1. Hashing: is an alternative search technique (earlier we had BST) Motivation: Try to access directly each possible keys! Suggestion: Enumerate possible keys. This means an ordering.
HASHING

2. Example: key space: 3 binary digits 1 2 3 4 7 8 N=2³1 1 1 1 1

3. Problem: in real life this is not possible.
Names : ~20 characters long needs
26²º (26 letters).
2. Digital numbers: 20 digits needs
10²º (0,..,9: 10 letters).
Binary Numbers: 20 spaces needs
2²º.
=> Too many => we cannot consider all.

4. We cannot create such long vectors such as 2²º or…. 26²º!
But: 
We don’t need all such long vectors because not all combinations occur in practice!
For example: A vocabulary contains maybe 100,000 ~ words.
Personal names (memphis).
White pages: 300pp*300 ~ 100,000
Or ~ ½ million ²º.

5. Idea: Assume that,
There are ~ N keys occur (approx.)
Define a vector of length ~ 2N.
Assign integers [1,..,2N] to the keys!
Look up according to the serial number (integer).

6. A Dynamical System Perspective:
Hashing,
Chopping up,
Granulating,
Coarse graining information!
This is done:
Continuously,
Autonomously,
Reliably…
in Bio-Systems!
(worms, ants….., humans….alike)

7. The Major Challenge Of Life:
How the delicately defined living substances can exist in an infinitely complex world?
How animals can survive and succeed?
How they separate the important from the useless?
=>There is/are mechanisms to complete ‘hashing’ very efficiently and promptly.
=>This course is far from that but indicates
a few main principles.

8. Major issues in Hashing:
How to assign the hashing?
Using the hashing function.
2. Sometimes the hashing function gives
the same number to different keys.
We have to resolve.
_________________________________

9. Complete space Hashing HASH TABLEK L
Eg: 26²º elements 2N

10. Example: dates: 1055, 1492, 1776, 1812, 1918, 1945.
Q: What is complete sp?
Hash Function:
HashCode(x) = (5x mod 8)
(hash code)
1776
1055
1492
1812
1945
1918

11. Evaluation of closed address or
chained hashing
Costs of Search:
Compute hash code I : costs ‘a’.
Search through linked list H[i].
Linked lists
H[1]………H[h]
hashing L1 L2 L3 … Lh ~

12. Average total cost of search k: T(n)=a + 1/n (h-1,i=0)(L1 + 1)/2
Worst case:Bad Distribution: All are in the same bucket. Needs n/2 comparisons in average same as search unordered array.
Better:Good Distribution: Equally distributed among cells.
Load factor  = n/h const. f cells
average O(1) computations.
Search # [i]

13. Hashing evaluation continued:
For uniform distribution: there is very good performance.
But a hashing function is required that gives uniform distribution independently of actual data structure!
Randomization: computer pseudo- random generator.
Eg: multiplicative congruent.

14. HashCode (K) = (aK) mod h.
Strategy:
multiply with constant a and take the modulus (i.e. remainder after division).
HashCode (K) = (aK) mod h.

15. Open Address Hashing:
this is really dynamic.
does not allow collisions as linked lists (before in closed hashing).
load factor;
 = n/h <1 (if  >= 0.5, array doubling)
If there is a collision: Rehashing.
Linear Probing.
Simple:
Rehash (j) = (j+1) mod h (j is the most recent probed location, start with j = i, go until empty cell found. Eg: 6.10)

16. Rehashing:
2. Double Hashing:
Rehash (j,d) = (j+d) mod h
Here d-increment of rehashing.
If d = 1  linear rehashing  it is determined separately.

17. Schedule 18-month schedule highlights Timing
Isolate timing dependencies critical to success
Jan
Feb
Mar
Apr
May
Jun
July
Sep
Oct
Nov
Dec
Task 2
Task 3
Task 4
Task 1
Milestone