Insertions and deletions in B+ trees - IT-Universitetet i...

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Rasmus Ejlers Møgelberg Insertions and deletions in B+ trees Introduction to Database Design 2011, Lecture 11 Supplement to lecture slides

Transcript of Insertions and deletions in B+ trees - IT-Universitetet i...

Page 1: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Insertions and deletions in B+ trees

Introduction to Database Design 2011, Lecture 11Supplement to lecture slides

Page 2: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Initial setup

• We consider the B+ tree below

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rickjudy

tompetemikejanebob

eddiebob joejane karenjudy philpetenanmike solrick tomalabe

Page 3: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Observations

• Observe that the tree has fan out 3

• Invariants to be preserved

- Leafs must contain between 1 and 2 values

- Internal nodes must contain between 2 and 3 pointers

- Root must have between 2 and 3 pointers

- Tree must be balanced, i.e., all paths from root to a leaf must be of same length

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Page 4: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Inserting Vince

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rickjudy

tompetemikejanebob

eddiebob joejane karenjudy philpetenanmike solrick vincetomalabe

Page 5: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Inserting Vera

• Leaf consisting of tom and vince is split and extra pointer is inserted in parent

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rickjudy

vincetompetemikejanebob

eddiebob joejane karenjudy philpetenanmike solrick veratomalabe vince

Page 6: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Inserting rob

• Inserting rob is more difficult. We first create a new leaf node and insert it as below

• The node above is temporarily extended to contain 4 pointers

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rickjudy

tomsolpetemikejanebob

eddiebob joejane karenjudy philpetenanmike robrick veratomalabe vince

vince

sol

Page 7: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Inserting rob

• The overfull internal node is then split in 2

• The new pointer is inserted into the root node which then becomes overfull

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rickjudy

solpetemikejanebob

eddiebob joejane karenjudy philpetenanmike robrick veratomalabe vincesol

vince

tom

Page 8: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Inserting rob

• Finally the overfull root is split in 2

• At this point the tree satisfies the requirements of slide 3 and so the insertion procedure ends

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judy

solpetemikejanebob

eddiebob joejane karenjudy philpetenanmike robrick veratomalabe vincesol

vince

rick

tom

Page 9: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Deleting jane

• Is straight forward

• Note that the node above the leaf where jane was deleted must also be updated

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judy

solpetemikejoebob

eddiebob joe karenjudy philpetenanmike robrick veratomalabe vincesol

vince

rick

tom

Page 10: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Deleting bob

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judy

solpetemikejoeeddie

eddie joe karenjudy philpetenanmike robrick veratomalabe vincesol

vince

rick

tom

Page 11: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Deleting joe

• Leads to a leaf being deleted and the parent being updated

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judy

solpetemikeeddie

eddie karenjudy philpetenanmike robrick veratomalabe vincesol

vince

rick

tom

Page 12: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Deleting eddie

• Leads to deletion of a leaf

• At this point the parent becomes underfull

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judy

solpetemike

karenjudy philpetenanmike robrick veratomalabe vincesol

vince

rick

tom

Page 13: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Deleting eddie

• When a node becomes underful the algorithm will try to redistribute some pointers from a neighbouring sibling to it.

• Since this is possible in this case we do it

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mike

solpetejudy

karenjudy philpetenanmike robrick veratomalabe vincesol

vince

rick

tom

Page 14: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Deleting abe and al

• Leads to deletion of a leaf

• This makes the parent underfull.

• We cannot redistribute pointers again since this will make the neighbour underfull

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mike

solpete

karenjudy philpetenanmike robrick veratom vincesol

vince

rick

tom

Page 15: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Deleting abe and al

• Instead we must merge with the neighbouring sibling

• But this makes the parent underfull

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solpetemike

karenjudy philpetenanmike robrick veratom vincesol

vince

rick

tom

Page 16: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Deleting abe and al

• Since we can not solve this problem by redistributing pointers we must merge siblings again

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tomrick

solpetemike

karenjudy philpetenanmike robrick veratom vincesol

vince

Page 17: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

Deleting abe and al

• Since the root is underfull it can be deleted

• The resulting tree satisfies the requirements and so the deletion algorithm ends

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tomrick

solpetemike

karenjudy philpetenanmike robrick veratom vincesol

vince

Page 18: Insertions and deletions in B+ trees - IT-Universitetet i …itu.dk/people/mogel/SIDD2011/lectures/BTreeExample.pdfInsertions and deletions in B+ trees Introduction to Database Design

Rasmus Ejlers Møgelberg

General remarks

• When a node becomes underfull the algorithm will try to redistribute pointers from the neighbouring sibling either on the left or the right

• If this is not possible, it should merge with one of them

• The value held in an internal node or the root should always be the smallest value appearing in a leaf of the subtree pointed to by the pointer after the value

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