Hash Tables

1. Hash Tables

2. Hash Tables • Many applications require a dynamic set that only supports the dictionary operations: Insert, Delete, Search • A hash table is an efficient implementation of a dictionary • Worst case – same as linked list – O(n) • Under reasonable assumptions – O(1)

3. Direct-address Tables • Assumptions • The universe U of keys is reasonably small:U = { 0, 1, 2, …, m-1 }, for some small m • No two elements have the same key • Implementation • Allocate an array of size m • Insert the kth element into the kth slot in the array

4. Direct-address Tables

5. Direct-address Tables • Advantage • O(1) time for all operations • Disadvantages • Wasteful if the number of elements actually inserted is significantly smaller than the size of the universe (m) • Only applicable for small values of m,i.e. a limited range of keys

6. Hash Tables • Performance is almost similar to that of a direct-address table, but without the limitations • The universe U may be very large • The storage requirement is O(|K|), where K is the set of keys actually used • Disadvantage – O(1) performance is now average case, not worst case

7. Hash Tables • We need a hash function to map keys from the universe U into the hash tableh: U  { 0, 1, …, m-1 } • For each key k, the hash function computes a hash value h(k) • If two keys hash to the same value:h(k1) = h(k2), we call this a collision

8. Hash Tables

9. Collisions • Can we avoid collisions altogether? • No. Since |U| > m, some keys must have the same hash value • A good hash function will be as ‘random’ as possible • Still, collisions must be resolved

10. Collision Resolution • Chaining (also called open hash) • Elements stored in their ‘correct’ slot • Collisions resolved by creating linked lists • Open addressing (also calledclosed hash) • All elements stored inside the table • Maybe rehashed if their slot is full

11. Chaining – Open Hash

12. Collision resolution – Chaining • All keys that have the same hash value are placed in a linked list • Insertion can be done at the beginning of the list in O(1) time • Searching is proportional to the length of the list – with a good hash function, will also be O(1)

13. Hash Function Requirements • A hash function must be deterministic – the hash value generated for each key cannot change during the life of the hash table • Equal keys must always be mapped to the same hash value

14. Hash Function Properties • Properties of a good hash function • Easy to evaluate – h(x) can be computed very quickly (not only in O(1), but also with a small constant) • Uniform distribution over all the table slots • Different keys are mapped to different slots (as much as possible)

15. Simple Uniform Hashing • The quality of the hash function strongly influences the efficiencyof the hash table • Simple Uniform Hashing assumption: The hash function will hash any keyinto any slot with equal probability • It is possible to define hash functions that almost satisfy this assumption

16. Analysis • The load factor of a hash table is defined as the number of elements stored in the table, divided by the total number of slots: • A search will take under the assumption of simple uniform hashing • Therefore, all hash operations can be performed in O(1)

17. Computing Key Values • The first step is to represent the key as a natural integer number • For example if S is a string then we can interpret it as an integer value using the following formula:

18. The Division Method • Key k is mapped into one of m slots by taking the remainder of k divided by m: • Choosing the value of m • Preferably prime • Not too close to a power of 2

19. Example – the Division Method • Let h be a hash table of 9 slots andh(k) = k mod 9. insert the elements:6, 43, 23, 62, 1, 13, 34, 55, 25 h(6) = 6 mod 9 = 6 h(43) = 43 mod 9 = 7 h(23) = 23 mod 9 = 5 h(62) = 62 mod 9 = 8 h(1) = 1 mod 9 = 1 h(13) = 13 mod 9 = 4 h(34) = 34 mod 9 = 7 h(55) = 55 mod 9 = 1 h(25) = 25 mod 9 = 7

20. Open Addressing • Each element occupies a single slot in the hash table – no chaining is done • To insert an element, we probe the table according to the hash function until an empty slot is found • The hash function is now a function of both the key and the number of attempts in the insertion process

21. Linear Probing • A hash value is computed using any hash function h’, and then the number of the current attempt is added to it: • Slots are examined sequentially, until an empty one is found

22. Linear Probing • Easy to implement but suffers from primary clustering • Clusters tend to grow: • If an empty slot is preceded by i full slots, the probability that it will be the next one filled is (i+1)/m • If an empty slot is preceded by another empty slot, the probability is only 1/m

23. Exercise • You are given a hash table H with 11 slots • Demonstrate inserting the following elements using linear probing and a hash function h(k) = k mod m 10, 22, 31, 4, 15, 28, 17, 88, 59

24. Solution • h(10, 0) = (10 mod 11 + 0) mod 11 = 10 • h(22, 0) = (22 mod 11 + 0) mod 11 = 0 • h(31, 0) = (31 mod 11 + 0) mod 11 = 9 • h(4, 0) = (4 mod 11 + 0) mod 11 = 4 • h(15, 0) = (15 mod 11 + 0) mod 11 = 4 • h(15, 1) = (15 mod 11 + 1) mod 11 = 5 • h(28, 0) = (28 mod 11 + 0) mod 11 = 6 • h(17, 0) = (17 mod 11 + 0) mod 11 = 6

25. Solution • h(17, 1) = (17 mod 11 + 1) mod 11 = 7 • h(88, 0) = (88 mod 11 + 0) mod 11 = 0 • h(88, 1) = (88 mod 11 + 1) mod 11 = 1 • h(59, 0) = (59 mod 11 + 0) mod 11 = 4 • h(59, 1) = (59 mod 11 + 1) mod 11 = 5 • h(59, 2) = (59 mod 11 + 2) mod 11 = 6 • h(59, 3) = (59 mod 11 + 3) mod 11 = 7 • h(59, 4) = (59 mod 11 + 4) mod 11 = 8

26. Quadratic Probing • In this case, the second attempt is a more complex function of i: • Tries to avoid primary clustering • However, suffers from secondary clustering • The entire probing sequence is determined by the initial probe:

27. Double Hashing • Given two hash functions • One of the best methods for open addressing collision resolution • Permutations are almost random • For the entire hash to be searched,m and h2(k) must be relatively prime

28. Double Hashing • Possible selections of h2(k) • Select m to be a power of 2, and design h2(k) to produce odd numbers • Select m to be prime, and m’ to be m-1

29. Double Hashing

30. Issues in Open Addressing • Search may fail if items are deleted • Solution: • Mark deleted items with a special symbol • Search treats this symbol as full, while insert treats it as empty • Table may be filled up • Solution: • Rehashing (copy into a larger table)