File Organizations
•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
Fall 2015
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ACS - 3902
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File Organizations
MySQL
(simplified syntax)
CREATE [UNIQUE] INDEX index_name
ON tbl_name (index_col_name,...)
index_type: USING {BTREE | HASH}
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File Organizations
Indexes are automatically created for:
PRIMARY KEY
Is 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.
UNIQUE
Is a constraint that provides entity integrity for a given column or columns
through a unique index. A table can have multiple UNIQUE constraints.
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File Organizations
CLUSTERED
Creates an object where the physical order of rows is the same as the
indexed order of the rows.
Index entries that are logically close also means the data will be close
together physically.
A table or view is allowed one clustered index.
Primary key index is normally clustered
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File Organizations
Motivation (finding one record given its key)
•Scanning a file is time consuming
•B+-tree provides a short access path
file of records
page1
B+-tree
page2
page3
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File Organizations
Motivation
•A B+-tree is a tree, in which each node is a bucket.
•A B+-tree for a file is stored in a separate file.
•A file could have many B+-trees
file of records
bucket 1
B+-tree
bucket 2
bucket 3
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File Organizations
b+-tree
•based on b-tree (Bayer, balanced, Boeing)
•dynamic
Root
Internal
nodes
...
...
Leaf nodes
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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)
• keys are in sequence
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
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ACS - 3902
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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
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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
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File Organizations
Only leaf nodes at this point
– need a split before there
are internal nodes
insert Diane
Pointer to next leaf
in ascending key
sequence
Diane
Pointer to data
(wherever the
record for Diane
is actually stored)
insert Cory
Cory , Diane
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File Organizations
Example
insert Ramon
Only leaf nodes
at this point
Cory , Diane , Ramon
inserting Amy will cause the node to overflow:
Amy
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This must split
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File Organizations
Example
This is logically correct but it exceeds the space available …..
it must split into two leafs:
Amy
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, Cory , Diane , Ramon
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File Organizations
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
Splitting the
above results
in
Tree has grown one
level, from the
bottom up
Cory
Amy
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, Cory
Diane , Ramon
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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.
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File Organizations
Leaf splitting
When 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 lesser 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
22 33
33
12 22 33
44 48 55
12 22
31 33
44 48 55
insert 31
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
...
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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 the left and right siblings)
26 55
55
22 33
22
33
Insert
26
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File Organizations
Internal node splitting
When a new root is formed, a key value and two pointers must
be placed into it.
55
26 55
26
56
Insert
56
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File Organizations
A sample trace
Diane, Cory, Ramon, Amy, Miranda,
Marshall, Zena, Rhonda, Vincent, Simon, Mary
into a b+-tree with p=3.
Cory
Amy
, Cory
Diane , Ramon
Miranda
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File Organizations
Cory
Amy
, Cory
Diane , Miranda , Ramon
Marshall
Cory
Amy
, Cory
Marshall
Diane , Marshall
Miranda , Ramon
Zena
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File Organizations
Cory
Amy
, Cory
Marshall
Diane , Marshall
Miranda , Ramon
, Zena
Rhonda
Cory Marshall Ramon
Amy , Cory
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Diane , Marshall
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Miranda , Ramon
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Rhonda , Zena
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File Organizations
Marshall
Ramon
Cory
Amy , Cory
Diane , Marshall
Miranda , Ramon
Rhonda , Zena
Vincent
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File Organizations
Marshall
Ramon
Cory
Amy , Cory
Diane , Marshall
Miranda , Ramon
Rhonda , Vincent ,Zena
Simon
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File Organizations
Marshall
Ramon Simon
Miranda , Ramon
Rhonda , Simon
Vincent , Zena
Mary
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File Organizations
A sample b+-tree
p = 3,
pleaf = 2.
5
3
1
3
5
7
6
7
8
8
9
12
Records
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File Organizations
Searching a b+-tree
- search a record with key = 8:
5
3
1
3
5
7
6
7
8
8
9
12
The cost of retrieving this record is at most 4 disk accesses: 3 to the index component
and 1 to the data component. In an actual system it would usually be less because the
system would keep as much of the index as it can in main memory.
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File Organizations
Entry deletion
- deletion sequence: 8, 12, 9, 7
5
3
1
3
5
7
6
7
9
9
12
Deleting 8 results in underflow and
key redistribution
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File Organizations
Entry deletion
- deletion sequence: 8, 12, 9, 7
5
3
1
3
5
7
6
7
9
12 is removed.
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File Organizations
Entry deletion
- deletion sequence: 8, 12, 9, 7
5
3
1
3
5
6
6
7
9 is removed.
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File Organizations
Entry deletion
- deletion sequence: 8, 12, 9, 7
5
3
1
3
5
6
6
Deleting 7 makes this pointer useless.
Therefore, a merge at the level above
the leaf level occurs.
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File Organizations
Entry deletion
- deletion sequence: 8, 12, 9, 7
5
3
A
55
This pointer becomes useless.
The corresponding node
should also be removed.
B
1
3
5
6
C
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.
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File Organizations
Entry deletion
- deletion sequence: 8, 12, 9, 7
3
1
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3
5
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6
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File Organizations
b+-tree operations
•search - always the same search length - tree height+1
•retrieval - sequential access is facilitated - how?
•insert - may cause overflow - tree may grow
•delete - may cause underflow - tree may shrink
Aside:
What do you expect for storage utilization?
http://en.wikipedia.org/wiki/B-tree
https://en.wikipedia.org/wiki/B%2B_tree
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