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CSIS 7101:CSIS 7101:Spatial Data (Part 1)

The R*-tree：An Efficient and Robust Access

Method for Points and RectanglesRollo Chan

Chu Chung ManMak Wai YipVivian LeeEric Lo

Sindy ShouHugh Wang

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Spatial Access Method (SAM) Handle spatial data efficiently Query Build Index

Retrieve data item from a database system quickly

Dynamic Update Why not use B-tree?

1 dimensional Designed for multi-dimensional points

E.g. 2D for Map

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R-tree and R*-tree R-tree [Guttman84] R*-tree [Beckmann90] Height-balanced tree (Similar to B-tree) Leaf-nodes has format:

<I, tuple-identifier> I is the Minimum Bounding Rectangle of a

spatial object Tuple-identifier id to retreive the spatial

object in the database (name, address, etc)

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The Spatial Data Minimum Bounding Box

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R-tree and R*-tree properties Leaf: <I, tuple-identifier> Non-leaf:

<I’, child-pointer> I’ covers all

rectangles in the children nodes entries Parameters:

M (max no of entries per node) m (min no of entries per node) m <= M/2

Root has at least two children All leaves in same level 1 node 1 disk page (minimize no. of I/O)

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Outline Introduction

Motivation R-tree and R*-tree structure

Searching of R*-tree Construction of R*-tree Conclusions References

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Searching May search more than one sub-tree

(why?) Try to search a rectangle S Search (S):

Search from root Find all index records overlap with S If not a leaf, check overlap, if yes Search

(subTree) Else it is a leaf, check all entries in that leaf

which entries overlap with S

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Searching examples

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Spatial Data Introduction

Motivation R-tree and R*-tree structure

Searching of R*-tree Construction of R*-tree Conclusions References

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R*-tree Optimization Criteria:

Minimize the area covered by an index rectangle

Minimize overlap between bounding rectangles Minimizes the number of paths to be traversed

Minimize the margin of a directory rectangle Creates less overlap, using same amount of area Allows for better, more structured clustering

Optimize the storage utilization Nodes in tree should be filled as much as possible

Sometimes it is impossible to optimize all the above criteria at the same time!

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R*-tree Insertion To insert a new entry, you need to choose

which leaf entry to insert ChooseSubTree: Select a leaf in which to place

a new index entry E: Start from Root If non-leaf node (children are leaves), choose the

node using the following criteria:1)Least overlap enlargement2)Least area enlargement3)Smaller area

If non-leaf node (children are not leaves), use 1 and 2 Invoke ChooseSubTree recursively If leaf, return this node to be inserted

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Splitting Node How about if a new entry E going to

add to a node N which is full? Split the full node? Reinserted?

How to split?1. Determine the axis2. Distribute the entries into 2 groups along

that axis3. Distribution may not evenly distributed!

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1. Determine the axis (M+1) entries

For each axis (i.e. x and y axis) sort entries by the lower value, then by upper

value E.g. X axis, sort by lower value, then generate

M-2m+2 = 3 distributions (M=3, m=1) kst distribution: [(m-1)+k] [the rest] E.g. 2nd distribution (1-1)+2: [E1 E2] [E3 E4] 3rd distribution (1-1)+3: [E1 E2 E3] [E4]

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1. Determining split axis (cont.) Compute S sum of all margin-value of

all (1, 2… M-2m+2) distributions Margin-value = perimeters of rectangles Choose the axis with lower S E.g. the S of 6 x-axis distributions (3 for

lower value, 3 for higher-value) < that of y-axis

Return x-axis as splitting axis

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2. Distribute entries along axis How to split?

1.Determine the axis2.Distribute the entries into 2

groups along that axis3.Distribution may not evenly distributed!

Along that axis, choose the distribution (out of 3) that with minimum overlap-value

Overlap-value: area[rect(group1)] area[rect(group2)]

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Who call split? R*-tree Insertion Algorithm Insert: Add a new entry into the level

specified Begin

End

1. Find appropriate node

Invoke ChooseSubtree to find node N in which to place the new entry E.

2. Check for space in node to insert entry

If N has less entries then M, insert E.

Else3. Split or Reinsert Invoke OverflowTreatment4. Propagate changes

upward If a split was performed, propagate upward. If a split of root node

occurred, Create new root.

5. Adjust covering rectangles

Adjust all rectangles in the insertion path to be minimum bounding box.

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R*-tree Insertion (cont.) Algorithm OverflowTreatment: Determine

whether to split the current node or try reinsertion.

Begin

End

1. Check condition

If level is not root level and this is the first call of OverflowTreatment in the given level during the insertion of one data rectangle,

2. Do Reinsert Invoke ReInsertElse

3. Do Split Invoke Split

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R*-tree Insertion (cont.) Algorithm ReInsert. Begin

End

1.Compute Distance

For all M+1 entries of a node N, compute the distance between the centers of their rectangles and the center of the bounding rectangle of N.

2.Sort entries Sort entries in decreasing order of their distances computed in 2.

3.Remove entries Remove the first p entries from N and adjust bounding rectangle

4.Reinsert entries Invoke Insert starting with maximum or minimum distance as defined in 3.

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R*-tree Split Example

R-tree R*-treeQuadratic Split m = 40% m = 40%

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R*-tree Forced Reinsert:

When R*-tree node p overflows, instead of splitting p immediately, try to see if some entries in p could possibly fit better in another node

As splitting only contribute to local re-organization of the directory rectangles

Reinsert increase slightly the construction time, BUT resulting less overlap improve query response time

Remove 30% (p) yield best performance

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Performance Comparison Using forced reinsert increases storage

efficiency, decreases overlap, causes fewer spits, and makes rectangles more quadratic (square).

CPU cost is higher when implementing forced reinsert, but due to fewer splits, the increase in disk access for insertions is only 4% (remains the lowest of all R-tree variants)!

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Outline Introduction

Motivation R-tree and R*-tree structure

Searching of R*-tree Construction of R*-tree Conclusions References

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Conclusions R*-trees performs significantly better than the

other R-tree variants. It is the most robust of the trees – requires less disk

access Gain is higher for smaller rectangles because strong

utilization is more important for larger query rectangles

400% gain over Linear, 180% gain over Quadratic split in R-tree

The best storage utilization Even with forced reinsertion, insertion cost is

decreased, due to fewer splits Spatial join has the highest gain

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References Guttman ， A., “R-Trees: A Dynamic

Index Structure for Spatial Searching”, Proceedings, ACM SIGMOD, pp47-57, June 1984.

Beckmann, N., Kriegel, H.P., Schneider, R., Seeger, B. ”The R*-Tree: An Efficient and Robust Access Method for Points and Rectangles”,Proceedings, ACM SIGMOD International Conferences on Management of Data, May 23-25, 1990.