This document discusses different techniques for handling collisions in open addressing hash tables: linear probing, quadratic probing, and double hashing. Linear probing searches sequentially through the hash table for the next empty slot when a collision occurs. This can lead to clustering issues as the table size increases. Quadratic probing uses a quadratic function to determine the next slot to search, reducing clustering. Double hashing uses a second hash function to determine the next slot, making it faster than other methods. The document provides examples and explanations of how each technique works to resolve collisions in open addressing hash tables.

1. 11.4 Open Addressing

2. Separate chaining has the disadvantage of using
linked lists.
Requires the implementation of a second data structure.
In an open addressing hashing system, all
the data go inside the table.
Thus, a bigger table is needed.
Generally the load factor should be below 0.5.
If a collision occurs, alternative cells are tried until an empty cell is found.

3. 0
1
2
3
4
5
6
7
8
9
h(k) = k mod m
m = 10
h(10) = 10 mod 10 = 0
h(20) = 20 mod 10 = 0
h(30) = 30 mod 10 = 0
h(40) = 40 mod 10 = 0
h(50) = 50 mod 10 = 0

4. 0
1
2
3
4
5
6
7
8
9
10 ←-- 20 ←- 30 ←-- 40 ←-- 50
All the nodes are
on the chain of
linked list
Take extra space for
linked list
but
not use available space in
hash table

5. 0 10
1
2
3
4
5
6
7
8
9
h(10) = 10 mod 10 = 0

6. 0 10
1
2
3
4
5
6
7
8
9
h(20) = 20 mod 10 = 0
But there is a collision so we
search the next free space

7. 0 10
1 20
2
3
4
5
6
7
8
9
h(20) = 20 mod 10 = 0

8. 0 10
1 20
2
3
4
5
6
7
8
9
h(30) = 30 mod 10 = 0
Again we search for free space
when collision occur

9. 0 10
1 20
2 30
3
4
5
6
7
8
9
h(30) = 30 mod 10 = 0

10. 0 10
1 20
2 30
3
4
5
6
7
8
9
h(40) = 40 mod 10 = 0

11. 0 10
1 20
2 30
3 40
4
5
6
7
8
9
h(40) = 40 mod 10 = 0

12. 0 10
1 20
2 30
3 40
4
5
6
7
8
9
h(50) = 50 mod 10 = 0

13. 0 10
1 20
2 30
3 40
4 50
5
6
7
8
9
h(50) = 50 mod 10 = 0

14. h(x)i
= Hash(x) + f(i)
Where f(0) = 0

15. This is called Linear
probing

16. In linear probing, collisions
are resolved by sequentially
scanning an array (with
wraparound) until an empty cell is
found.
i.e. f is a linear function of i, typically f(i)= i

19. But we have an issue on this Linear
Probing is clustering
As long as table is big
enough, a free cell can always
be found, but the time to do
so can get quite large.

20. Summary

We have solve the issue of extra memory and utilize the avaliable
memory in the hash table

Insertion take more time on to find the free space

Finding the element also following the insertion algorithm, so it
also take too time

Deletion cannot be done in a single shot

21. Quadratic Probing

22. “Make it simple with the understanding of
Linear probing”

23. In Linear Probing
h(x)i
= Hash(x) + f(i)
Where f(i) = i
In Quadratic Probing
h(x)i
= Hash(x) + f(i)
Where f(i) = i2

24. Now the clustering problem is removed from
this quadratic probing

25. Try this in your own style! :)

26. Double Hashing

27. A second hash function is used to drive the
collision resolution.
h(x)i
= Hash(x) + f(i)
f(i) = i * hash2(x)
This is the second hash function

28. Our hash2 function is
hash2(x) = R – ( x mod R)
Where R is the prime which R < Table Size (m)

29. Important! note is the 2nd
hashing
must not be equal to zero

30. Double Hashing is much faster than others :)

31. Some Hashing Applications
Compilers use hash tables to implement the symbol table
Game programs use hash tables to keep track of
positions it has encountered
On line spelling checkers

32. Thank you for watching:)