R-Trees 2-dimensional indexing structure. R-trees 2-dimensional version of the B-tree: B-tree of maximum degree 8; degree between 3 and 8 Internal nodes.

R-Trees2-dimensional indexing structure

R-trees• 2-dimensional version of the B-tree:

B-tree of maximum degree 8; degree between 3 and 8

Internal nodes with k children have k-1 split values

R-trees• Can store:

– a set of polygons (regions of a subdivision)– a set of polygonal lines (or boundaries)– a set of points– a mix of the above

• Stored objects may overlap

R-trees

• Originally by Guttman, 1984• Dozens of variations and optimizations since• Suitable for windowing, point location and

intersection queries• Heuristic structure, no order bounds ( O(..) )• Tree with higher degree: suitable for

background storage (short search paths);one node per disk block

Definition R-tree

• Every internal node contains entries (rectangle, pointer to child node)

• All leaves contain entries (rectangle, pointer to object) in database or file

• Rectangles are minimal bounding rectangles (MBR)

• The root has 2 and M entries

• All other nodes have at least m and at most M entries

• All leaves have the same depth

• m > 1 and M > 2m(e.g. m = 200; M = 1000)

Object descriptions

Grouping of objects

Windowing query: the fewer rectangles intersected, the fewer subtrees to descend into

Grouping of objects

• Objects close together in same leaves small rectangles queries descend in only few subtrees

• Group the child nodes under a parent node such that small rectangles arise

Heuristics for fast queries

• Small area of rectangles• Small perimeter of rectangles• Little overlap among rectangles• Well-filled nodes (tree less deep fewer disk

accesses on each search path)

Example R-tree

Object descriptions

point containment query

point containment query

Searching in an R-tree

• Q is query object (point, window, object)• For each rectangle R in the current node,

if Q and R intersect,– search recursively in the subtree under the pointer

at R (at an internal node)– get the object corresponding to R and test for

intersection with R (at a leaf)

Inserting in an R-tree

• Determine minimal bounding rectangle (MBR) of new object

• When not yet at a leaf (choose subtree):– determine rectangle whose area increment after

insertion of R is smallest– increase this rectangle if necessary and insert R

• At a leaf:– if there is space, insert, otherwise Split Node

Split Node

• Divide the M+1 rectangles into two groups, each with at least m and at most M rectangles

• Make a node for each group, with the rectangles and corresponding subtrees as entries

• Hang the two new nodes under the parent node in the place of the overfull node; determine the new MBRs (if the root was overfull, make a new root with two child nodes)

• If the parent has M+1 children, repeat Split Node with this parent

Split Node, example

New MBRs

Strategies for Split Node, I

• Determine R1 and R2 with largest MBR: the seeds for sets S1 and S2

• While |S1| , |S2| < M - m and not all rectangles distributed:– Take not yet distributed rectangle Rj , add to

the set whose MBR increases least

Linear R-tree of Guttman, 1984

Example Split Node I

Strategies for Split Node, II

• Determine R1 and R2 with largest area(MBR) -area(R1) - area(R2): the seeds for sets S1 and S2

• While |S1| , |S2| < M - m and not all distributed:– Determine of every not yet distributed rectangle Rj :

d1 = area increment of MBR(S1 Rj) (* w.r.t. MBR(S1) *)d2 = area increment of MBR(S2 Rj) (* w.r.t. MBR(S2) *)

– Choose Ri with maximal | d1 - d2 | ; add it to the set with smallest area increment

Quadratic R-tree of Guttman, 1984

Example Split Node, II

Strategies for Split Node, III

• Determine R1 and R2 with largest area(MBR) - area(R1) - area(R2): the seeds for sets S1 and S2 (* same as quadratic R-tree *)

• Determine axis with largest normalized separation of R1 and R2

( x-separation / x-range of MBR(R1 R2), ory-separation / y-range of MBR(R1 R2) )

• Sort rectangles according to that axis (lower left corner) and split evenly in subsets of size (M+1) / 2

Greene’s split, 1989

Example Split Node, III

Y-axis has largestnormalizedseparation

Deletion from an R-tree

• Find the leaf (node) and delete object; determine new (possibly smaller) MBR

• If the node is too empty (< m entries):– delete the node recursively at its parent– insert all entries of the deleted node into the R-tree

• Note: Insertions of entries/subtrees always occurs at the level where it came from

Insert as rectangle on middle level

Insert in a leaf

object

R*-trees

• Experimentally determined measures for choices at insertion (Choose Subtree, Split Node)

• Experimentally determined algorithms for:– Choose Subtree– Split Node

R*-trees; Choose Subtree

• At nodes directly above leaves: Choose entry (rectangle) with smallest overlap-increase

• At higher nodes: Choose entry (rectangle) with smallest area-increase (same as before)

p

kiiikk RRR

;1

)(area)(overlap R ,…, R are the entry rectangles

1 p

R*-trees; Split Node

Determine split axis:• For both the x- and the y-axis:

– sort the rectangles by smallest and largest coordinate– determine the M - 2m + 2 allowed distributions into two

groups– determine for each: the perimeter of the two MBRs– add the M - 2m + 2 perimeter lengths

• Choose the axis with smallest sum of perimeters

m mM - 2m + 1

R*-trees; Split Node

Determine split index (given the split axis):• Choose the distribution, among the M - 2m + 2,

with the smallest area of intersection of the MBRs

Nearest neighbor queries

• An R-tree can be used for nearest neighbor queries

• The idea is to perform a DFS, maintain the closest object so far and use the distance for pruning

closest object so far

queried

pruned

1

2

3

4

5

Forced reinsert

• Build R-tree by repeated insertion: first inserted rectangles are possibly badly placed

• Experiment:– make R-tree by inserting 20.000 rectangles– again, but afterwards, delete the first inserted

10.000 and insert them again!

• Search time improvement of 20-50% !

Summary R-trees

• Versatile 2-dimensional search tree (referred to as: indexing structure, or spatial index)

• Some variant used in most GIS• Well-suited for windowing, point location,

intersection, and nearest neighbor queries• Heuristic structure, no order bounds ( O(..) )• Dynamic; insertions and deletions supported• Tree with higher degree: well-suited for

background storage (short search paths)