Introduction%20to%20Hashing%20-%20Hash%20Functions

1
Introduction to Hashing - Hash Functions

• Sections 5.1, 5.2, and 5.6

2
Hashing

• Data items stored in an array of some fixed size
• Hash table
• Search performed using some part of the data item
  
• key
• Used for performing insertions, deletions, and
  finds in constant average time
• Operations requiring ordering information not
  supported efficiently
• Such as findMin, findMax

3
Hash Table Example
4
Hash Table Applications

• Comparing search efficiency of different data
  structures
• Vector, list O(N)
• AVL search tree O(log(N))
• Hash table O(1) expected time
• Compilers to keep track of declared variables
• Symbol tables
• Mapping from name to id
• On-line spelling checkers

5
Hash Functions

• Map keys to integers (which represent table
  indices)
• Hash(Key) Integer
• Evenly distributed index values
• Even if the input data is not evenly distributed
• What happens if multiple keys mapped to the same
  integer (same position)?
• Collision management (discussed in detail later)
• Collisions are likely to be reduced if keys are
  evenly distributed over the hash table

6
Simple Hash Functions

• Assumptions
• K an unsigned 32-bit integer
• M the number of buckets (the number of entries
  in a hash table)
• Goal
• If a bit is changed in K, all bits are equally
  likely to change for Hash(K)
• So that items are evenly distributed in the hash
  table

7
A Simple Function

• What if
• Hash(K) K M
• Where M is of any integer value
• What is wrong?
• Values of K may not be evenly distributed
• But Hash(K) needs to be evenly distributed
• Suppose
• M 10,
• K 10, 20, 30, 40
• Then K M 0, 0, 0, 0, 0

8
Another Simple Function

• If
• Hash(K) K P, P prime number
• Suppose
• P 11
• K 10, 20, 30, 40
• K P 10, 9, 8, 7
• More uniform distribution
• So hash tables often have prime number of entries

9
A Simple Hash for Strings

• unsigned int Hash(const string Key)
• unsigned int hash 0
• for (int j 0 j ! Key.size() j)
• hash Keyj
• return hash
• Problem Small sized keys may not use a large
  fraction of a large hash table

9
10
Another Simple Hash Function

• unsigned int Hash(const string Key)
• return Key0 27Key1 729Key2
• Problem English does not use random strings
  so, the hash values are not uniformly distributed
• Using more characters of the key can improve the
  hash function

11
A Better Hash Function

• unsigned int Hash(const string Key)
• unsigned int hash 0
• for (int j 0 j ! Key.size() j)
• hash 37hash (Keyj-a1)
• return hashTableSize
• The for loop computes ?ai37n-i using Horners
  rule, where ai has the value 1 for a, 2 for
  b, etc
• a3 37a2 372a1 373a0 37(37(37a0 a1) a2)
  a3
• The for implicitly performs arithmetic modulo 2k,
  where k is the number of bits in an unisigned int

12
STL Hash Tables

• STL extensions
• hash_set
• hash_map
• The key type, hash function, and equality
  operator may need to be provided
• Available in new standard as unordered set and
  map
• lttr1/unordered_mapgt or ltunordered_mapgt
• lttrl/unordered_setgt or ltunordered_setgt
• Example Lec24/hashmapex.cpp
• Reference
• www.open-std.org/jtc1/sc22/wg21/docs/papers/2003/
  n1456.html