Hash Tables• Hash function h: search key

[0…B-1]. • Buckets are blocks,

numbered [0…B-1]. • Big idea: If a record with

search key K exists, then it must be in bucket h(K). - Cuts search down by a

factor of B. - One disk I/O if there is

only one block per bucket.

Hash Table Lookup: For record(s) with search key K, compute h(K); search that bucket.

Hash Table Insertion• Put in bucket h(K) if it fits; otherwise create an overflow block.

- Overflow block(s) are part of bucket. Example: Insert record with search key g.

What if the File Grows too Large?• Efficiency is highest if

#records < #buckets #(records/block) • If file grows, we need a dynamic hashing method to maintain

the above relationship. - Extensible Hashing: double the number of buckets when

needed. - Linear hashing: add one more bucket as appropriate.

Dynamic Hashing Framework• Hash function h produces a sequence of k bits. • Only some of the bits are used at any time to determine

placement of keys in buckets.

Extensible Hashing (Buckets may share blocks!)• Keep parameter i = number of bits from the beginning of h(K)

that determine the bucket. • Bucket array now = pointers to buckets.

- A block can serve as several buckets. - For each block, a parameter ji tells how many bits of

h(K) determine membership in the block. - I.e., a block represents 2i-j buckets that share the first j bits

of their number.

Example• An extensible hash table when i=1:

Extensible Hash table Insert• If record with key K fits in the block pointed to by h(K), put it

there. • If not, let this block B represent j bits.

1. j<i: a) Split block B into two and distribute the records (of B)

according to (j+1)st bit; b) set j:=j+1; c) fix pointers in bucket array, so that entries that formerly

pointed to B now point either to B or the new block How? depending on…(j+1)st bit

2. j=i: 1. Set i:=i+1; 2. Double the bucket array,

so it has now 2i+1 entries; 3. proceed as in (1).

Let w be an old array entry. Both the new entries w0 and w1 point to the same block that w used to point to.

Example• Insert record with h(K) = 1010.

Before

Now, after the insertion

Example: Next

• Next: records with h(K)=0000; h(K)=0111. - Bucket for 0... gets split, - but i stays at 2.

• Then: record with h(K) = 1000. - Overflows bucket for 10... - Raise i to 3.

After the insertions

Currently

Extensible Hash Tables:Advantages:• Lookup; never search more than one data block.

- Hope that the bucket array fits in main memory

Defects:• Substantial amount of work to double the bucket array

- Interrupts access to data file- Makes certain insertions appear to take very long

• Doubling the bucket array soon is going to make the array to not fit in main memory.

• Problem with skewed key distributions. - E.g. Let 1 block=2 records. Suppose that three records have

hash values, which happen to be the same in the first 20 bits.- In that case we would have i=20 and and one million bucket-

array entries, even though we have only 3 records!!

Linear Hashing• Use i bits from right (low order) end of h(K). • Buckets numbered [0…n-1], where 2i-1<n2i.

• Let last i bits of h(K) be m = (a1,a2,…,ai)

1. If m < n, then record belongs in bucket m.

2. If nm<2i, then record belongs in bucket m-2i-1, that is the bucket we would get if we changed a1 (which must be 1) to 0.

i=1

n=2

r=3

This is also part of the structure

#of records

#of buckets

Linear Hash Table Insert• Pick an upper limit on capacity,

- e.g., 85% (1.7 records/bucket in our example). • If an insertion exceeds capacity limit, set n := n + 1.

- If new n is 2i + 1, set i := i + 1. No change in bucket numbers needed --- just imagine a leading 0.

- Need to split bucket n - 2i-1 because there is now a bucket numbered (old) n.

Example• Insert record with h(K) = 0101.

- Capacity limit exceeded; increment n.

r=3

n=2

i=1

#of records

#of buckets

r=4

n=3

i=2

#of records

#of buckets

Example • Insert record with h(K) = 0001.

- Capacity limit not exceeded. - But bucket is full; add overflow bucket.

r=5

n=3

i=2

Example• Insert record with h(K) = 1100.

- Capacity exceeded; set n = 4, add bucket 11. - Split bucket 01.

r=7

n=4

i=2

Lookup in Linear Hash Table• For record(s) with search key K, compute h(K); search the

corresponding bucket according to the procedure described for insertion.

• If the record we wish to look up isn’t there, it can’t be anywhere else.

• E.g. lookup for a key which hashes to 1010, and then for a key which hashes to 1011.

r=4

n=3

i=2

Exercise• Suppose we want to insert keys with hash values: 0000…

1111 in a linear hash table with 100% capacity threshold. • Assume that a block can hold three records.