1. 1 Designing Hash Tables Sections 5.3, 5.4, 5.5

2. 2 Designing a hash table 1.Hash function: establishing a key with an indexed location in a hash table –E.g. Index = hash(key) % table_size; 2.Resolve conflicts: –Need to handle case where multiple keys mapped to the same index. –Two representative solutions Chaining with separate lists Probing open addressing

3. 3 Separate Chaining Each table entry stores a list of items Multiple keys mapped to the same entry maintained by the list Example –Hash(k) = k mod 10 –(10 is not a prime, just for illustration)

4. 4 Separate Chaining Implementation Type Declaration for Separate Chaining Hash Table

5. 5 HashedObj Needs to provide –Hash function Provided for string and int (the two non-member functions) –Equality operators ( operator== or operator!= )

6. 6 An example class for HashedObj

7. 7 Chaining

8. 8 Chaining (contd.)

9. 9

10. 10 Analysis of Chaining Consider an array of size M with N records –Worst case insert without uniqueness check = O(1) Find location to insert and push_back/front –Worst case remove/find/unique insert = O(N) –Expected case unique insert/find/remove 1 + O(N/M) –Let us resize the table is N/M exceeds some constant –Expected time = 1 + O( ) = O(1)

11. 11 Hash Tables Without Chaining Try to avoid buckets with separate lists How use Probing Hash Tables –If collision occurs, try another cell in the hash table. –More formally, try cells h 0 (x), h 1 (x), h 2 (x), h 3 (x)… in succession until a free cell is found. h i (x) = hash(x) + f(i) And f(0) = 0

12. 12 f(i)=i Insert (assume no duplicated keys) 1.Index = hash(key) % table_size; 2.If table[index] is empty, put information (key and others) in entry table[index]. 3.If table[index] is not empty then Index ++; index = index % table_size; goto 2. Search (key) 1.Index = hash(key) % table_size; 2.If (table[index] is empty) return –1 (not found). 3.Else if (table[index].key == key) return index; 4.Index ++; index = index % table_size; goto 2. Linear Probing

13. 13 Example Insert 89, 18, 49, 58, 69 (hash(k) = k mod 10)

14. 14 Linear probing Delete –Can be tricky, must maintain the consistency of the hash table. –What is the simplest deletion strategy you can think of??

15. 15 Quadratic Probing f(i) = i 2

16. 16 Probing strategy hash table

17. 17 Double Hashing f(i) = i*hash 2 (x) E.g. hash 2 (x) = 7 – (x % 7) What if hash 2 (x) = 0 for some x?

18. 18 Analysis of Hash Table Without Chaining Expected case analysis of insertion into a table of size M containing n records – i=1 i probability of trying i buckets –= 1 (M-n)/M + 2(n/M)(M-n)/M + 3(n/M) 2 (M-n)/M +... Let = n/M –Time = 1*(1- ) + 2 (1- ) + 3 2 (1- ) +... –= 1 - + 2 - 2 2 + 3 2 - 3 3 + 4 3 - 4 4... –= 1 + + 2 + 3 +... = 1/(1- ) Assume < 1 –Keep bounded by some constant < 1

19. 19 Rehashing Hash Table may get full –No more insertions possible Hash table may get too full –Insertions, deletions, search take longer time Solution: Rehash –Build another table that is twice as big and has a new hash function –Move all elements from smaller table to bigger table Cost of Rehashing = O(N) –But happens only when table is close to full –Close to full = table is X percent full, where X is a tunable parameter

20. 20 Rehashing Example After Rehashing Original Hash Table After Inserting 23

21. 21 Rehashing Implementation

22. 22 Rehashing implementation