give me all rectangles that are contained in the input rectangle. – Give me all rectangles intersecting this rectangle.

Insert Similar to insertion into B+-tree but may insert into any leaf; leaf splits in case capacity exceeded. – Which leaf to insert into? (Choose Leaf) – How to split a node? (Node Split)

Insert: Choose Leaf m n op

m

n

o

Insert: Choose leaf p

Node Splitting Quadratic method – Select max area gradient in the nodes as seeds. – Start clustering from the seeds Linear method – Select seeds with max separation using max x, y – Randomly assign rectangles to seeds

Delete Search for the rectangle If the rectangle is found, remove it. If the node is deficient, – Put the remaining entries in a re-insert queue. – Adjust the parent rectangle if needed. – Continue this till you reach the root. – Re-insert in such a way that all internal nodes remain above the leaf nodes. Adjust the rectangles making them smaller. Alternative sibling combination like a B-tree. – But re-insertion shows similar performance and is simple to implement.

Performance Tests R-Trees in C under UNIX on VAX11/780 computer running on 2D data(1057) for 5 page sizes – Linear node split was better than quadratic as expected. – CPU time unchanged with page sizes, indicating that when one side became full all split algorithms simply put everything in the other side. – Delete is affected by the fill factor. – Search insensitive to the fill factor and split algorithm used. – Storage space is a function of the fill factor, page size and split algorithm – All split algorithms came in 10% of the best exhaustive search and split algorithm.

Performance: 2 nd Innings Same configuration but on various data sizes 1057, 2238, 3295 and 4559 rectangles. – Low CPU cost, close to 150 micro seconds. – Comparable performance of split algorithms – Most space was used by the leaf nodes

Conclusions from the paper. R-Tree perform well for spatial data with non zero node sizes. With smaller node structure can be used as an in-memory spatial data index. – CPU performance of in-memory R-tree index is comparable and there is no IO cost. Linear split was almost as good as others. – It was fast. – Node split quality was a bit off-target, but it did not hurt the search performance noticeably. Possible use with abstract data types and abstract indexes to streamline handling of spatial data.