1. Alternative method for dictionary building : Hashing
Chapter 3.5
Alternative method for
dictionary building :
Hashing

2. Hashing (= Key Transformation)
Space
(e.g. All
possible
English
words)
2725
Address
Space
(array
index)
20000
Mapping Function
Many to one function !

3. Hashing Mapping Function H The most effective : A = ORD(Key) MOD n
with n = Size of Address Space
n preferably a prime number
n certainly not = 2k
Exotic functions based on bit manipulations quite common but sometimes statistically ineffective

4. The Hashing Function A = ORD(Key) MOD n e.g. Key = "Key"
ORD (Key) = ORD(K) * 20
+ ORD(e) * 28
+ ORD(y) * 216
If n were chosen equal to 256, all words beginning with K would have the same address. In fact, only the first character would determine the address !
To be sure all characters play a role chose n prime !

5. Hashing Collisions (two keys map into one address)
Detection : Key included in record
Handling : Various techniques
Linked lists with pointers
Open addressing
Linear
Quadratic

6. Linked List with Pointers
Hashing
Address space

7. Open Addressing Linear : h0 = H(Key) hi = (h0+i) MOD n Quadratic :
Kathy
Linear :
h0 = H(Key)
hi = (h0+i) MOD n
Quadratic :
hi = (h0+i2) MOD n
Linda

8. Open Addressing Linear : h0 = H(Key) hi = (h0+i) MOD n Quadratic :
Kathy
Linear :
h0 = H(Key)
hi = (h0+i) MOD n
Quadratic :
hi = (h0+i2) MOD n
Jim
Jim
Linda

9. Open Addressing Linear : h0 = H(Key) hi = (h0+i) MOD n Quadratic :
Kathy
Linear :
h0 = H(Key)
hi = (h0+i) MOD n
Quadratic :
hi = (h0+i2) MOD n
Jim
Bob
Linda

10. Open Addressing Linear : h0 = H(Key) h1 = (h0+1) MOD n Quadratic :
Kathy
Linear :
h0 = H(Key)
h1 = (h0+1) MOD n
Quadratic :
hi = (h0+i2) MOD n
Jim
Bob
Linda

11. Open Addressing Linear : h0 = H(Key) h2 = (h0+2) MOD n Quadratic :
Kathy
Linear :
h0 = H(Key)
h2 = (h0+2) MOD n
Quadratic :
hi = (h0+i2) MOD n
Bob
Jim
Bob
Linda

12. Open Addressing Linear : h0 = H(Key) hi = (h0+i) MOD n Quadratic :
Kathy
Linear :
h0 = H(Key)
hi = (h0+i) MOD n
Quadratic :
hi = (h0+i2) MOD n
Jim
Bob
Linda

13. Open Addressing Linear : h0 = H(Key) h1 = (h0+1) MOD n Quadratic :
Kathy
Linear :
h0 = H(Key)
h1 = (h0+1) MOD n
Quadratic :
h1 = (h0+12) MOD n
Jim
Bob
Linda

14. Open Addressing Linear : h0 = H(Key) h2 = (h0+2) MOD n Quadratic :
Kathy
Linear :
h0 = H(Key)
h2 = (h0+2) MOD n
Quadratic :
h2 = (h0+22) MOD n
Bob
Jim
Bob
Linda

15. Analysis of Hashing All possible keys equally likely
Uniformly distributed over range 0..n-1
Number of keys already in table = k
Probability of hitting free place = 1- k/n
Pi = probability to need i attempts
P1 = (n-k)/n
P2 = (k/n)*[(n-k)/(n-1)]
P3 = (k/n)*[(k-1)/(n-1)]*[(n-k)/(n-2)]
Pi = (k/n)*[(k-1)/(n-1)]*[(k-2)/(n-2)]*...
*[(k-i+2)/(n-i+2)]*[(n-k)/(n-i+1)]

16. Analysis of Hashing (2) E = Average number of probes  = k/n
E = [- ln(1- )] /  
0.1
0.5
0.9
0.99 E
1.05
1.39
2.56
4.66

17. Hashing Main advantage : Speed Main disadvantages :
Size of table FIXED
Data in random order
Removal extremely difficult