CS102-07 Indexed Sequential_1

8/7/2019 CS102-07 Indexed Sequential_1

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Perfect Binary TreesPerfect Binary tree :

a perfect binary tree has all nodes with 2 childrenexcept the leaf nodes which are all at level n.

a perfect binary tree has 2l-1

nodes at level l.a perfect binary tree of height h has 2 h -1 nodes

the height h of a perfect binary tree with n nodesis log 2(n+1) .

Binary trees have height at least log 2(n+1) .

a tree is balanced if the height is a minimum forthe number of nodes. (Other non-equivalent andapproximate definitions of "balanced" exist)

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Example Perfect and Balanced Trees

E

MI

B

D

L N

F G

C

A

OH KJ

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Binary Search TreesBinary Search Tree :

an organized binary tree such that

the key value of every node is greater thanthe key value of its left child and less than thekey value of its right child.

if the tree is balanced, searching a BST could beas efficient as binary search

balanced binary search trees are used toorganize data for quick access.

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AVL TreesCreated by Adelson, Velskii, Landis in 1962

height balanced binary search trees

AVL trees definition

the heights of the left and right subtrees ofevery node of T do not differ by more than 1

Random retrieval in an AVL tree

maximum height h 1.4404 log 2 (n+2) - 0.328or roughly 1.5 log 2 n

for n=1,000,000, max h is 29.

proven by AVL themselves.

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AVL Tree InsertionUse binary search to determine insertion point ofnew key, k. Insert new node x containing k.

Update the subtree heights, h , of x, h(x)=1, andupwards to all its ancestors (no change or add 1)

While updating subtree heights, if there is animbalance at node a : |h(a.left) h(a.right)| > 1,

Let p be a.parent, p is null if a is the root.

Let node b be child of a on path from x.

Let node c be child of b on path from x.

Fix imbalance using appropriate rotations.

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AVL Tree Rotations (1)To maintain the AVL tree after insertions anddeletions, local rotations are performed.

20

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Initial BST Insert 30

10 30

20

RR-RotationL2 null

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R2

R1

a

b

c


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RR Rotationa.right b.left

b.left a

if p = NULL

then root belse if p.left = a

then p.left = b

else p.right = b

h(a) = h(a) - 1

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AVL Tree Rotations (2)

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Insert 40 Insert 50

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RR-RotationL2 null

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R2

R1

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a

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Insert 60RR-RotationL2 not null

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30 60

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R1

R2

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a

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Without 60,Insert 35 or 25

RL-Rotation40s left 30s right20s right 30s left30s left 2030s right 40

Note : LL- and LR-Rotations are symmetric toRR- and RL-Rotations

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4020

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3525

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R1

L2

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RL Rotationa.right c.left

b.left c.right

c.left a

c.right b

if p = NULL

then root c

else if p.left = a

then p.left = c

else p.right = c

h(a) = h(a) 2

h(b) = h(b) 1

h(c) = h(c) + 1CJD

AVL Updates and DeletionsWhat about key updates ?

Delete record, and insert record with new key

How do you handle deletions?

Swap with successor (next in linear ordering)Successor is a leaf node.

Delete this node after swap.

Handle rotations from thereupdate subtree heights towards root.

do rotations if it becomes imbalanced.

Details left to the student


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AVL Trees Analysis

If the AVL tree is created in main memory, theremay not be enough space for all the key values.

If the AVL tree is created on disk, the number ofdisk accesses required to access a record is atmost proportional to the height of the tree.

for 1,000,000 records, no more than 29 diskaccess are required for the search of anyrecord in the file.

This beats sequential access.

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m-Way Search TreesDefinition :

All nodes have m or less children

Each node contains the following information:

t = the number of key values in the node, 1 t < mS 0 = pointer to a subree containing key values lessthan K 1(K1, A1, S 1), , (K t, At, S t) = t triples described below

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m-Way Search TreesDefinition (continued):

Ki (1 i t) are key values where K i < Ki+1 for1 i < t

S i (1 i < t) are pointers to subtree containingkey values between K i and K i+1 exclusive

S t is a pointer to subtree containing key valuesstrictly greater than K tS i (0 i t) are pointers to balanced m-waysearch subtrees

Ai (1

i

t) are addresses in the file of the recordwith key K iCJD

Access

Accessing the m-way search tree :

Each node of the tree is stored in a block ofstorage

All computations within a node are done inmain memory

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Illustration

1, 0, (100, addr[100], 0)f1, 0, (60, addr[60], 0)e2, 0, (80, addr[80], 0), (90, addr[90], f)d2, 0, (40, addr[40], 0), (50, addr[50], e)c2, 0, (10, addr[10], 0), (20, addr[20], 0)b2, b, (30, addr[30], c), (70, addr[70], d)aFormatNode

30 70

40 50 80 9010 20

60 100

a

b c d

e f

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m-Way Analysis (1)

Since there are more key values per node, thereare less nodes

Since a node could have more than 2 children, theheight is generally smaller than binary searchtrees with the same number of nodes

If balanced, the height of an m-way search treewith n nodes is about log m n which is smallerthan for a balanced binary search tree when m=2.


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Example

50 70

80 87 95

10 20 40

5730

55 60 75 10090

partial insertion of87 and 85

85

adjust overflow node

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Example

50 70 87

95

10 20 40

5730

55 60 75 10090

partial insertion of87 and 85

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adjust overflow node

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Example

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95

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5730

55 60 75 10090

complete insertion of87 and 85

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Deletion (1)search for key value and delete it from containingnode

if node is a leaf and does not underflow, youredone

if node is a leaf and underflows,

search for adjacent sibling with more thanminimum number of keys

if sibling found, adjust all nodes f rom this sibling tocurrent node and parent, to transfer an excess keyto the current node using simple rotations

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Deletion (2)if sibling not found,

merge the node with a sibling node. Demote theintervening key value in the parent node to themerged node.

handle deletion of a key value from parent as ifparent is a leaf (i.e. r epeat this step for parent)

if node is a non-leaf,

replace the deleted key by its predecessorkey in the tree

handle the deletion of the predecessor keyfrom the containing leaf node

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Example : m=5

B-Trees where m=5

Root node has 2 to 5 children

all non-terminal nodes has at least 5/2 = 3children: 3, 4 or 5

all non-terminal nodes has 2, 3 or 4 keys

all terminal nodes are on the same level


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Example

90

30 70

10 20 40 60 75 80 100 110 130 140 150 170 180 200 210 230 240

120 160 190 220

Lets delete 20 - its in a leaf node;- results in underflow in 10;- adjacent sibling(s) minimal

merge with adjacent sibling

B-Treeof order 5

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Example

90

70

10 30 40 60 75 80 100 110 130 140 150 170 180 200 210 230 240

120 160 190 220

partial deletionof 20

adjust underflow node rotate with non-minimal sibling

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Example

120

70 90

10 30 40 60 75 80 100 110 130 140 150 170 180 200 210 230 240

160 190 220

20 deleted

Lets delete 160 its a non-leaf- replace it with its predecessor 150

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Example

120

70 90

10 30 40 60 75 80 100 110 130 140 170 180 200 210 230 240

150 190 220

160 deleted

Lets delete 90 replace it with its predecessor 80 results in underflow in 75

rotate 75 with non-minimal sibling

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Example

120

60 80

10 30 40 70 75 100 110 130 140 170 180 200 210 230 240

150 190 220

90 deleted

Lets delete 120 replace it with its predecessor 110- results in underflow in 100- adjacent sibling is minimal merge !

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Example

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60

10 30 40 70 75 80 100 130 140 170 180 200 210 230 240

150 190 220


adjust underflow node rotate with non-minimal sibling


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Example

150

60 110

10 30 40 70 75 80 100 130 140 170 180 200 210 230 240

190 220

120 deleted

Lets delete 190 replace with its successor 180- results in underflow in 170;- adj sibling minimal; merge !

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Example

150

60 110

10 30 40 70 75 80 100 130 140 170 180 200 210 230 240

220


adjust underflow node adjacent sibling minimalmerge !

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Example

60 110 150 220

10 30 40 70 75 80 100 130 140 170 180 200 210 230 240

190 deleted

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AnalysisSome nodes may not be full (i.e. not havemaximum number of children)

This increases the number of nodes but does notsubstantially increase the height.

This allows easy insertions because there maybe space available in the nodes

Each node is at least half-full, so the height isless than about logm/2 n

Very efficient to use as an indexing structure of

files and compares very well to a perfectlybalanced tree

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B*-Trees

Description :

This is a modification of B-Trees using anoverflow handling technique

Insertion into a full node causes rotations thatoverflow excess keys to sibling nodes.

Nodes are at least 2/3 full instead of at least half-full

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B*-TreesB*-Trees Definition :

An empty tree is a special m-way search tree with nonodes

All non-empty subtrees are m-way search tr ees

The root node has at least two children and a maximum of2 (2m-2)/3 + 1 children. Note: This could exceed m.

All non-terminal nodes (S i 0 for some i) other than theroot has at least (2m-1)/3 children; therefore the non-terminal nodes store K i values i (2m-1)/3 - 1

All terminal nodes (S i = 0 for all i) are on the same levelTerminal nodes are initially empty.


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Illustrative ComparisonWith m=7, B-Trees require each (non-root)internal node to have 4 to 7 children or 3 to 6keys per node.

With m=7, B*-Trees require each (non-root)internal node to have 5 to 7 children or 4 to 6keys per node.

With m=31, B-Trees require each (non-root)internal node to have 16 to 31 children or 15 to30 keys per node.

With m=31, B*-Trees require each (non-root)internal node to have 20 to 31 children or 19 to30 keys per node.

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InsertionInserting a key to a full non-root node causes overflow ofan excess key into its siblings using rotations throughtheir parent.

If all siblings are also full, then the overflow node, a

nearest sibling, and the dividing parent key are merged.This is then divided into three n odes as evenly aspossible with the 2 dividing keys moved u p to theirparent, keeping each new node at least about 2/3 full.

If the parent overflows, then adjust that node with itssiblings and parent.

Inserting a key to a full root node creates a new root withthe first half of keys as its left child node, and the secondhalf of keys as its right child node.

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Deletionskipped

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Analysis

Some nodes may not be full

Nodes are more full than B-trees

This reduces the number of nodes required.

Each node is at least two-thirds full, so the heightis less than about log 2m/3 n

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B+-TreesDescription :

modification of B-Trees where each node is atleast half full

internal nodes do not contain addresses of keyvalues: just keys and child pointers

Each K i in internal nodes represent the largestkey value in the left subtree pointed by S i-1. S tpoints to the rightmost subtree whose key valuesare greater than K tall keys and their addresses are in the leaves

leaves are linked to each otherCJD

B+-TreesDescription (continued):

Often used as indexing technique where eachleaf is stored as a block in the file.

Each leaf node contains the key values, theaddress of the block in the disk, and theaddress of the next leaf.

The index is usually stored as a separate indexfile from the data file.


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Example B +-TreeNode Layout : (n, S 0, (K 1, S 1), (K 2, S 2), (K t, S t))

60 8020

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..............................OtherInfo

801006090301070205040Key

9876543210RelAddr

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Implementations

CDC machines use a variation of B +-Trees toimplement its indexed sequential file organizationcalled SIS (Scope Indexed Sequential)

IBM uses a variation of B+-Trees to implementits indexed sequential file organization calledVSAM (Virtual Sequential Access Method)

IBM also implemented ISAM (IndexedSequential Access Method) using cylinders andsurfaces to organize its indices, prime area andoverflow areas. It is a variation of m-way searchtrees.

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End

CS102-07 Indexed Sequential_1

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