1. B-Trees

2. Motivation for B-Trees
So far we have assumed that we can store an entire data structure in main memory
What if we have so much data that it won’t fit?
We will have to use disk storage but when this happens our time complexity fails
The problem is that Big-Oh analysis assumes that all operations take roughly equal time
This is not the case when disk access is involved

3. Applications: Databases
A database is a collection of data organized in a fashion that facilitates updating, retrieving, and managing the data.
The data can consist of anything, including, but not limited to names, addresses, pictures, and numbers.
Databases are commonplace and are used everyday.
For example, an airline reservation system might maintain a database of available flights, customers, and tickets issued.
A teacher might maintain a database of student names and grades.
Because computers excel at quickly and accurately manipulating, storing, and retrieving data, databases are often maintained electronically using a database management system.

4. Applications: Databases
Database management systems are essential components of many everyday business operations. Database products like Microsoft SQL Server, Sybase Adaptive Server, IBM DB2, and Oracle serve as a foundation for
accounting systems,
inventory systems,
medical recordkeeping systems,
airline reservation systems, and
countless other important aspects of modern businesses

5. Applications: Databases
It is not uncommon for a database to contain millions of records requiring many gigabytes of storage.
TELSTRA, an Australian telecommunications company, maintains a customer billing database with 51 billion rows (yes, billion) and 4.2 terabytes of data.
For a database to be useful, support the desired operations, such as retrieval and storage, quickly.
Databases cannot typically be maintained entirely in memory, b-trees are often used to index the data and to provide fast access.
For example, searching an non-indexed and unsorted database containing n key values will have a worst case running time of O(n);
Data indexed with a b-tree, the same operation will run in O(log n).
A search for a single key on a set of one million keys (1,000,000),
a linear search will require at most 1,000,000 comparisons.
If the same data is indexed with a b-tree of minimum degree 10,
114 comparisons will be required in the worst case.

6. Motivation (cont.) Assume that a disk spins at 3600 RPM
In 1 minute it makes 3600 revolutions, hence one revolution occurs in 1/60 of a second, or 16.7ms
On average what we want is half way round this disk – it will take 8ms
This sounds good until you realize that we get 120 disk accesses a second – the same time as 25 million instructions
In other words, one disk access takes about the same time as 200,000 instructions
It is worth executing lots of instructions to avoid a disk access

7. Motivation (cont.)
Assume that we use an AVL tree to store all the car driver details in beautiful Santa Barbara (about 20 million records)
We still end up with a very deep tree with lots of different disk accesses; log2 20,000,000 is about 24, so this takes about 0.2 seconds (if there is only one user of the program)
We know we can’t improve on the log n for a binary tree
But, the solution is to use more branches and thus less height!
As branching increases, depth decreases

8. Definition of a B-tree
A B-tree of order m is a tree, where each node may have up to m children, and in which:
1. the number of keys in each non-leaf node is one less than the number of its children and these keys partition the keys in the children in the fashion of a search tree
2. all leaves are on the same level
3. all nodes except the root have at least children ( keys)
4. the root is either a leaf node, or it has from 2 to m children
5. No node can contain more than m children (m-1 keys)
The number m should always be odd?

9. An example B-Tree

10. Constructing a B-tree
Suppose we start with an empty B-tree and keys arrive in the following order:
We want to construct a B-tree of order m=5. So every node (except root) contains at least 2 keys, and at most 4 keys.
The first four items go into the root:
To put the fifth item in the root would violate condition 5
Therefore, when 25 arrives, pick the middle key to make a new root 1 2 8 12

11. Constructing a B-tree (contd.)8 1 2 12 25
6, 14, 28 get added to the leaf nodes: 1 2 8 12 14 6 25 28

12. Constructing a B-tree (contd.)
Adding 17 to the right leaf node would over-fill it, so we take the middle key, promote it (to the root) and split the leaf 8 17 1 2 6 12 14 25 28
7, 52, 16, 48 get added to the leaf nodes 8 17 12 14 25 28 1 2 6 16 48 52 7

13. Constructing a B-tree (contd.)
Adding 68 causes us to split the right most leaf, promoting 48 to the root, and adding 3 causes us to split the left most leaf, promoting 3 to the root; 26, 29, 53, 55 then go into the leaves 3 8 17 48 1 2 6 7 12 14 16 25 26 28 29 52 53 55 68
Adding 45 causes a split of 25 26 28 29
and promoting 28 to the root then causes the root to split

14. Constructing a B-tree (contd.)17 3 8 28 48 1 2 6 7 12 14 16 25 26 29 45 52 53 55 68

15. Inserting into a B-Tree
Attempt to insert the new key into a leaf
If this would result in that leaf becoming too big, split the leaf into two, promoting the middle key to the leaf’s parent
If this would result in the parent becoming too big, split the parent into two, promoting the middle key
This strategy might have to be repeated all the way to the top
If necessary, the root is split in two and the middle key is promoted to a new root, making the tree one level higher

16. Exercise in Inserting a B-Tree
Insert the following keys to a 5-way B-tree:
3, 7, 9, 23, 45, 1, 5, 14, 25, 24, 13, 11, 8, 19, 4, 31, 35, 56

17. Simple leaf deletion
Assuming a 5-way
B-Tree, as before... 12 29 52 2 7 9 15 22 56 69 72 31 43
Delete 2: Since there are enough
keys in the node, just delete it

18. Simple leaf deletion
Assuming a 5-way
B-Tree, as before... 12 29 52 2 7 9 15 22 56 69 72 31 43

19. Simple non-leaf deletion
Assuming a 5-way
B-Tree, as before... 12 29 52 2 7 9 15 22 56 69 72 31 43
Delete 52

20. Simple non-leaf deletion
Assuming a 5-way
B-Tree, as before... 12 29 2 7 9 15 22 56 69 72 31 43

21. Simple non-leaf deletion12 29 52 7 9 15 22 56 69 72 31 43 56
Delete 52
Borrow the predecessor
or (in this case) successor

22. Simple non-leaf deletion12 29 56 7 9 15 22 69 72 31 43

23. Too few keys in node and its siblings12 29 56 7 9 15 22 69 72 31 43
Delete 72

24. Too few keys in node and its siblings12 29 56 7 9 15 22 69 72 31 43
Too few keys!
Delete 72

25. Too few keys in node and its siblings
Join back together 12 29 56 7 9 15 22 69 72 31 43
Too few keys!
Delete 72

26. Too few keys in node and its siblings12 29 7 9 15 22 69 56 31 43

27. Enough siblings 12 29 7 9 15 22 69 56 31 43
Delete 22

28. Enough siblings 12 29 7 9 15 22 69 56 31 43
Delete 22

29. Enough siblings 12 29 7 9 15 22 69 56 31 43 Demote root key and
promote leaf key
Delete 22

30. Enough siblings 12 31 69 56 43 7 9 15 29

31. Exercise in Removal from a B-Tree
Given 5-way B-tree created by these data (last exercise):
3, 7, 9, 23, 45, 1, 5, 14, 25, 24, 13, 11, 8, 19, 4, 31, 35, 56
Delete these keys: 4, 5, 7, 3, 14

32. Analysis of B-Trees
The maximum number of items in a B-tree of order m and height h:
root m – 1
level 1 m(m – 1)
level 2 m2(m – 1)
level h mh(m – 1)
So, the total number of items is (1 + m + m2 + m3 + … + mh)(m – 1) = [(mh+1 – 1)/ (m – 1)] (m – 1) = mh+1 – 1
When m = 5 and h = 2 this gives 53 – 1 = 124

33. Reasons for using B-Trees
When searching tables held on disc, the cost of each disc transfer is high but doesn't depend much on the amount of data transferred, especially if consecutive items are transferred
If we use a B-tree of order 101, say, we can transfer each node in one disc read operation
A B-tree of order 101 and height 3 can hold 1014 – 1 items (approximately 100 million) and any item can be accessed with 3 disc reads (assuming we hold the root in memory)
B-Trees are always balanced (since the leaves are all at the same level), so 2-3 trees make a good type of balanced tree

34. Comparing Trees Binary trees Multi-way trees
Can become unbalanced and lose their good time complexity (big O)
AVL trees are strict binary trees that overcome the balance problem
Heaps remain balanced but only prioritise (not order) the keys
Multi-way trees
B-Trees can be m-way, they can have any (odd) number of children
B-Tree approximates a permanently balanced binary tree, exchanging the AVL tree’s balancing operations for insertion and (more complex) deletion operations

35. B-Trees in Databases

36. Concurrent Access to B-Trees
Databases typically run in multi-user environments where many users can concurrently perform operations on the database.
For example, imagine a database storing bank account balances.
Someone attempts to withdraw $40 from an account containing $60.
First, the current balance is checked to ensure sufficient funds.
After funds are disbursed, the balance of the account is reduced.
This approach works flawlessly until concurrent transactions are considered.

37. Concurrent Access to B-Trees
Suppose that another person simultaneously attempts to withdraw $30 from the same account.
At the same time the account balance is checked by the first person, the account balance is also retrieved for the second person.
Since neither person is requesting more funds than are currently available, both requests are satisfied for a total of $70.
After the first person's transaction, $20 should remain ($60 - $40), so the new balance is recorded as $20. Next, the account balance after the second person's transaction, $30 ($60 - $30), is recorded overwriting the $20 balance. Unfortunately, $70 have been disbursed, but the account balance has only been decreased by $30. Clearly, this behavior is undesirable, and special precautions must be taken.

38. Concurrent Access to B-Trees
A B-tree suffers from similar problems in a multi-user environment. If two or more processes are manipulating the same tree, it is possible for the tree to become corrupt and result in data loss or errors.
The simplest solution is to serialize access to the data structure. In other words, if another process is using the tree, all other processes must wait. Locking, introduced by Gray and refined by many others, provides a mechanism for controlling concurrent operations on data structures in order to prevent undesirable side effects and to ensure consistency.

39. References B-Trees: Balanced Tree Data Structures by Peter Neubauer
B-Trees. The University of Wales.