1. Indexing

2. Physical Disk Structure

3. Disk Example Four platters providing eight surfaces
213 = 8192 tracks per surface
28 = 256 sectors per track
29 = 512 bytes per sector
Sector is physical unit while block is logical unit dependent on the DBMS
Typically a block is as large as a sector

4. Disk Access Characteristics
To read a block
Head has to move to the track containing the block (seek time)
The block has rotate under the head (rotational latency)
Transfer time (negligible)

5. I/O model of computation
If a block needs to be moved between disk and main memory then the time taken to perform the read or write is much larger than the time likely to be used manipulating the data in main memory. Thus number of blocks accessed is a good approximation of the time needed by the algorithm and should be minimized
Minimize seek time + rotational latency

6. Sorting data in Secondary Storage
Suppose a relation consists of 10,000,000 records
Want to sort this relation on a “key”
100 bytes per record. Approx 1 gigabyte
Assume 50 MB of main memory available
Disk block is 4096 bytes.
Relation takes 250,000 blocks.
12,800 blocks can fit in memory.

7. Sorting…cont
If data fits in main memory, the fastest algorithms for sorting are variants of Quicksort.
Preferred method is to minimize the number of times a block is brought into main memory

8. Merge-Sort example Step List 1 List 2 Output Start 1,3,4,9 2,5,7,8
none 1)
3,4,9 1 2)
5,7,8
1,2 3)
4,9
1,2,3 4) 9
1,2,3,4 5)
7,8
1,2,3,4,5 6) 8
1,2,3,4,5,7 7)
1,2,3,4,5,7,8 8)
1,2,3,4,5,7,8,9

9. Two-Phase Multiway Merge Sort
Phase 1: Sort main-memory sized pieces of data and store them as sorted lists.
Fill all memory with blocks from orig. list
Time:read and write 250,000 blocks; 15 millisecond per block; 7500 seconds, 125 min
Phase 2: Merge all the sorted lists into a single sorted list.

10. Phase 2
If we use the main-memory merge then we would have to read the data 2log(n) times where n is the number of sorted lists.
The common strategy is
Bring the first block of each sorted list into memory and have one output buffer
Find smallest key among remaining keys in all lists
Move the smallest element to the first available position in the buffer
If output block is full write buffer to disk and reinitialize to empty
If the block from where the smallest element was taken is full, then bring in the next block from the same sorted list into the buffer
Cost of phase 2 is again is 125 min; Total cost is 250 min

11. Motivation SQL is declarative
How should the following queries be processed:
SELECT * from R
SELECT * from R where R.A = ’10’

12. Index Function
Block
Holding
records
IIndex
value
Matching
Records

13. Book Analogy Just remember the book index
Index is a set of pages (a separate file) with pointers (page numbers) to the data page which contains the value
Also note difference between “search key” and “primary key”

14. Types of Indexes Simple indexes on sorted files
Secondary indexes on unsorted files
B-trees on any type of file

15. Sequential File A file sorted on the attribute(s) of the index.
Very useful when the search attribute is the primary key.
Build a dense index on the file.
Called dense because every key from the data file is represented in the index.
Note the index only contains the key and pointer of the data file and thus is usually much smaller than the data file.

16. Example Suppose a relation has 1,000,000 records
A block is of size 4096 bytes.
10 records fit in one block.
Thus size of data is > 400 MB
An index will have a 12 byte representation for each record.
Thus will fit 100 index entries in a block.
Index will fit in blocks (40 MB).
Log2(10000) ~= 14 (Thus 14 I/O’s for lookup)

17. Sparse Index
Instead of one index record per data record, use one index record per block of data record.
This is called a sparse index.
Suppose query is “Is there a record with key value K”.
Just check in dense index
For sparse index a data block has to be retrieved

18. Secondary Indexes
When you go to a library, books are sorted by Call Number.
The call numbers ranges on the shelves are like a sparse index.
Now what if you want to search by “last name” and not call number.
You build a secondary structure on the books.
Secondary structure does not determine or influence the place of data records.
You can have several secondary structures on one file.

19. Example SELECT book.title FROM books WHERE books.lastname=“Codd”
Create secondary index by SQL statement
CREATE INDEX LNIndex on Books (lastname)’

20. Question?
Can a secondary index be sparse?

21. B-Trees
B-Trees are the most commonly used indexing structure in commercial systems.
Several variants are available, but the most popular is called the B+ tree.
Roughly speaking
Ord Array (search: O(Log(n)), update:O(n))
Linked List (search:O(n), update: O(1))
Tree (search: Log(n), update:Log(n))

22. Structure of B-Tree Organizes its blocks into a tree
Tree is Balanced: all paths from leaf to root have the same length
There are three types of nodes
Root,
Interior
Leaf
Associated with each tree is a layout parameter n (n search keys, n+1 pointers)

23. Example
Interior Node
Leaf Node 20 31 52 57 81 95
Next leaf
in sequence
To keys
K>=52
To record
with key 57
To keys
K < 20
To keys
31 <= K < 52
To record
with key 81
To record
with key 95
To keys
20 <= K < 31
Suppose block size is 4096 bytes. Each key is 4 bytes and pointer is 8 bytes.
Want 4*n + 8*(n+1) <= 4096; n = 340

24. Example 13 7 23 31 43 2 3 5 11 17 19 29 37 41 47

25. 17 7 - 37 43 2 3 5 7 13 13 17 23 23 23 23 37 41 43 47

26. Efficiency of B-Trees Earlier we saw up to 340 key-pointer pairs.
Assume average block has half-occupancy =~ 255
1 root block, 255 child nodes and 255*255 leaf nodes.
In the leaf we will have 2553 = 16.6 million records
Thus upto 4 I/O to access any record!

27. B-Tree Animation
Go to to see B-Tree animation.