1. 1 Hash table

2. 2 Objective To learn: Hash function Linear probing Quadratic probing Chained hash table

3. 3 A basic problem We have to store some records and perform the following:  add new record  delete record  search a record by key Find a way to do these efficiently!

4. 4 Unsorted array Use an array to store the records, in unsorted order  add - add the records as the last entry fast O(1)  delete a target - slow at finding the target, fast at filling the hole (just take the last entry) O(n)  search - sequential search slow O(n)

5. 5 Sorted array Use an array to store the records, keeping them in sorted order  add - insert the record in proper position. much record movement slow O(n)  delete a target - how to handle the hole after deletion? Much record movement slow O(n)  search - binary search fast O(log n)

6. 6 Linked list Store the records in a linked list (sorted / unsorted)  add - fast if one can insert node anywhere O(1)  delete a target - fast at disposing the node, but slow at finding the target O(n)  search - sequential search slow O(n) (if we only use linked list, we cannot use binary search even if the list is sorted.)

7. 7 Array as table 9903030 9802020 9801010 0056789 0012345 0033333 tom mary peter david andy betty 73 100 20 56.8 81.5 90 studidnamescore 9908080bill49... Consider this problem. We want to store 1000 student records and search them by student id.

8. 8 Array as table : 33333 : 12345 0 : : betty : andy : : 90 : 81.5 : studidnamescore 56789 david56.8 : 9908080 : : : bill : : : 49 : : 9999999 One ‘naive’ way is to store the records in a huge array (index 0..9999999). The index is used as the student id, i.e. the record of the student with studid 0012345 is stored at A[12345]

9. 9 Array as table Store the records in a huge array where the index corresponds to the key  add - very fast O(1)  delete - very fast O(1)  search - very fast O(1) But it wastes a lot of memory! Not feasible.

10. 10 Hash function function Hash(key: KeyType): integer; Imagine that we have such a magic function Hash. It maps the key (stud_id) of the 1000 records into the integers 0..999, one to one. No two different keys maps to the same number. H(‘0012345’) = 134 H(‘0033333’) = 67 H(‘0056789’) = 764 … H(‘9908080’) = 3

11. 11 Hash table : betty : bill : : 90 : 49 : studidnamescore andy81.5 : : david : : : 56.8 : : 0033333 : 9908080 : 0012345 : : 0056789 : 3 67 0 764 999 134 To store a record, we compute Hash(stud_id) for the record and store it at the location Hash(stud_id) of the array. To search for a student, we only need to peek at the location Hash(target stud_id).

12. 12 Hash table with Perfect Hash Such magic function is called perfect hash  add - very fast O(1)  delete - very fast O(1)  search - very fast O(1) But it is generally difficult to design perfect hash. (e.g. when the potential key space is large)

13. 13 Hash function A hash function maps a key to an index within in a range Desirable properties:  simple and quick to calculate  even distribution, avoid collision as much as possible function Hash(key: KeyType);

14. 14 Division Method Certain values of m may not be good:  Good values for m are prime numbers which are not close to exact powers of 2. For example, if you want to store 2000 elements then m=701 (m = hash table length) yields a hash function: h (k) = k mod m h (key) = k mod 701

15. 15 Collision For most cases, we cannot avoid collision Collision resolution - how to handle when two different keys map to the same index H(‘0012345’) = 134 H(‘0033333’) = 67 H(‘0056789’) = 764 … H(‘9903030’) = 3 H(‘9908080’) = 3

16. 16 The problem arises because we have two keys that hash in the same array entry, a collision. There are two ways to resolve collision:  Hashing with Chaining: every hash table entry contains a pointer to a linked list of keys that hash in the same entry  Hashing with Open Addressing: every hash table entry contains only one key. If a new key hashes to a table entry which is filled, systematically examine other table entries until you find one empty entry to place the new key Hash Tables

17. 17 Open Addressing The key is first mapped to a slot: If there is a collision subsequent probes are performed: If the offset constant, c and m are not relatively prime, we will not examine all the cells. Ex.:  Consider m=4 and c=2, then only every other slot is checked. When c=1 the collision resolution is done as a linear search. This is known as linear probing. 0 123

18. 18 Linear Probing example1 89 18 89 49 18 89 49 58 18 89 49 58 9 18 89 Insert: 89, 18, 49, 58, 9 to table size=10, hash function is: %tablesize 0 1 2 3 4 5 6 7 8 9

19. 19 Linear Probing Example-2 Single character keys, table size, m=8 Hash function (map characters to range 0...7): k: APQ BORCNS DMT ELU FKN GJWZ HIXY h 1 (k): 0 1 2 3 4 5 6 7 Action Store A Store C Store D Store G Store P Store Q Delete P Delete Q Load B Load R Load Q 0AAAAAAAAAAA0AAAAAAAAAAA 1PP±±BBB1PP±±BBB 2CCCCCCCCCC2CCCCCCCCCC 3DDDDDDDDD3DDDDDDDDD 4QQ±±RR4QQ±±RR 5Q5Q 6GGGGGGGG6GGGGGGGG # probes 1 2 5 2 5 1 4 6 7

20. 20 Choosing a Hash Function Notice that the insertion of Q required several probes (5). This was caused by A and P mapping to slot 0 which is beside the C and D keys. The performance of the hash table depends on a having a hash function which evenly distributes the keys.  The statistics of the key distribution needs to be accounted for. For example, choosing the first letter of a surname will cause problems depending on the nationality of the population; the variable names in a compiler often differ by one character, eg., t1, t2, t3, etc. Consult computer science texts, such as Knuth’s “The Art of Computer Programming”.

21. 21 Clustering Even with a good hash function, linear probing has its problems:  The position of the initial mapping i 0 of key k is called the home position of k.  When several insertions map to the same home position, they end up placed contiguously in the table. This collection of keys with the same home position is called a cluster.  As clusters grow, the probability that a key will map to the middle of a cluster increases, increasing the rate of the cluster’s growth. This tendency of linear probing to place items together is known as primary clustering.  As these clusters grow, they merge with other clusters forming even bigger clusters which grow even faster.

22. 22 Performance Analysis If n slots in a table of size m are occupied, the load factor is defined as where  =1 means the table is full, and  =0 means the table is empty. It can be shown that the number of probes in a successful search, C, and the number of probes in an unsuccessful search, C’ is given by:

23. 23 Quadratic Probing  h(k)=h(k) + f(i) ( i=0,1,2,…)%TS  h(k)=Rmod TS  f(i)=i 2 Theorem 20.4: If quadratic probing is used and the table size is prime, then a new element can always be inserted if the table is at least half empty. Furthermore, in the course of the insertion, no cell is probed twice.

24. 24 Quadratic probing-example 89 18 89 49 18 89 49 58 18 89 49 58 9 18 89 Insert: 89, 18, 49, 58, 9 to table size=10, hash function is: %tablesize 0 1 2 3 4 5 6 7 8 9

25. 25 Double Hashing Recall that in open addressing the sequence of probes follows We can solve the problem of primary clustering in linear probing by having the keys which map to the same home position use differing probe sequences. In other words, the different values for c should be used for different keys. Double hashing refers to the scheme of using another hash function for c Note that h 1 and h 2 need to be evaluated only once per key.

26. 26 Chained Hash Table 2 4 1 0 3 nil 5 : HASHMAX Key: 9903030 name: tom score: 73 One way to handle collision is to store the collided records in a linked list. The array now stores pointers to such lists. If no key maps to a certain hash value, that array entry points to nil.

27. 27 Chained Hash table Hash table, where collided records are stored in linked list  good hash function, appropriate hash size Few collisions. Add, delete, search very fast O(1)  otherwise… some hash value has a long list of collided records.. add - just insert at the head fast O(1) delete a target - delete from unsorted linked list slow search - sequential search slow O(n)

28. 28 Common errors (page 749) Providing a poor hash function Not rehashing when load factor reaches 0.5. More errors listed on page 749 of the book

29. 29 In class exercises 20.1, 20.2 and 20.5 in the book on page 750.