Slideshow:
The hash function used in Extensible Hashing
Comment of the terminology
Logical and Physical buckets
The traditional hashing technique
Example:
E.g.: Logical bucket # = physical bucket #
The Extensible Hashing technique
Example:
Important property achieved by the
logical → physical bucket mapping
Important property achieved by the
logical → physical bucket mapping
In other words:
Example Extensible Hashing
Example Extensible Hashing
Example:
Example Extensible Hashing
Notice that:
Hash values that ends in
000 and
100
are stored in physical bucket 00
Hash values that ends in
001 and
101
are stored in physical bucket 01
Hash values that ends in
010 and
110
are stored in physical bucket 10
Hash values that ends in
011 and
111
are stored in physical bucket 11
Example Extensible Hashing
Example Extensible Hashing
Result: the logical hash table size will increase
Example Extensible Hashing
We can "fix" the mapping errors in the new hash function using a modified mapping:
Example Extensible Hashing
Verify that:
Handling overflow in Extensible Hashing by
increasing physical bucket size
Overflow situation:
Handling overflow in Extensible Hashing by
increasing physical bucket size
Overflow situation:
Handling overflow in Extensible Hashing by
increasing physical bucket size
Allocate a new disk block:
Handling overflow in Extensible Hashing by
increasing physical bucket size
Re-hash the keys in the overflow block:
Handling overflow in Extensible Hashing by
increasing physical bucket size
Adjust logical → physical mapping:
Increasing physical bucket size:
a concrete example
Initial state:
Increasing physical bucket size:
a concrete example
We can increase the physical bucket size by spliting a (physical) bucket
Increasing physical bucket size:
a concrete example
We first re-hash all the keys using an additional bit in the hash key:
Increasing physical bucket size:
a concrete example
Adjust the logical → physical mapping:
Increasing physical bucket size:
a concrete example
Final state: now we have 5 physical buckets
- Hash function used
in traditional hash techniques:
Hash( key ) = RandomNum( key ) % B = "Hash value" % Bwhere:
- B =
number of
hash buckets
B is usually a prime number (so that Hash(key) is uniformly distributed)
- B is fixed !!!
- B =
number of
hash buckets
- Hash function used
in extensible hashing:
Hash( key ) = RandomNum( key ) % 2i "Hash value" % 2iGraphically:
In other words:
Hash( key ) = last i-bits of the Hash-value
- Number of
hash buckets
is Extensible hashing:
- Number of
hash bucket B =
2i
- i can change !!!
- Number of
hash bucket B =
2i
- Comment:
- These concepts are not in the text book....
I introduced the concepts of
- Logical hash bucket
- Physical hash bucket
- I think this make the material easier to understand....
- These concepts are not in the text book....
- Physical hash bucket:
- Physical hash bucket = a disk block used to store the index entries
- Logical hash bucket:
- Logical hash bucket = a entry in the block number array
Graphically:
- In "normal" hashing:
- The number of
logical hash bucket
is equal to the
number of
physical hash bucket
and
- Every logical hash bucket is mapped to a different physical hash bucket
Graphically:
- The number of
logical hash bucket
is equal to the
number of
physical hash bucket
and
- In Extensible hashing:
- The number of
logical hash bucket
is
≥ the
number of
physical hash bucket
and
- Every logical hash bucket
is mapped to
a physical hash bucket
- Multiple logical hash buckets can be mapped to the same physical hash bucket !!!
Graphically:
- The number of
logical hash bucket
is
≥ the
number of
physical hash bucket
and
- Important property
of Extensive hashing:
- In Extensive hashing,
we can double the
size of the
logical hash table
without having
to re-hash
any keys !!!
In other words:
- In Extensive hashing, we can double the range of the hash function without having to re-hash any keys !!!
- Meaning:
- We can change the
hash function from:
Hash( key ) = RandomNum( key ) % 2i "Hash value" % 2i
to a new hash function:
Hash( key ) = RandomNum( key ) % 2i+1 "Hash value" % 2i+1
without having to re-hash any search key !!!!
- We can change the
hash function from:
- In Extensive hashing,
we can double the
size of the
logical hash table
without having
to re-hash
any keys !!!
- Example:
- Consider the
following example using
extensive hashing
where the hash function uses the
last 2 bits of the
hash value:
Hash( K ) = RandomNum( K ) % 4Graphically:
Notice that:
- Hash values that ends in 000 and 100 are stored in physical bucket 00
- Hash values that ends in 001 and 101 are stored in physical bucket 01
- Hash values that ends in 010 and 110 are stored in physical bucket 10
- Hash values that
ends in
011
and 111
are stored in
physical bucket 11
- Suppose we
double the
size of the
logical hash table by
using a different hash function:
Hash( K ) = RandomNum( K ) % 8Graphically:
- We can change the
hash function
(using B = 8)
without
having to
re-hash the
data records by
mapping the
new index buckets
as follows:
Notice that:
- Hash values that ends in 000 and 100 are directed to physical bucket 00
- Hash values that ends in 001 and 101 are directed to physical bucket 01
- Hash values that ends in 010 and 110 are directed to physical bucket 10
- Hash values that ends in 011 and 111 are directed to physical bucket 11
- Consider the
following example using
extensive hashing
where the hash function uses the
last 2 bits of the
hash value:
- Important observation:
- We can
locate
all the index records using
the
new hash function:
Hash( K ) = RandomNum( K ) % 8Example:
- Before we
increase the
logical hash table, we
use the last 2 bits in
the hash value to
find the logical hash bucket:
The last 3 bits that ends in 00 can be 000 or 100
The last 3 bits that ends in 01 can be 001 or 101
The last 3 bits that ends in 10 can be 010 or 110
The last 3 bits that ends in 11 can be 011 or 111
- We construct the
appropriate mapping to:
- 000 and 100 to the block that store hash key values ending in 00
- 001 and 101 to the block that store hash key values ending in 01
- 010 and 110 to the block that store hash key values ending in 10
- 011 and 111 to the block that store hash key values ending in 11
- We construct the
appropriate mapping to:
- After we
increase the
logical hash table, we
(must)
use the last 3 bits in
the hash value to
find the logical hash bucket:
The index record for every search key can still be found using the new ("larger") hash function !!
- Before we
increase the
logical hash table, we
use the last 2 bits in
the hash value to
find the logical hash bucket:
- We can
locate
all the index records using
the
new hash function:
- Conclusion:
- We can change the
hash function:
Hash( key ) = RandomNum( key ) % 2i "Hash value" % 2ito:
Hash( key ) = RandomNum( key ) % 2i+1 "Hash value" % 2i+1 (New Hash function will use a new B that is 2 × the old B)by:
- Double the
size of the
logical bucket buckets
- Map the
new
logical bucket buckets
as follows:
- Double the
size of the
logical bucket buckets
- We can change the
hash function:
- How to handle
overflow in
Extensive Hashing:
- When we
want to insert a
new search key K into
a full physical hash bucket:
We will get an overflow condition:
- To fix the
overflow condition,
we allocate
a new disk block (= physical bucket):
And re-hash (only) the keys in the overflow block (using one more bit):
We still need to "link" the new hash bucket to the logical hash table.....
- When we
want to insert a
new search key K into
a full physical hash bucket:
- Important reminder:
- In Extensive Hahsing,
it is possible that
multiple logical hash buckets
are mapped into
the same physical hash bucket
Example:
- Logical buckets 000 and 100 are mapped to the physical bucket 00
- Logical buckets 001 and 101 are mapped to the physical bucket 01
- Logical buckets 010 and 110 are mapped to the physical bucket 10
- Logical buckets 011 and 111 are mapped to the physical bucket 11
- In Extensive Hahsing,
it is possible that
multiple logical hash buckets
are mapped into
the same physical hash bucket
- Very important property of
Extensible Hashing:
- Extensive hashing
(using a map from
logical hash buckets to
physical hash buckets)
allow us to:
- Split a
physical bucket
by re-hash
only the
keys
in the
physical bucket (disk block)
without having
to rehash
any other buckets (disk blocks) !!!
- Link the new data block to the logical hash table by re-assigning the block pointers !!!
- Split a
physical bucket
by re-hash
only the
keys
in the
physical bucket (disk block)
without having
to rehash
any other buckets (disk blocks) !!!
- Extensive hashing
(using a map from
logical hash buckets to
physical hash buckets)
allow us to:
- Example:
- Consider the
physical bucket 01
The logical buckets 001 and 101 maps to physical bucket 01:
- We can split the
physical bucket 01
by:
- Re-hash
only the
key in the
physical bucket
using their last 3 bits:
-
Then
adjust the
logical ⇒ physical
hash bucket mapping:
Final state:
- Re-hash
only the
key in the
physical bucket
using their last 3 bits:
- Consider the
physical bucket 01