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## PowerPoint Slideshow about ' R-Trees' - miette

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R-Trees

- Extension of B+-trees.
- Collection of d-dimensional rectangles.
- A point in d-dimensions is a trivial rectangle.

Non-rectangular Data

- Non-rectangular data may be represented by minimum bounding rectangles (MBRs).

Operations

- Insert
- Delete
- Find all rectangles that intersect a query rectangle.
- Good for large rectangle collections stored on disk.

R-Trees—Structure

- Data nodes (leaves) contain rectangles.
- Index nodes (non-leaves) contain MBRs for data in subtrees.
- MBR for rectangles or MBRs in a non-root node is stored in parent node.

R-Trees—Structure

- R-tree of order M.
- Each node other than the root has between m <= ceil(M/2) and M rectangles/MBRs.
- Assume m = ceil(M/2) henceforth.

- Typically, m = ceil(M/2).
- Root has between 2 and M rectangles/MBRs.
- Each index node has as many MBRs as children.
- All data nodes are at the same level.

- Each node other than the root has between m <= ceil(M/2) and M rectangles/MBRs.

Example

- R-tree of order 4.
- Each node may have up to 4 rectangles/MBRs.

Example

- Possible partitioning of our example data into 12 leaves.

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Example- Possible R-tree of order 4 with 12 leaves.

Leaves are data nodes that contain 4 input rectangles each.

a-p are MBRs

Query

- Report all rectangles that intersect a given rectangle.

Query

- Start at root and find all MBRs that overlap query.
- Search corresponding subtrees recursively.

Insert

- Similar to insertion into B+-tree but may insert into any leaf; leaf splits in case capacity exceeded.
- Which leaf to insert into?
- How to split a node?

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Insert—Leaf Selection- Follow a path from root to leaf.
- At each node move into subtree whose MBR area increases least with addition of new rectangle.

Insert—Split A Node

- Split set of M+1 rectangles/MBRs into 2 sets A and B.
- A and B each have at least m rectangles/MBRs.
- Sum of areas of MBRs of A and B is minimum.

Insert—Split A Node

- Split set of M+1 rectangles/MBRs into 2 sets A and B.
- A and B each have at least m rectangles/MBRs.
- Sum of areas of MBRs of A and B is minimum.

M = 8, m = 4

Insert—Split A Node

- Split set of M+1 rectangles/MBRs into 2 sets A and B.
- A and B each have at least m rectangles/MBRs.
- Sum of areas of MBRs of A and B is minimum.

M = 8, m = 4

m!(M+1-m)!

Insert—Split A Node- Exhaustive search for best A and B.
- Compute area(MBR(A)) + area(MBR(B)) for each possible A.
- Note—for each A, the B is unique.
- Select partition that minimizes this sum.

- When |A| = m = ceil(M/2), number of choices for A is

Impractical for large M.

Insert—Split A Node

- Grow A and B using a clustering strategy.
- Start with a seed rectangle a for A and b for B.
- Grow A and B one rectangle at a time.
- Stop when the M+1 rectangles have been partitioned into A and B.

Insert—Split A Node

- Quadratic Method—seed selection.
- Let S be the set of M+1 rectangles to be partitioned.
- Find a and b inS that maximize

area(MBR(a,b)) – area(a) – area(b)

Insert—Split A Node

- Quadratic Method—seed selection.
- Let S be the set of M+1 rectangles to be partitioned.
- Find a and b inS that maximize

area(MBR(a,b)) – area(a) – area(b)

Insert—Split A Node

- Quadratic Method—assign remaining rectangles/MBRs.
- Find an unassigned rectangle c that maximizes

|area(MBR(A,c)) – area(MBR(A))

- (area(MBR(B,c)) – area(MBR(B)))|

Insert—Split A Node

- Quadratic Method—assign remaining rectangles/MBRs.
- Find an unassigned rectangle c that maximizes

|area(MBR(A,c)) – area(MBR(A))

- (area(MBR(B,c)) – area(MBR(B)))|

Insert—Split A Node

- Quadratic Method—assign remaining rectangles/MBRs.
- Assign c to partition whose area increases least.

Insert—Split A Node

- Quadratic Method—assign remaining rectangles/MBRs.
- Continue assigning in this way until all remaining rectangles must necessarily be assigned to one of the two partitions for that partition to have m rectangles.

Insert—Split A Node

- Linear Method—seed selection.
- Choose a and b to have maximum normalized separation.

Insert—Split A Node

- Linear Method—seed selection.
- Choose a and b to have maximum normalized separation.

Separation in x-dimension

Insert—Split A Node

- Linear Method—seed selection.
- Choose a and b to have maximum normalized separation.

M = 8, m = 4

Rectangles with max x-separation

Insert—Split A Node

- Linear Method—seed selection.
- Choose a and b to have maximum normalized separation.

M = 8, m = 4

Divide by x-width to normalize

Insert—Split A Node

- Linear Method—seed selection.
- Choose a and b to have maximum normalized separation.

Separation in y-dimension

Insert—Split A Node

- Linear Method—seed selection.
- Choose a and b to have maximum normalized separation.

M = 8, m = 4

Rectangles with max y-separation

Insert—Split A Node

- Linear Method—seed selection.
- Choose a and b to have maximum normalized separation.

M = 8, m = 4

Divide by y-width to normalize

Insert—Split A Node

- Linear Method—assign remainder.
- Assign remaining rectangles in random order.
- Rectangle is assigned to partition whose MBR area increases least.
- Stop when all remaining rectangles must be assigned to one of the partitions so that the partition has its minimum required m rectangles.

M = 8, m = 4

Delete

- If leaf doesn’t become deficient, simply readjust MBRs in path from root.
- If leaf becomes deficient, get from nearest sibling (if possible) and readjust MBRs.
- Combine with sibling as in B+ tree.
- Could instead do a more global reorganization to get better R-tree.

Variants

- R*-tree
- Leaf selection and node overflows in insertion handled differently.

- Hilbert R-tree

Related Structures

- R+-tree
- Index nodes have non-overlapping rectangles.
- A data object may be represented in several data nodes.
- No upper bound on size of a data node.
- No bounds (lower/upper) on degree of an index node.

Related Structures

- Cell tree
- Combines BSP and R+-tree concepts.
- Index nodes have non-overlapping convex polyhedrons.
- No lower/upper bound on size of a data node.
- Lower bound (but not upper) on degree of an index node.

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