1. CS121 Data Structures CS121 © JAS 2004 Tables An abstract table, T, contains table entries that are either empty, or pairs of the form (K, I) where K is a key and I is some information associated with key K.

2. CS121 Data Structures CS121 © JAS 2004 The operations we may require are: 1.Initialise the table, T, to be the empty table. The empty table can be seen as filled with entries (K o, I o ) where K o is a special empty key, distinct from all other nonempty keys. 2.Determine whether or not the table, T, is full. 3.Insert a new entry (K,I), into the table, T, provided T is not already full. 4.Delete the table entry (K,I) from the table T. 5.Given, K, retrieve the information, I, from the entry (K,I) 6.Update the table entry (K,I) in table, T, by replacing it with a new table entry (K,I') 7.Enumerate the table entries (K,I) in table, T, in increasing order of their keys, K.

3. CS121 Data Structures CS121 © JAS 2004 Implementing a Table using a (Binary Search) Tree Initialise – create empty tree Empty – test for empty tree (root is nil) Full – tree is full only when memory exhausted Insertion – add entry to tree Deletion – delete a node (and reshape tree) Retrieve – search tree Update – search for entry and modify it Enumerate – traverse tree in appropriate order Initialise, Empty – O(1) Insert, Delete, Retrieve, Update – O(logn) Enumerate – O(n)

4. CS121 Data Structures CS121 © JAS 2004 Hash Tables Principle – calculate a value (hash value) from the key to determine the position of the entry in the table If the key field is alphabetic a simple hash function is to associate an integer value with each letter; sum them and return a value modulo the table size

5. CS121 Data Structures CS121 © JAS 2004 Associate the value 1 with a, 2 with b, and so on up to 26 with z Let the table size be 10 (indexes 0..9) cat 3+1+20 = 24 mod 10 = 4 cat dog 4+15+7 = 26 mod 10 = 6 test 20+5+19+20 = 64 mod 10 = 4 dog test

6. CS121 Data Structures CS121 © JAS 2004 Unless our table size is such that it is big enough to contain all possible entries we will get collisions

7. CS121 Data Structures CS121 © JAS 2004 Probability of Collisions von Mises Birthday Paradox for 23 or more people, there is a greater than 50% chance that two or more will have the same birthday. for 88 or more – three or more will have the same birthday. Hence for a 365 entry table (based on birthdays), after 23 insertions there will be a greater than 50% chance that an insertion will result in a collision. This is counter-intuitive because the table is in fact only 23/365 = 6.3% full. This is known as the load factor.

8. CS121 Data Structures CS121 © JAS 2004 The Theory For M people – probability same birthday P(M) – probability they don’t Q(M). Q(M) = 1 – P(M). For M = 1 Q(1) = 1 (nobody to share with!) For M = 2 Q(2) = 364/365 (only 364 days not the other person’s birthday) For M = 3 Q(3) = (364x363) / (365 2 ) For M > 1Q(M) = (364x353x…x(366-M)) / (365 (M-1) ) P(22) = 0.476 P(23) = 0.507

9. CS121 Data Structures CS121 © JAS 2004 When a collision occurs we need to find an alternative space for the entry The simplest method is a linear search – look forward for the next empty space – open addressing

10. CS121 Data Structures CS121 © JAS 2004 Primary clustering

11. CS121 Data Structures CS121 © JAS 2004 Solution to clustering is to use a different search path for different key values - double hashing If the initial hash function is based on remainder then a typical second hash would be based on the quotient if a <> b and a mod t = b mod t then a div t <> b div t

12. CS121 Data Structures CS121 © JAS 2004 cat 3+1+20 = 24 mod 10 = 4 cat dog 4+15+7 = 26 mod 10 = 6 dogtest Recall our initial example and let the rehash be divide by table size test 20+5+19+20 = 64 mod 10 = 4 div 10 = 6

13. CS121 Data Structures CS121 © JAS 2004 The same principles of hashing and collision resolution are used for both insertion and retrieval/update. The series of entries that are examined to see if they are empty is called the probe sequence. Insertion terminates when we find an empty entry to insert into. Retrieval/update terminates when either we find the entry we are looking for, or we have searched the whole table.

14. CS121 Data Structures CS121 © JAS 2004 Need to ensure that all possible entries are probed. Obvious for a linear probe sequence (increment by 1). For a double hash probe sequence it is less obvious unless the table size is a prime number (or table is even and probe step size is odd). To check whole table has been searched we can count the number of probes. Alternative is define a full table as one with one empty space. A search thus ends when an empty space has been found. This requires the insertion routine to know that it doesn’t fill the last space.

15. CS121 Data Structures CS121 © JAS 2004 Some code:- Hashing by Open Addressing public class OpenAddress { private Object[] Elt; private int Tablesize; private Object EmptyCell;

16. CS121 Data Structures CS121 © JAS 2004 OpenAddress(int tablesize) { int index; Tablesize = tablesize; EmptyCell = null; Elt = new Object[tablesize]; for (index=0;index<tablesize;++index) Elt[index] = EmptyCell; }

17. CS121 Data Structures CS121 © JAS 2004 public void Store (Object newElt) throws OAException { int index = Hash (newElt.key); probe = Hash2 (newElt.key); for(int cnt = 1; cnt < Tablesize; cnt++) { if (Elt[index].equals(EmptyCell)) {Elt[index] = newElt; return;} else if (Elt[index].key.equals(newElt.key)) throw new OAException(“already in”); else index = (index+probe)%Tablesize; } throw new OAException(“table full”); }

18. CS121 Data Structures CS121 © JAS 2004 public Object Retrieve(Object SearchKey) throws OAException { int index = Hash(Searchkey); probe = Hash2(Searchkey); for (int cnt = 1; cnt < Tablesize; cnt++0 { if (Elt[index].equals(EmptyCell)) throw new OAException(“not in table”); else if (Elt[index].key.equals(SearchKey)) return Elt[index]; else index = (index + probe)%Tablesize; } throw new OAException(“not in table”); }

19. CS121 Data Structures CS121 © JAS 2004 cat dog testcat dog test Alternative approach is to use chaining

20. CS121 Data Structures CS121 © JAS 2004 Operations on a table Initialisation Chaining – set chain pointers to nil – O(n) Open addressing – set keys to empty key – O(n) Insert, Retrieval, Update Chaining – use hash function, then search linked list – O(s) where s is average list length Open addressing – use hash functions to find entry – O(k) where k is some constant based on complexity of hash functions

21. CS121 Data Structures CS121 © JAS 2004 Deletion Chaining – delete entry from list – O(s) Open addressing – must mark entry has having been deleted to avoid stopping search early – O(k) Enumeration Not normally feasible

22. CS121 Data Structures CS121 © JAS 2004 Design of Hash functions – step one generate a key value – preconditioning (1) Using codes for letters – a=1 asciiposn cat 3+1+20=2499+97+116=3123*128 2 +2*128 1 +20*128 0 =49428 tac 20+1+3=24116+97+99=31220*128 2 +2*128 1 +3*128 0 =327939

23. CS121 Data Structures CS121 © JAS 2004 (2) Using bit patterns cat 1100011 1100001 1110100 = 1634548 tac 1110100 1100001 1100011 = 1913059 If the bit patterns get too long then a section of the bit pattern can be selected to use for the hash function.

24. CS121 Data Structures CS121 © JAS 2004 Step two – generate an address from the key value Division by table size Folding – The key is divided into sections, and the sections are added together. – if Key = 013402122 ; hash(Key) = 013+402+122 = 537 –addition, subtraction and multiplication could be used Midsquare – square the key and select a number of bits from the middle –if Key = 12345; Key 2 = 152399025; select middle – 399 Truncate – select last n digits –if Key = 13402122; select last 3 digits – 122

25. CS121 Data Structures CS121 © JAS 2004 Hash Table Reordering When a hash table is nearly full many items are not at the locations given by their hash address so many comparisons are made to locate them, and if an item is not present an entire list of rehash positions must be searched. To address this problem the table can be reordered.

26. CS121 Data Structures CS121 © JAS 2004 Ordered Hash Tables (Amble and Knuth) All items that hash to the same address are maintained in descending order of the key. Assume that a nil entry is less than all possible keys. A search can then terminate whenever a key less than that being searched for is found. On insertion when a collision occurs the keys are compared and the rehash/search for free space process continues with the smaller key.

27. CS121 Data Structures CS121 © JAS 2004 012345678012345678 H(cat)=(3+1+20)/9=24/9=6; rh=2 H(act)=(1+3+20)/9=24/9=6; rh=2 H(tac)=(20+1+3)/9=24/9=6; rh=2 cat tac act

28. CS121 Data Structures CS121 © JAS 2004 Ordered Hash Tables reduce the average number of probes to determine an entry is in the table. However, the number of probes for insertion is not reduced and equals the number required for an unsuccessful search in an unordered table. Also ordered tables require significant data movement.

29. CS121 Data Structures CS121 © JAS 2004 Brent’s Method Works with double hashing. Rehash search key until empty slot found. Then consider all keys in rehash path and determine if placing one of these in an empty slot would require fewer rehashes. If so place search key in this position and move existing key to empty rehash slot.

30. CS121 Data Structures CS121 © JAS 2004 012345678012345678

31. CS121 Data Structures CS121 © JAS 2004 Brent’s method reduces the average number of probes for a successful search, but not for an unsuccessful search. Insertion is also more complex. Brent’s method can be applied recursively – the displaced item is again inserted using Brent’s method (checking keys on rehash path for possible replacement – but if carried to extreme would yield unacceptably long insertion times.

32. CS121 Data Structures CS121 © JAS 2004 Consider a hash function which used the initial letter of the key word and inserted into a table of size 26 If collisions are resolved by chaining we have an index table – indexed by initial letter If the chains are replaced by the same hash structure using the second letter of the key word as the hash function - we have a 26-ary tree

33. CS121 Data Structures CS121 © JAS 2004 With an index table (or a 26-ary tree!) we regain the property that we can list the entries in sorted form However, it is also possible to combine hash tables, trees and index tables EgUse an index table according to first letter, then use hash tables for each letter sub-table

34. CS121 Data Structures CS121 © JAS 2004