File Organizations March 2007R McFadyen ACS - 39021 In SQL Server 2000 Tree terms root, internal, leaf, subtree parent, child, sibling balanced, unbalanced

March 2007 R McFadyen ACS - 3902 1

File Organizations

•In SQL Server 2000

•Tree terms

•root, internal, leaf, subtree

•parent, child, sibling

•balanced, unbalanced

•b+-tree

- split on overflow; merge on underflow

- in practice it is usually 3 or 4 levels deep

•search, insert, delete algorithms


File Organizations

Using Enterprise Manager to create an index


File Organizations

Using Query Analyzer to create a create table / index

Create an index on the employeeID column of the emp_pay table that enforces uniqueness. This index physically orders the data on disk because the CLUSTERED clause is specified.

CREATE TABLE emp_pay ( employeeID int NOT NULL, base_pay money NOT NULL, commission decimal(2, 2) NOT NULL )

INSERT emp_pay VALUES (1, 500, .10) INSERT emp_pay VALUES (2, 1000, .05) …

CREATE UNIQUE CLUSTERED INDEX employeeID_ind ON emp_pay (employeeID) GO


File Organizations

Using Query Analyzer to create a create table / index

Create an index on the orderID and employeeID columns of the order_emp table.

CREATE TABLE order_emp ( orderID int IDENTITY(1000, 1), employeeID int NOT NULL, orderdate datetime NOT NULL DEFAULT GETDATE(), orderamount money NOT NULL )

INSERT order_emp (employeeID, orderdate, orderamount) VALUES (5, '4/12/98', 315.19) INSERT order_emp (employeeID, orderdate, orderamount) VALUES (5, '5/30/98', 1929.04) INSERT order_emp (employeeID, orderdate, orderamount) VALUES (1, '1/03/98', 2039.82) … CREATE INDEX emp_order_ind ON order_emp (orderID, employeeID)


File Organizations

Using Enterprise Manager to create a create table / index

PRIMARY KEYIs a constraint that enforces entity integrity for a given column or columns through a unique index. Only one PRIMARY KEY constraint can be created per table.

UNIQUEIs a constraint that provides entity integrity for a given column or columns through a unique index. A table can have multiple UNIQUE constraints.


File Organizations

Using Enterprise Manager to create table / index

CLUSTEREDCreates an object where the physical order of rows is the same as the indexed order of the rows, and the bottom (leaf) level of the clustered index contains the actual data rows. (note this is a variation from the text’s discussion)A table or view is allowed one clustered index at a time. (another variation … indexing a view)A view with a clustered index is called an indexed view.

FILLFACTORSpecifies how full SQL Server should make each index page used to store the index data. User-specified fillfactor values can be from 1 through 100, with a default of 0. A lower fill factor creates the index with more space available for new index entries without having to allocate new space.


File Organizations

Motivation (finding one record given its key)

•Scanning a file is time consuming

•B+-tree provides a short access path

file of recordspage1

page2

page3

B+-tree


File Organizations

Motivation

•A B+-tree is a tree, in which each node is a page.

•A B+-tree for a file is stored in a separate file.

•A file could have many B+-treesfile of records

page1

page2

page3

B+-tree


File Organizations

b+-tree

•based on b-tree (Bayer, balanced, Boeing)

•dynamic

Root

Internal

nodes

Leaf nodes

... ...


File Organizations

Node structure for b+-tree of order p

non-leaf node (internal node or a root)

•< P1, K1, P2, K2, …, Pq-1, Kq-1, Pq > (q p)

•K1 < K2 < ... < Kq-1 (i.e. it’s an ordered set)

•for any key value, X, in the subtree pointed to by Pi

•Ki-1 < X Ki for 1 < i < q•X K1 for i = 1•Kq-1 < X for i = q

•each internal node has at most p pointers•each node except root must have at least p/2 pointers•the root, if it has some children, must have at least 2 pointers

One more

pointer than

there are keys

All but root

must be at

least half full


File Organizations

Node structure for b+-tree of order p

leaf node (terminal node)

•< (K1, Pr1), (K2, Pr2), …, (Kq-1, Prq-1), Pnext >

•K1 < K2 < ... < Kq-1

•Pri points to a record with key value Ki or

•Pri points to a block containing a record with key value Ki

•each leaf has at least p/2 keys •maximum of p keys•all leaves are at the same level (balanced)•Pnext points to the next leaf for key sequencing

Pairs of

key/pointer plus

one next pointer

Dense or non-

dense index


File Organizations

Example

•insert records with key values

Diane, Cory, Ramon, Amy, Miranda, Ahmed,

Marshall, Zena, Rhonda, Vincent, Hok

into a b+-tree with p=3.

internal node will have minimum 2 pointers and maximum 3 pointers - inserting a fourth will cause a split

leaf can have at least 2 key/pointer pairs and a maximum of 3 key/pointer pairs - inserting a fourth will cause a split

Typically p is very

large … 100, 200, …


File Organizations

insert Diane

Diane

Pointer to data

Pointer to next leaf in ascending key sequence

insert Cory

Cory , Diane


File Organizations

Example

insert Ramon

Cory , Diane , Ramon

inserting Amy will cause the node to overflow:

Amy , Cory , Diane , Ramon This leaf must split

see next =>


File Organizations

continuing with insertion of Amy - split the node and promote a key value upwards (this must be Cory because it’s the highest key value in the left subtree)

Amy , Cory , Diane , Ramon

Amy , Cory Diane , Ramon

Cory

Tree has grown one level, from the bottom up


File Organizations

Splitting Nodes

Any value being promoted upwards will come from the node that is splitting.

•When a leaf splits, a ‘copy’ of a key value is promoted. •When an internal node splits, the middle key value ‘moves’ from a child to a parent node.

There are three situations to be concerned with: •a leaf splits, •an internal node splits, •a new root is generated.


File Organizations

Leaf splittingWhen a leaf splits, a new leaf is allocated

•the original leaf is the left sibling, the new one is the right sibling •key and pointer pairs of the overflowing node are redistributed: the left sibling will have smaller keys than the right sibling•a 'copy' of the key value which is the largest of the keys in the left sibling is promoted to the parent

Two situations arise: the parent exists or not. If the parent exists, then a copy of the key value and the pointer to the right sibling are promoted upwards. Otherwise, the b+-tree is just beginning to grow ...

33

12 22 33 44 48 55 12 22 44 48 5531 33

22 33

insert 31


File Organizations

Internal node splitting

If an internal node splits and it is not the root,•insert the key and pointer and then determine the middle key•a new 'right' sibling is allocated•everything to its left stays in the left sibling•everything to its right goes into the right sibling •the middle key value along with the pointer to the new right sibling is promoted to the parent (the middle key value 'moves' to the parent to become the discriminator between this left and right sibling)

22 33

55

22

26 55

Insert 26

33


File Organizations

Internal node splitting

When a new root is formed, a key value and two pointers must be placed into it.

26 55

Insert 56

26 56

55


File Organizations

A sample trace

Diane, Cory, Ramon, Amy, Miranda,

Marshall, Zena, Rhonda, Vincent, Simon, Mary

into a b+-tree with p=3.

Amy , Cory Diane , Ramon

Cory

Miranda


File Organizations

Amy , Cory

Cory

Diane , Miranda , Ramon

Marshall

Amy , Cory Diane , Marshall Miranda , Ramon

Cory Marshall

Zena


File Organizations

Amy , Cory Diane , Marshall Miranda , Ramon , Zena

Cory Marshall

Rhonda

Amy , Cory Diane , Marshall Rhonda , Zena

Cory Marshall Ramon

Miranda , Ramon


File Organizations

Amy , Cory Diane , Marshall Rhonda , Zena

Marshall

Miranda , Ramon

Cory Ramon

Vincent


File Organizations

Amy , Cory Diane , Marshall

Rhonda , Vincent ,Zena

Marshall

Miranda , Ramon

Cory Ramon

Simon


File Organizations

Marshall

Miranda , Ramon

Ramon Simon

Rhonda , Simon Vincent , Zena

Mary


File Organizations

A sample b+-tree

5

3 7 8

6 7 9 125 81 3

Records

p = 3,pleaf = 2.


File Organizations

Searching a b+-tree

- search a record with key = 8:

5

3 7 8

6 7 9 125 81 3


File Organizations

Entry deletion

- deletion sequence: 8, 12, 9, 7

5

3 7 9

6 7 125 91 3

Deleting 8 results in underflow and key redistribution


File Organizations

Entry deletion


5

3 7

6 75 91 3

12 is removed.


File Organizations

Entry deletion


5

3 6

65 71 3

9 is removed.


File Organizations

Entry deletion


5

3 6

651 3

Deleting 7 makes this pointer no use.Therefore, a merge at the level abovethe leaf level occurs.


File Organizations

Entry deletion


For this merge, 5 will be taken as a key value in A since any key value in B is less than or equal to 5 but any key value in C is larger than 5.

651 3

53 5A

B

C

5

This pointer becomes useless.The corresponding nodeshould also be removed.


File Organizations

Entry deletion


51 3 6

53 5


File Organizations

b+-tree operations

•search - always the same search length - tree height

•retrieval - sequential access is facilitated - how?

•insert - may cause overflow - tree may grow

•delete - may cause underflow - tree may shrink

What do you expect for storage utilization?

Documents

File Organizations March 2007R McFadyen ACS - 39021 In SQL Server 2000 Tree terms root, internal, leaf, subtree parent, child, sibling balanced, unbalanced