1. B-Tree
Michael Tsai
2017/06/06

2. B-Tree Overview Balanced search tree Very large “branching factor”
Height = O(log n), but much less than that of RB tree
Usage: Large amount of data to be stored -- Partially in memory, and partially in secondary storage (e.g., hard drive)
Goal:
Minimizing disk I/O operation
Minimizing CPU time

3. Typical Storage Speed / Capacity
Read speed
Capacity
Hard Drive
Typically ~100 MB/s
Up to 10 TB
SSD
~500 MB/s
Up to 1 TB SD
30 MB/s (UHS-3) 10 MB/s (class 10)
(Typically 32 GB or 64 GB)
Memory
6400 MB/s (DDR3)
Desktop/laptop has 4~16 GB

4. Typical B-Tree (keys) Internal node x has x.n keys (3)
Keys in x separate the ranges of keys in its sub trees.
Internal node x has x.n+1 children (4)

5. Typical B-Tree (search)
Internal node x has x.n keys (3)
Keys in x separate the ranges of keys in its sub trees.
Internal node x has x.n+1 children (4)

6. A more realistic B-tree
Usually only a node’s keys/data is read from the disk at a time.
Root is always kept in the memory.

7. B-Tree Definition (1) B-Tree is a rooted tree For each node x:
x.n is the number of keys in x
Keys: are stored in non-decreasing order.
x.leaf, TRUE if x is a leaf and FALSE otherwise.
Each internal node x contains x.n+1 pointers
Leaves have these undefined.

8. B-Tree Definitions (2)
The keys separate the ranges of keys stored in each subtree: is any key stored in the subtree with root ,
All leaves have the same depth: the tree’s height h.
Minimum degree of B-tree:
Every node other than root have at least t-1 keys (thus t children)
Every node can have at most 2t-1 keys (thus 2t children) (In this case, this node is full)

9. Proof: B-Tree Height
If , then for any n-key B-tree T of height h and minimum degree ,
Proof: Consider the case with each node having the least # of node

10. Proof: B-Tree Height

11. Disk Operation
DISK-READ(x) : if x is not in memory, then we require this before accessing x. “no-op” if x is already in the memory.
DISK-WRITE(x): this is required for putting any changes of x back to the disk.
Root is always stored in the memory
Typical work flow:
x = pointer to an object
DISK-READ(x)
(operations to modify x)
DISK-WRITE(x)
Operations to access x (but no modifications)

12. Search in B-Tree CPU time: Disk I/O:
Input: x: search from this node k: key to be searched
Return value: (x,i): key k is found at node x’s i-th key
CPU time: Disk I/O:
i=1,2,3,4, 5
x.key_i =

13. Create an empty B-Tree CPU time: O(1) Disk I/O: O(1)
Allocate-Node() is a O(1) operation to allocate a disk page to store a new node
CPU time: O(1) Disk I/O: O(1)

14. B-Tree Insertion: Overview
Cannot simply create a new leaf node and insert it: this will violate the B-tree definitions
Sol: insert into an existing leaf node
Problem: what if that leaf node is already FULL?
FULL: having 2t-1 keys and 2t children
Sol: split a full node y around its median key
Then move up to y’s parent node.
What if y’s parent is also full? We split it too
Workflow: Start from root (search), split all traversed full nodes
t keys smaller than median
t keys larger than median

15. B-Tree Insertion: Overview

16. B-Tree: Split Full Child
Split node x’s i-th child, which is full
CPU time: O(t) Disk I/O: O(1)

17. B-Tree: Split the Root
Splitting the root is the only way to increase the height of a B-tree
Height is increased at the top, not at the bottom

18. B-Tree: Split the Root
Please review the pseudo code below yourself.

19. Insertion Example

20. B-Tree Insertion:pseudo code
If x is a leaf, insert k at the right location
If x is not a leaf, then..
Find the right child node
If the child node is full, first split it! (its median key will come back to this node)
Finally, recursive call to continue to the child node

21. B-Tree Insertion: running time
CPU time: Disk I/O:

22. Reading Assignment (Real)
Chapter 18.3 Deleting a key from a B-tree