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An Efficient and Robust Access

Method forPoints and Rectangles

Norbert Beckmann, Hans-Peter Kriegel,

Ralf Schneider, Bernhard Seeger

Presented By :

Alok Kumar Yadav2009cs10177

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Overview Introduction

Rtree and Its Optimization

R-tree Variants R*-tree

Experiments

Conclusions

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Introduction

Spatial Access Methods (SAMs):

Approximation of complex spatial object by MBR

Pros: Complex object can be represented by limited no of bytes

Preserves most essential geometric properties i.e. location andextension

Cons: (Obviously) A lot of information is lost

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Introduction (Cont..) R-tree:

B+-tree like structure

Popular access method for rectangles

Based on the heuristic optimization ofarea of enclosingrectangles in each inner node

R*-tree:

Combined optimization: Area, Margin & Overlap Outperforms exiting R-tree variants

Efficiently supports point and spatial data

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Introduction (Cont..)

Given a city map, index all university buildings in an efficient structure for quicktopological search.

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Introduction (Cont..)

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Introduction (Cont..)

MBRof the cityneighbourhoods.

MBRof the citydefining the

overall search

region.

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R-tree B+-tree like structure

Nodes : E (cp, rectangle)

cp : Child Pointer

For leaves it is a record in database

rectangle :

MBR of all rectangle in child node For leaves it is enclosing rectangle of spatial object

Max no of elements in a node is M, min is m

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R-tree (Cont..)Structure

I(A) I(B) I(M)

I(a) I(b) I(c) I(d)

B

a

b

c

d

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R-tree (Cont..) B+-tree like structure

Nodes : E (cp, rectangle) cp :

Child Pointer For leaves it is a record in database

rectangle : MBR of all rectangle in child node

For leaves it is enclosing rectangle of spatial object

Optimization criterion: Minimization of areaof enclosingrectangles in each inner node

Allows overlapping of directory rectangles, hence cannotguarantee a single search path

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Insertion Algorithm

ChooseSubtree Algorithm

Split Algorithm

If ends in a node

filled with M entries

R-tree Variants It is dynamic, hence all optimization

have to applied during insertion

Finds most suitable subtree fornew entry

If node is filled with more then M

entries, distribute in two nodes in a

appropriate manner

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R-tree Variants Original R-tree : Guttman

Greenes R-tree

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Original R-tree: Guttman Method of optimization is minimize directory

rectangle area

Split algos: Exponential, Linear, Quadratic Exponential best but cpu cost too high

Others are approximations

Quadratic outperforms linear

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Guttmans ChooseSubtreeCS1 Set N to be the root node

CS2 If N is a leaf,

return N

else

Choose the entry in N whose rectangle needs least areaenlargement to include the new data. Resolve ties bychoosing the entry with the rectangle of smallest area

endCS3 Set N to be the childnode pointed to by the childpointer of

the chosen entry. Repeat from CS2

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Guttmans Split AlgorithmQuadratic Split[Divide a set of M+1 index entries into two groups]

QS1 Invoke PickSeeds to choose two entries, each be first entryof each group

QS2 Repeat

DistributeEntry

until

all entries are distributed or one of the two groups hasMm+1 entries (so that the other group has m entries)

QS3 If entries remain, assign them to the other group so that ithas the minimum number m required

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Guttmans Split Algorithm (Cont..) PickSeedsPS1 For each pair of entries E1 and E2, compose a

rectangle R including E1 rectangle and E2 rectangle

Calculate d = area(R) - area(E1 rectangle) - area(E2 rectangle)PS2 Choose the pair with the largest d

DistributeEntry

DE1 Invoke PickNext to choose the next entry to be assignedDE2 Add It to the group whose covering rectangle will have tobe enlarged least to accommodate It. Resolve ties by addingthe entry to the group with the smallest area, then to theone with the fewer entries, then to either

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Guttmans Split Algorithm (Cont..)PickNextDE1 For each entry E not yet in a group, calculate

d1 = the area increase required in the coveringrectangle of Group 1 to include E Rectangle.

Calculate d2 analogously for Group 2.

DE2 Choose the entry with the maximum difference

between d1 and d2

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Problems Small Seeds:

If d-1 of the d axes of a far away rectangle is same as oneseed, needle like bounding rectangle may be formed

May initiate a bad split

R1 R2

R3

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Problems (Cont..) Prefer Bounding Rectangle:

Algo prefer the MBR created

from previous assignment Since it was enlarged, it

requires less area enlarge-

ment to include next entry

If a group reached M-m+1 entries, rest are assigned

to other without considering geometric properties

Z

YX

W

G2

G1

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Greenes R-tree ChooseSubtree is same as Guttmans

Alternative split algorithm

Invokes PickSeeds to find two most distant rectangles Picks a axis using these two rectangles depending

upon there separation distance

Sorts the remaining rectangles along chosen axis

Distributes half entries to one group and remaining toother

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Greenes R-tree (Cont..) In some situations cannot find right axis and bad split

may occur

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Inspiration For R*-tree Minimize overlap between directory rectangles :

Decrease no of path to be traversed

Minimize margin of a directory rectangle : Rectanglewould be shaped more quadratic

Optimize space utilization : Height will be low

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R*-tree Structure same as R-tree

For insertion R-tree versions only consider area

R*-tree consider area, margin & overlap in differentcombinations

Overlap is defined as

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R*-tree: ChooseSubtree Similar to original one, only difference is that it minimizes overlap enlargement

when N points to leaves

CS1 Set N to be the root node

CS2 If N is a leaf,return N

else Ifchildpointers in N point to leaves [determine theminimum overlap cost],choose the entry in N whose rectangle needs leastoverlap enlargement to include the new data rectangle.Resolve ties by choosing the entry whose rectangleneeds least area enlargement, then the entry with therectangle of smallest area

else [determine the minimum area cost]choose the entry in N whose rectangle needs least areaenlargement to include the new data rectangle. Resolveties by choosing entry with rectangle of smallest area

end

CS3 Set N to be the childnode pointed to by the childpointer ofthe chosen entry. Repeat from CS2

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R*-tree: Split Algorithm Three goodness values:

area-value area[bb(first group)] +

area[bb(second group)]

margin-value margin[bb(first group)] +margin[bb(second group)]

overlap-value area[bb(first group)

bb(second group)]

Depending on these values final distribution is determined

*bb: bounding box

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R*-tree: Split Algorithm (Cont..) Method for good split:

Along each axis entries are sorted first by lower and thenby upper value of their rectangles

For each sort M-2m+2 distribution of M+1 entries isdetermined

First group contains (m-1)+k entries while othercontains remaining (k=1,..,(M-2m+2))

For each distribution goodness value is measured

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R*-tree: Split Algorithm (Cont..) SplitS1 Invoke ChooseSplitAxis to determine the axis, perpendicular to

which the split is performedS2 Invoke ChooseSplitIndex to determine the best distribution

into two groups along that axis

S3 Distribute the entries into two groups ChooseSplitAxisCSA1 For each axis

Sort the entries by the lower then by the upper value oftheir rectangles and determine all distributions asdescribed above. Compute S, the sum of all margin-

values of the different distributionsend

CSA2 Choose the axis with the minimum S as split axis

ChooseSplitIndexCSI1 Along the chosen split axis, choose the distribution with the

minimum overlap-value. Resolve ties by choosing thedistribution with minimum area-value

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Reinsert Dealing with under filled nodes in R-tree: Remove its

entries and reinsert them

It improves retrieval performances

Deletion and reinsertions tunes R-tree but it is verystatic

To achieve dynamic reorganization R*-tree uses forced

reinsertion during insertion routine

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Forced Reinsert If a node is overfilled, R*-tree takes p entries based on

distance of their center from center of MBR

Removes p entries and adjust the MBR

Reinserts them to prevent splits

If they are reinserted in the same node again then itcalls split

Now cpu cost of insert is increased but if we takeaverage on large insert it is only increased about 4%due to reduced splits and better structure

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Forced Reinsert: Advantages More reconstruction, less split

Storage utilization is improved

Outer rectangles are reinserted, directory rectanglebecomes more quadratic which is a desired property

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Experiments Comparison between four R-tree variants

R-tree with quadratic split algorithm (qua.Gut)

R-tree with linear split algorithm (lin.Gut)

Greenes variant of R-tree (Greene)

R*-tree

Six data files containing about 100,000 2D rectangles

All experiments were measured in number of diskaccesses

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Experiments (Cont..) Types of queries

Rectangle intersection query given a rectangle S, find all rectangles R in the file with R S

Point query given a point P, find all rectangles R in the file with P R.

Rectangle enclosure query given a rectangle S, find all rectangles R in the file with R S

Spatial join Over two files as the set of all pairs of rectangles where rectangle

from f1 intersects rectangle from f2 Also measured the parameters insertion and storage

utilization

Seven query files were created

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Results The page access for queries to R*-tree are standardized

to 100%.

Here is the relative performance for all 4 variants forR-tree

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Results (Cont..) Unweighted average results over all distributions

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Results (Cont..) R*-tree are very efficient for PAM

Even outperforms very popular 2-level grid file

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Conclusions Since all three area, margin and overlap are reduced,

R*-tree is very robust against ugly data

Storage utilization is higher, insertion cost is low

Outperforms all existing R-tree variants

R*-tree can efficiently be used as an access method indatabase systems organizing both multidimensional

points and spatial data

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References The R*-Tree: An Efficient and Robust Access Method for Points and Rectangles

-N. Beckmann, H.-P. Kriegel, R. Schneider and B. Seeger. SIGMOD 1990

http://en.wikipedia.org/wiki/R-tree

Image Sources: Maps & R-tee: http://electures.informatik.uni-freiburg.de/portal/download/3/8534/thm03%20-%20rTtee%20p1.ppt Thumbs Up: http://www.ideachampions.com/weblogs/Peer%20to%20Peer%20Recognition.png

Gears: https://reader009.{domain}/reader009/html5/0422/5adcbd7ec56a1/5adcbd8dbeb01.png Tree:http://a01421.deviantart.com/art/tree-variants-304634600

Choose: http://www.transforming-technologies.com/blog/index.php/2011/06/16/how-to-choose-an-esd-mat/

Original: http://www.pixmac.com/picture/original+ink+stamp/000045168969

Split: http://www.clipartguide.com/_pages/0511-1001-2605-2460.html

Forced: http://thepoliticalcarnival.net/2011/05/ Advantages : http://www.webgraffiti.ca/advantages.html

Experiments: http://nilssmith.com/becoming-a-social-media-pastor-part-4-the-experiment/

Results: http://www.iconshock.com/icons/sigma/project_managment/results-icon.html

Conclusions: http://herbertjlkld.portrelay.com/conclusions-clip-art.html

Introduction: http://www.eng.fju.edu.tw/iacd_2010S/computer/introduction1.htm

http://www.google.com/imgres?um=1&hl=en&safe=off&biw=1280&bih=622&tbm=isch&tbnid=qvmYAWv5AJ1YnM:&imgrefurl=http://a01421.deviantart.com/art/tree-variants-304634600&docid=QWzpLMFK3Va1XM&itg=1&imgurl=http://www.deviantart.com/download/304634600/tree_variants_by_a01421-d51ddk8.png&w=1500&h=2095&ei=r2wyUNqgJsuIrAfxgoEg&zoom=1&iact=hc&vpx=421&vpy=114&dur=1228&hovh=265&hovw=190&tx=87&ty=100&sig=114563138652018660749&page=1&tbnh=128&tbnw=92&start=0&ndsp=19&ved=1t:429,r:2,s:0,i:78http://www.google.com/imgres?um=1&hl=en&safe=off&biw=1280&bih=622&tbm=isch&tbnid=qvmYAWv5AJ1YnM:&imgrefurl=http://a01421.deviantart.com/art/tree-variants-304634600&docid=QWzpLMFK3Va1XM&itg=1&imgurl=http://www.deviantart.com/download/304634600/tree_variants_by_a01421-d51ddk8.png&w=1500&h=2095&ei=r2wyUNqgJsuIrAfxgoEg&zoom=1&iact=hc&vpx=421&vpy=114&dur=1228&hovh=265&hovw=190&tx=87&ty=100&sig=114563138652018660749&page=1&tbnh=128&tbnw=92&start=0&ndsp=19&ved=1t:429,r:2,s:0,i:78

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Thanks

Q/A