1. Chapter 5: Hashing Hash Tables
Mark Allen Weiss: Data Structures and Algorithm Analysis in Java
Chapter 5: Hashing
Hash Tables
Lydia Sinapova, Simpson College

2. Hashing
What is Hashing?
Direct Access Tables
Hash Tables

3. Hashing - basic idea
A mapping between the search keys and indices - efficient searching into an array.
Each element is found with one operation only.

4. Hashing example Example: 1000 students,
identification number between 0 and 999,
use an array of 1000 elements.
SSN of each student a 9-digit number.
much more elements than the number of the students, - a great waste of space.

5. The Approach
Directly referencing records in a table using arithmetic operations on keys to map them onto table addresses.
Hash function: function that transforms the search key into a table address.

6. Direct-address tables
The most elementary form of hashing.
Assumption – direct one-to-one correspondence between the keys and numbers 0, 1, …, m-1., m – not very large.
Array A[m]. Each position (slot) in the array contains a pointer to a record, or NIL.

7. Direct-Address Tables
The most elementary form of hashing
Assumption – direct one-to-one correspondence between the keys and numbers 0, 1, …, m-1., m – not very large.
Array A[m]. Each position (slot) in the array contains either a reference to a record, or NULL.
Cost – the size of the array we need is determined the largest key. Not very useful if there are only a few keys

8. Hash Functions
Transform the keys into numbers within a predetermined interval to be used as indices in an array (table, hash table) to store the records

9. Hash Functions – Numerical Keys
Keys – numbers
If M is the size of the array, then
h(key) = key % M.
This will map all the keys into numbers within the interval
[0 .. (M-1)].

10. Hash Functions – Character Keys
Keys – strings of characters
Treat the binary representation of a key as a number, and then apply the hash function

11. How keys are treated as numbers
If each character is represented with m bits,
then the string can be treated as base-m number.

12. Example
A K E Y : = =
44271
Each letter is represented by its position in the alphabet. E.G, K is the 11-th letter, and its representation is ( 11 in decimal)

13. Long Keys If the keys are very long, an overflow may occur.
A solution to this is to apply the Horner’s method
in computing the hash function.

14. Horner’s Method 4x5 + 2x4 + 3x3 + x2 + 7x1 + 9x0 =
anxn + an-1.xn-1 + an-2.xn-2 + … + a1x1 + a0x0 =
x(x(…x(x (an.x +an-1) + an-2 ) + …. ) + a1) + a0
4x5 + 2x4 + 3x3 + x2 + 7x1 + 9x0 =
x( x( x( x ( 4.x +2) + 3) + 1 ) + 7) + 9
The polynomial can be computed by
alternating the multiplication and addition operations

15. Example
V E R Y L O N G K E Y
V E R Y L O N G K E Y

16. Example (continued) V E R Y L O N G K E Y h0 = (22.32 + 5) % M
((((((((( ) ) ) ) ) )32 + 7)32 +11)32 +5)
compute the hash function by applying the mod operation at each step, thus avoiding overflowing.
h0 = ( ) % M
h1 = (32.h ) % M
h2 = (32.h ) % M

17. Code int hash32 (char[] name, int tbl_size) {
key_length = name.length;
int h = 0;
for (int i=0; i < key_length ; i++)
h = (32 * h + name[i]) % tbl_size;
return h; }

18. Hash Tables sentinel value, or a special field in each slot.
Index - integer, generated by a hash function between 0 and M-1
Initially - blank slots.
sentinel value, or a special field in each slot.

19. Hash Tables Insert - hash function to generate an address
Search for a key in the table - the same hash function is used.

20. Size of the Table
Table size M - different from the number of records N
Load factor: N/M
M must be prime to ensure even distribution of keys