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 table h: 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 called closed 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 efficiency of the hash table
Simple Uniform Hashing assumption: The hash function will hash any key into 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 and h(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 ) mod 11 = 10
h(22, 0) = (22 mod ) mod 11 = 0
h(31, 0) = (31 mod ) mod 11 = 9
h(4, 0) = (4 mod ) mod 11 = 4
h(15, 0) = (15 mod ) mod 11 = 4
h(15, 1) = (15 mod ) mod 11 = 5
h(28, 0) = (28 mod ) mod 11 = 6
h(17, 0) = (17 mod ) mod 11 = 6 1 2 3 4 5 6 7 8 9 10 22 15 28 31

25. Solution
h(17, 1) = (17 mod ) mod 11 = 7
h(88, 0) = (88 mod ) mod 11 = 0
h(88, 1) = (88 mod ) mod 11 = 1
h(59, 0) = (59 mod ) mod 11 = 4
h(59, 1) = (59 mod ) mod 11 = 5
h(59, 2) = (59 mod ) mod 11 = 6
h(59, 3) = (59 mod ) mod 11 = 7
h(59, 4) = (59 mod ) mod 11 = 8 1 2 3 4 5 6 7 8 9 10 22 88 15 28 17 59 31

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
Rehashing (copy into a larger table)