multidim indexing operations

7/27/2019 multidim indexing operations

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Jaruloj Chongstitvatana 2006

K-D-B Tree

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K-D-B TreeMultidimensional Index


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Jaruloj Chongstitvatana 2006 K-D-B Tree 2

CharacteristicsMulti-way branch

Height-balanced tree

Repeatedly divide area of the domaininto disjoint sub-area

A node in a tree corresponds to a (set

of consecutive) disk page(s)


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Example of Data RecordsTable (stdntID, courseID, grade, year, smstr)

Table (accID, branchID, saving, name, addr)

Table (custID, age, gender, occupation,

salary, children, promotion, since)


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Nodes = PagesRegion pages

Contain a set of

Internal nodesPoint pages

Contain a set of Leaf nodes


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Region PagesXmax XminYmaxYmin

PAGE

Xmax XminYmaxYmin

PAGERegionThe branching factor is determined by thepage size and the size of each entry.


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Point PagesX Y

DATA RECORD

X Y

DATA RECORD

X Y

DATA RECORD

POINTThe branching factor of a point page isusually larger than that of a region page.


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Example

Point page Point pagePoint pagePoint page


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Search

Point page Point pagePoint pagePoint page

Pointquery


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Insert

Insert a point here

and the point pageoverflows.


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Split Split a region rwith page idp alongxiIfris on the right/page ofxi then put in the

right/left page.

Otherwise;For each childrenpcofp , splitpcalongxi

Split ralongxi into rleftand rright.

Create 2 new pages with page idpleftandpright.Move children ofp in the left region intopleftandchildren in the right region intopright.

Return and .


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Split: ExampleThe page overflows, and is splitted.

This region is splitted.

This region is also splitted.


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Split: ExampleThe region page is splitted.

The point page is also splitted.

Create a new region page.Children pages are transferred.


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How to find split axis Cyclic: x -> y -> x -> y ->

Priority: x -> x -> y -> x -> x -> y ->

Possible one


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Insert Insert a record with point a and location lin a

tree with root r

Ifris NIL, then create a point pagep and insert therecord with inp and returnp.

Otherwise;

Search fora in the tree with root runtil a pointpage, sayp, is reached.

Insert the record in the point pagep.


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Insert (contd) Insert a record with point a and location lin a

tree with root r

If the point pagep is overflowed, then find anappropriate axis to splitp intopleftandpright.

Ifp is not the root, then change top and pleft, and

insertpright into the parent ofp.Ifp is the root, then create a new root node withtwo children ofpleftandpright.


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Insert: Example

Insert here and splitpoint page if overflows.Divide region.

Search for the given point

until the point page is found.


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Insert: ExampleParent page overflows,then split the page.

This region is splitted.


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Insert: ExampleThe point page is splitted.

The region page is splitted.


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Insert: ExampleInsert the new region page in its parent.

The root node is overflowed, and then splitted.


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Insert: ExampleCreate the new root node


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Delete Simple, if storage utilization is ignored.

Otherwise, an underfull page should be

merged with another page.

When 2 pages are merged, the region of

the new page must be a valid region.

A number of regions are joinable if theirunion is also a region.


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Joinable Regions


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Unjoinable Regions


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Delete (contd) If a pagep is underfull, merge sibling

pages ofp whose regions are joinable.

If the newly-created page is overflowed,

then split the page.


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Further Discussion


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Splitting CriteriaAxis

Cyclic

Priority

Shape ?

ValueArea

Number of data points

Ratio ?

Random ?

Combine the two decisions ?

multidim indexing operations

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