The document discusses hashing techniques including: 1. Hashing keys to the same address is called synonyms and solutions include spreading out records or storing multiple records at one address. 2. Hashing algorithms work by taking a key, performing a calculation on it like taking the modulus of the size of the address space, and using that as the address. 3. Extendible hashing uses a trie structure and directory to dynamically expand the hash table as needed to accommodate more records without rehashing the entire table.

3. • Keys hashed to same address is called as synonyms
Solution:
• Spread out the records
• Use extra memory
• Put more than 1 records at a single address.

4. A simple Hashing Algorithm
Step1:

5. • Step 3: divide the size of address space
a= s mod n
A=13883 mod 101
A=83

7. Hashing Function and Record Distributions

8. Some other Hashing Methods
• Examine keys for a pattern
• Fold parts of the key
• Divide the key by a number
• Square the key and take the middle.
Key=453 square= 205209 so addr=52
• Radix transformation
Key= 453 base equivalent=382 addr= 382%99=85

9. Predicting the Distribution of Records
Poisson Distribution
• It helps to predict the probability of collisions likely to occur in a file.
• Using Poisson function,
 It predicts no. of collisions occur
 Consider, address in file is N
 It gives 2 outcomes, when a key is hashed
1. A- address not chosen : p(A) termed as a
2. B address chosen: p(B) termed as b
If N=10,
b=1/10=0.1
a=1-0.1= 0.9

10. How Much Extra Memory Should Be Used?
• Poisson function applied to hashing
• To reduce the collisions, use extra memory.
• Packing Density gives the measure of amount of space in a file is used.
• If there are 75 records and 100 addresses than packing density is 75/100=0.75=75%

11. • Predicting collision for different packing densities
• Suppose that 1000 addresses are allocated to hold 500 records in a randomly
hashed file, and that each address can hold one record. The packing density
for the file is
Let us answer the following questions about the distribution of records among
the available addresses in the file:
• How many addresses should have no records assigned to them?
• How many addresses should have exactly one record assigned (no synonyms)?
·. ·
• How many addresses should have one record plus· one or more synonyms? - -
• Assuming that only one record can be assigned to each home address, how
many overflow records an be expected?
• What percentage of records should be overflow records?

12. Progressive overflow

13. Extendable Hashing
• Direct access (hashing) files have static size, so not suitable for files whose size is
unknown in advance
• Dynamic file structure is desired which retains the feature of fast retrieval by primary key,
and which also expands and contracts as the number of records in the file fluctuates
(without reorganizing the whole file)
• Similar motivation!
• Indexed-sequential File ==> B tree
• Hashing ==> Extendible Hashing

14. • How Extendible Hashing works
• Idea from Tries file (radix searching)
• The branching factor of the tree is equal to the # of alternative symbols in
each position of the key
e.g.) Radix 26 trie - able, abrahms, adams, anderson, adnrews, baird
• Use the first n characters for branching

16. Turning the trie into a directory
• Using Trie for extendible hashing
(1) Use Radix 2 Trie :
Keys in A : beginning with 0
Keys in B : beginning with 10
Keys in C : beginning with 11
(2) Retrieving from secondary storage the buckets containing keys,
instead of individual keys

17. Representation of Ties
• Rather than representing tries as a tree , it will be represented as array of
contiguous records forming a
• Directory of hash key and
• pointers to the corresponding buckets.

18. Extendible Hashing Example
000
001
010
011
100
101
110
111
d’=1
d’=3
d’=3
d’=2
B0
B100
B101
B11
H(key)=0
H(key)=100
H(key)=101
H(key)=11
d=3

19. Splitting handle overflow
• When overflow occurs
e.g.1) Overflowing of bucket A
• Split A into A and D
• Come to use additional unused bits
• No need to expand the directory

20. • Overflowing of bucket B
• Do not have additional unused bits
(need to expand the directory)
1. Divide B using 3 bits of hash address
2. Make a complete full binary tree
3. Collapse it into the directory structure

22. Dynamic Hashing
• Similar to dynamic extendible hashing
• Use a directory to track bucket addresses
• Extend the directory through the use of tries
• Start with a hash function that covers an address space of a fixed size
• When overflow occurs
• splits forming the leaves of a trie that grows down from the original
address node makes a trie
Two kinds of nodes
 External node: reference a data bucket
 Internal node: point to two children index nodes
 When a node has split children, it changed from an external node to an internal nod

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