Hashing:%20is%20an%20alternative%20search%20technique%20(earlier%20we%20had%20BST)%20%20Motivation:%20Try%20to%20access%20directly%20each%20possible%20keys!%20%20Suggestion:%20Enumerate%20possible%20keys.%20This%20means%20an%20ordering.

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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
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Example key space 3 binary digits
1
2
3

4
7
8 N2³
0 0 1








0 1 0
0 1 1
1 0 0
1 1 1
0 0 0
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• 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.
• gt Too many gt we cannot consider all.

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• We cannot create such long vectors such as 2²º
  or. 26²º!
• But ?
• We dont 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 300pp300 100,000
• Or ½ million 26²º.

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• 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).

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

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• 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?
• gtThere is/are mechanisms to complete hashing
  very efficiently and promptly.
• gtThis course is far from that but indicates
• a few main principles.

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• 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.
• _________________________________

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• Complete space Hashing
• HASH TABLE
• 
  
• 
  
• 
  

K
L
Eg 26²º elements
2N
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• Example dates 1055, 1492, 1776, 1812, 1918,
  1945.
• Q What is complete sp?
• Hash Function
• HashCode(x) (5x mod 8)
• 0 1 2 3 4 5 6
  7
• (hash code)

1776 1055 1492 1812 1945 1918
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• Evaluation of closed address or
• chained hashing
• Costs of Search
• Compute hash code I costs a.
• Search through linked list Hi.

Linked lists H1Hh
hashing
L1 L2 L3 Lh

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• Average total cost of search k
• T(n)a 1/n (h-1,i0)?(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
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• 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.

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• Strategy
• ?multiply with constant a and take the modulus
  (i.e. remainder after division).

HashCode (K) (aK) mod h.
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• Open Address Hashing
• ?this is really dynamic.
• ?does not allow collisions as linked lists
  (before in closed hashing).
• ?load factor
• ? n/h lt1 (if ? gt 0.5, array doubling)
• If there is a collision Rehashing.
• Linear Probing.
• Simple
• Rehash (j) (j1) mod h (j is the most recent
  probed location, start with j i, go until empty
  cell found. Eg 6.10)

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• Rehashing
• 2. Double Hashing
• Rehash (j,d) (jd) mod h
• Here d-increment of rehashing.
• If d 1 ? linear rehashing ? it is determined
  separately.

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Schedule

• 18-month schedule highlights
• Timing
• Isolate timing dependencies critical to success