Hash Tables

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 2i-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. • j<i: • Split block B into two and distribute the records (of B) according to (j+1)st bit; • set j:=j+1; • 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 • j=i: • Set i:=i+1; • Double the bucket array, so it has now 2i+1 entries; • proceed as in (1). Letwbe an old array entry. Both the new entries,w0andw1,point to the same block that w used to point to.

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

8. Currently After the insertions 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.

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 2i-1<n2i. • Let last i bits of h(K) be m = (a1,a2,…,ai) • If m < n, then record belongs to bucket m. • If nm<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. #of buckets #of records This is also part of the structure

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 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.

12. i=1 i=1 n=1 0000 0 0 n=1 r=0 r=1 Example • Insert records with h(K) = 0000, 1010, 1111, 0101, 0001, 1100. Before After

13. i=1 i=1 0000 0000 1 0 0 n=1 n=2 1010 r=1 r=2 Example • Insert records with h(K) = 0000, 1010, 1111, 0101, 0001, 1100. Before After Capacity limit exceeded; increment n

14. i=1 i=1 0000 1111 0000 1 0 1 0 n=2 n=2 1010 1010 r=3 r=2 Example • Insert records with h(K) = 0000, 1010, 1111, 0101, 0001, 1100. Before After

15. i=2 i=1 1111 0000 0000 1010 00 1 10 0 n=3 n=2 1010 1111 01 r=4 r=3 0101 Example • Insert records with h(K) = 0000, 1010, 1111, 0101, 0001, 1100. Before After Capacity limit exceeded; increment n, which causes incrementing i as well.

16. i=2 i=2 1010 0000 1010 0000 10 00 10 00 n=3 n=3 1111 1111 01 01 r=4 r=5 0101 0101 0001 Example • Insert records with h(K) = 0000, 1010, 1111, 0101, 0001, 1100. Before After As long as capacity is not exceeded can add overflow blocks.

17. 0000 00 i=2 i=2 1111 0000 1010 1010 1100 11 00 10 10 n=3 n=4 1111 0001 01 01 r=5 r=6 0101 0101 0001 Example • Insert records with h(K) = 0000, 1010, 1111, 0101, 0001, 1100. Before After Capacity limit exceeded; increment n.

18. 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. i=2 n=3 r=4

19. 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.