1. 308-203A Introduction to Computing II Lecture 11: Hashtables Fall Session 2000

2. Dictionary An Abstract class which defines data-structures which support: void put(Object key, Object value) Object get(Object key) void remove(Object key)

3. Implementations for Dictionary? If we use an unsorted linked list: putO( 1 ) getO( n ) removeO( n ) Naïve solution: must search all possibilities

4. Implementations for Dictionary? If we use a binary tree (assume depth = d): putO( 1og d ) getO( log d ) removeO( log d ) Good, unless the tree is unbalanced…

5. Implementations for Dictionary? If we use a heap: putO( log n ) getO( n ) removeO( n ) Insert is easy, but finding arbitrary elements is hard…

6. Implementations for Dictionary? If we use a sorted array: putO(n ) getO( log n ) removeO( n ) Binary search is easy, but lots of copying is needed

7. Implementations for Dictionary? If we use an array with enough space for every possible key (not realistic): putO( 1 ) getO( 1 ) removeO( 1 ) All operations are quick and easy, but requires enormous (i.e. infinite) memory

8. Hashtables We can try to patch this “perfect solution” so that it is feasible.

9. The “Perfect” Solution If we had an array that was infinitely large and each key had it’s own slot, every access would be O( 1 ) [ and we would waste a lot of space on null pointers] 1234 j-1jj+1j+ 2 Key = 3 Key = j    … …

10. Hash Function Definition: A hash function is a function which maps keys to a finite range of integers, called hashcodes: f: keys [ 0, (m-1) ]

11. Example Let the keys be non-negative integers: { 0, 1, … } Let the hash function be f(x) = x mod 7 For the keys (4, 15, 26): f(4) = 4 f(15) = 1 f(26) = 5

12. Example Let the keys be non-negative integers: { 0, 1, … } Let the hash function be f(x) = x mod 7 For the keys (4, 15, 26): f(4) = 4 f(15) = 1 f(26) = 5 426 Fits in an array of size 7 15 0123456

13. Collisions Problem: When two or more keys hash to the same slot, there is a possiblity of collision.

14. Open-Addressing A simple way to handle collisions When a collision occurs look for an empty slot elsewhere Some elements may end up in the slot corresponding a different hashcode

15. Linear Probing Find an alternative slot after collision by stepping sequentially through the slots, for example: 42615 0123456 Insert 18 : f(18) = 18 mod 7 = 4 18 Collision in slot 4!

16. Linear Probing Find an alternative slot after collision by stepping sequentially through the slots, for example 42615 0123456 Insert 18 : f(18) = 18 mod 7 = 4 18 Slot 5 is also taken

17. Linear Probing Find an alternative slot after collision by stepping sequentially through the slots, for example 42615 0123456 Insert 18 : f(18) = 18 mod 7 = 4 18 Slot 6 is free

18. Disadvantages In open-addressing, the table can fill up; Must have (n < m) Linear-probing leads to “primary clustering:” A run of filled slots is more likely to receive more collisions Although best-case access is O( 1 ), worst-case access O( m )

19. Chaining A (Better) Solution to Collisions: Use the flexibility of the linked-list, but only when needed, i.e. within a single slot where collisions may occur.

20. Example (chaining) 0123456 15 426 Insert 39 into the previous hashtable:

21. Example (chaining) 0123456 15 426 f(39) = 39 mod 7 = 4collision 39

22. Worst-Case If all elements hash to the same entry we get a linked list: Therefore put, get and remove are O(n) worst-case.

23. Best-Case 0123456 Equal distribution to each slot

24. Best Case Definition: The load factor for a hashtable with n elements hashed into m slots is the average number of elements per slot:  = n / m

25. Best Case If every slot contains  elements (uniformly distributed hashing): put, get and remove are O(  )

26. Best Case If every slot contains  elements (uniformly distributed hashing): put, get and remove are O(  ) If the number of slots is allowed to grow as O( n ) :  = n/m = n /O( n ) = O( 1 ) put, get and remove are O( 1 )

27. Average-Case More realistic analysis involves determination of statistics of the data and how well it will be hashed. Example: hashing olympic years by f(x) = x mod 4 would be a bad idea (always hash to the same slot)

28. Java Hashtable Class Constructor: Hashtable(int initialCapacity, float loadFactor) Default: initialCapacity = 101, loadFactor = 0.75f Collision resolution with chaining

29. Java Hashtable Class hashcode(): defined in java.lang.Object equals(): assumed defined for the entries Keys can be objects of any class provided the following is appropriately defined:

30. Java Hashtable Class Hashtables grow multiplicatively: Put() checks if the hashtable contains more than (m  ) elements and if so m  2m+1 Hashtables only grow, never shrink, no matter how many elements you delete

31. Java Hashtable Class Other Features: elements() returns an enumeration of everything in the table. This works by keeping references into the table rather than by copying the table itself.

32. Any questions?