01-b trees

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Analysis and Design of Algorithms II

B-Trees

Arif Nurwidyantoro

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Overview

Motivation Definition of B-Trees

Basic operations of B-Trees

Search

Create

Insert Delete

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Motivation

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B-Trees

Balanced search trees Designed for disks and other direct access secondary storag

Similar to Red-Black Trees

Branching factor

Internal nodes may have many children

Every n-node B-Tree has height O(lg n)

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Analysis and Design of Algorithms IIAnalysis and Design of Algorithms II

B-Trees

Internal node x contains x.n keys x has x.n+1 children

Keys in node x as dividing pointsseparating range of keys handledby x into x.n+1 subranges

Each handled by one child of x

Search key in B-Trees

Make (x.n+1)-way decisions

Based on comparisons with the x.nkeys stored at node x

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Data structures on secondary storages

Memory in computer Main memory

Secondary storage

Main memory

Faster

Silicon memory chips Expensive

Secondary storage

Magnetic disks

Bigger capacity

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Disk Drive

Platters One or more

Rotate at constant speedspindle

Covered by magnetizable

Drive reads and writes phead at the end of an ar

Armscan move heads toaway from spindle

Track Surface underneath head

stationary

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Disk

Slow 5400-15000 RPM

5 orders of magnitude lower than memory acces times

Have mechanical parts

Mechanical

Platter rotation

Arm movement

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Disk

Information divided into equalpages of bit Appear consecutively within tracks

Each disk read/write is of one or more entire pages

211to 214 bytes in length

Running time

Number of disk accesses

CPU (computing) time

Number of disk accesses

Number of pages of information need to be read/written to the di

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B-Tree and Disk

Amount of data handled large Didnt fit into main memory at once

B-Tree algorithms

copy selected pages from disk into main memory as needed

write back into disk the pages that have changed

Objective To optimize disks access by using the full amount of information re

disk access

i.e. one page -> size of 1 node of the B-Tree = 1 page

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B-Tree and Disk

x = pointer to some object Disk-Read(x)

Read object x into main memory

Disk-Write(x)

Save any changes made to the attributes of object x

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B-Tree and Disk

Branching factors between 50-2000

Depends on the size of a key relativeto the size of a page

Large branching factors

Reduce height of tree and number of

disk accesses required to find any key

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Definition of B-Trees

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B-Trees Properties

Every node x has the following attributes: x.n, the number of keys currently stored in node x

x.nkeys themselves,x.key1, x.key2, , x.keyx.n, stored in nondecrea

so thatx.key1 x.key2 x.keyx.n

X.leaf,

TRUE, if x is a leaf

FALSE, if x is an internal node

Each internal node x also containsx.n+1 pointersx.c1, x.c2,its children

Leaf node have no children, ciattributes are undefined

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B-Trees Properties

The keysx.keyiseparate the ranges of keys stored in each suis any key stored in the subtree with rootx.ci, then

k1 x.key1 k2 x.key2 x.keyx.n kx.n+1

All leaves have the same depth, which is the trees height h

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B-Trees Properties

Nodes have lower and upper bounds on the number of keyscontain. Bounds expressed in terms of fixed integer t 2 calminimum degree of the B-Tree

Every node other than the root must have at least t-1 keys

Every internal node othet than the root thus has at least t children

If the tree is nonempty, the root must have at least one key

Every node may contain at most 2t-1 keys

An internal node may have at most 2t children

Node is full if it contains exactly 2t-1 keys

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Height of B-Tree

Root of B-Tree T contains at least one key, all other nodes coleast t-1 keys

T has at least 2 nodes at depth 1, 2t nodes at depth 2, at leadepth 3, so on until depth h has at least 2th-1nodes

l i d i f l i h

l i d i f l i h

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Height of B-Tree

The number n of keys satisfies

A l i d D i f Al ith II


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Height of B-Tree

Altough the height of the tree grows as O(lg n)

The base of the logarithm can be many times larger

B-Trees save a factor about lg t over red-black tree in the nunodes examined for most tree operations


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Basic Operations of B-Tree



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Basic Operations

Search

Create

Insert

Delete



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Convention

The root of the B-Tree is always in main memory, so we nevDisk-Read on the root

Any nodes that are passed as parameters must already haveDisk-Read operation performed on them

Procedure presented here are all one pass algorithm that pdownward from the root of the tree, without having to back u



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Search a B-Tree

Similar to search a binary search tree Multiway branching decision At each internal node x, make an (x.n + 1)-way branching decision

Input Pointer to the root node x of a subtree Key k to be searched for in that subtree

Top level call example

B-TREE-SEARCH (T.root, k) Output

If k is in B-tree, return (y,i) node y and an index i such that y.keyi= k

Otherwise, return NIL



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Search a B-Tree



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Search a B-Tree

B-TREE-SEARCH procedure accesses O(h) = O(logt

n)disk pag

h is the height of the B-tree

n is the number of keys in the B-tree

Since x.n < 2t, while loop of lines 2-3 takes O(t) time within

Total CPU time is O(th) = O(t logtn)



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Create an empty B-Tree

use B-TREE-CREATE to create an empty root node

Then, call B-TREE-INSERT to add new keys

Use an auxiliary procedure ALLOCATE-NODE

Allocates one disk page to be used as a new node in O(1)time

B-TREE-CREATE requires O(1) disk operations and O(1)CPU



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y g gy g g

Create an empty B-Tree



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y g gy g g

Insert a key into B-Tree

Insert the new key into existing leaf node

Can not insert a key into a leaf node that is full Split operations

Split full node y (having 2t-1 keys) arounds its median key y.keyt Into two nodes having t-1 keys each

The median key moves up into ys parent to identify the dividingbetweent two new trees

If the parent is also full, must split it before insert new key

Could end up splitting full nodes all the way up the tree



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Insert a key into B-Tree

Another approach: single pass down from the root to a leaf

As travel down the tree searching for the position where thebelongs, split each full node we come along the way (includleaf itself)

We are assured that its parent is not full, in case we want to split a

Procedure involved

Splitting a node in a B-tree

Inserting a key into a B-tree in a single pass down the tree



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Splitting a node in a B-Tree

Procedure B-TREE-SPLIT-CHILD

Input

Nonfull internal node x

Index i such thatx.ciis afull child of x

The procedure splits this child in two and adjusts x so that itadditional child

To split a full root

Make the root a child of a new empty root node



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Split the full node y=x.ciabout its median key S, which moveys parent node x

Those keys in y that are greater than the median key move inode z, which becomes a new child of x



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CPU time used (t)

Due to the loops on lines 5-6 and 8-9

The other loops run for O(t) iterations

Disk operations O(1)



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Inserting a key into a B-Tree in a single pass dowtree



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Inserting a key into a B-Tree in a single passdown the tree



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Inserting a key into a B-Tree in a single pass dowtree

For a B-tree of height h

B-TREE-INSERT performs O(h) disk accesses

Only O(1) DISK-READ and DISK-WRITE operations occur between cTREE-INSERT-NONFULL

Total CPU time O(th) = O(t logtn)



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Deleting a key from a B-Tree

Delete a key in internal node, need to rearrange the nodes

Must not violate the B-tree properties

Ensure that the node size doesnt get too small during delet

Back up if a node (other than the root) along the path to whkey is to be deleted has the minimum number of keys



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Deleting a key from a B-Tree

When call the procedure recursively on a node x

The number of keys in x is at least the minimum degree t

Requires one more key than the minimum required by the usual B

So that sometimes a key may have to be moved into a child node brecursion descends to that child

If the root node x ever becomes an internal node having no

We delete x, and xs only childx.c1becomes the new root of the tr

Decreasing the height of the tree by one and preserving the properoo of the tree contains at least one key



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Various cases deleting a key from B-Tree

1. If the key k is in node x and x is a leaf, delete the key k fro

2. If the key k is in node x and x is an internal node, do the foa. If the child y that precedes k in node x has at least t keys, then f

predecessor k of k in the subtree rooted at y. Recursively deletereplace k by k in x (we can find k and delete it in a single down

b. If y has fewer than t keys, then, symmetrically, examine the chilfollows k in node x. If z has at least t keys, then find the successo

the subtree rooted at z. Recursively delete k, and replace k by kcan find k and delete in in a single downward pass)

c. Otherwise, if both y and z have only t-1 keys, merge k and all of that x loses both k and the pointer to z, and now contains 2t-1 free z and recursively delete k from y



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Various cases deleting a key from B-Tree

3. If the key k is not present in internal node x, determine th

of the appropriate subtree that must contain k, if k is in thall. Ifx.cihas only t-1 keys, execute step 3a or 3b as necesguarantee that we descend to a noe containing at least t kfinish by recursing on the appropriate child of xa. Ifx.cihas only t-1 keys but has an immediate sibling with at least

x.ci an extra key by moving a key from x down into x.ci, moving ax.c

is immediate left or right sibling up into x, and movint the app

child pointer from the sibling intox.cib. Ifx.ciand bothx.cis immediate sibling have t-1 keys, mergex.ciw

sibling, which involves moving a key from x down into the new mto become the median key for that node



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Deleting a key from B-Tree

Since most of the keys in a B-tree are in the leaves

In practice, deletion operations are most often used to delete keysleaves

B-TREE-DELETE acts downward pass through the tree without havup

When deleting a key in an internal node, the procedure mak

downward pass through the tree but may have to return to from which the key was deleted to replace the key with itspredecessor or successor (cases 2a and 2b)



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Deleting a key from B-Tree

Involves only O(h)disk operations

CPU time O(th) = O(t logtn)

01-b trees

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