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: Goal: 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 What is wrong?
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 FunctionIf
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 += Key[j] }
return hash;
Problem: Small sized keys may not use a large fraction of a large hash table 9 9

10. Another Simple Hash Function
unsigned int Hash(const string& Key) {
return Key[0] + 27*Key[1] + 729*Key[2]; }
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) {
for (int j = 0; j != Key.size(); ++j)
hash = 37*hash + (Key[j]-’a’+1);
return hash%TableSize; }
The for loop computes ai37n-i using Horner’s rule, where ai has the value 1 for ‘a’, 2 for ‘b’, etc
a3 + 37a a a0 = 37(37(37a0 + a1)+ a2) + a3
The for loop 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
<tr1/unordered_map> or <unordered_map>
<trl/unordered_set> or <unordered_set>
Example: Lec24/hashmapex.cpp
Reference