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 nodes
  entries.

7
R-Tree A 2-D (n2) 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 nodes
  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 nodes
  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 nodes
  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 nodes
  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 nodes
  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 lt 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
Searching

• 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.

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
Insertion ChooseLeaf

• 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.

17
SplitNode Node Splitting

• A full node contains M entries. Divide the
  collection of M1 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.

Total area is larger!
18
SplitNode Node Splitting

• A full node contains M entries. Divide the
  collection of M1 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.

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
Quadratic-Cost algorithm

• A heuristic to find a small-area split.
• Cost is quadratic in M and linear in the number
  of dimensions.
• Pick two of the M1 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.

23
Quadratic-Cost algorithm

• A heuristic to find a small-area split.
• Cost is quadratic in M and linear in the number
  of dimensions.
• Pick two of the M1 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.

24
Quadratic-Cost algorithm

• A heuristic to find a small-area split.
• Cost is quadratic in M and linear in the number
  of dimensions.
• Pick two of the M1 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.

25
Linear Cost Algorithm

• Identical to Quadratic with the following
  differences
• Uses a different version of PickSeeds.
• PickNext simply chooses any of the remaining
  entries.

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 objects 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.