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

2. Hash­Table 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.

3. 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.

4. 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 j  i tells how many bits of h(K) determine membership in the block. - I.e., a block represents 2 i-j buckets that share the first j bits of their number.

5. Example An extensible hash table when i=1:

6. Extensible Hash­table 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 2 i+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.

7. Example Insert record with h(K) = 1010. Before Now, after the insertion

8. 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

9. 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!!

10. Linear Hashing Use i bits from right (low­order) end of h(K). Buckets numbered [0…n-1], where 2 i-1 <n  2 i. Let last i bits of h(K) be m = (a 1,a 2,…,a i ) 1.If m < n, then record belongs in bucket m. 2.If n  m<2 i, then record belongs in bucket m-2 i-1, that is the bucket we would get if we changed a 1 (which must be 1) to 0. i=1 n=2 r=3 This is also part of the structure #of records #of buckets

11. Linear Hash­Table 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 2 i + 1, set i := i + 1. No change in bucket numbers needed --- just imagine a leading 0. - Need to split bucket n - 2 i-1 because there is now a bucket numbered (old) n.

12. 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

13. Example Insert record with h(K) = 0001. - Capacity limit not exceeded. - But bucket is full; add overflow bucket. r=5 n=3 i=2

14. Example Insert record with h(K) = 1100. - Capacity exceeded; set n = 4, add bucket 11. - Split bucket 01. r=7 n=4 i=2

15. 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

16. 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.