COMP 171 Data Structures and Algorithms

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COMP 171Data Structures and Algorithms

• Tutorial 10
• Hash Tables

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Data Dictionary

• A data structure that supports
• Insert
• Search
• Delete
• Examples
• Binary Search Tree
• Red Black Tree
• B Tree
• Link List

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Hash Table

• An effective data dictionary
• Worst Case T(n) time
• Under assumptions O(1) time
• Generalization of an array
• Size proportional to the number of keys actually
  stored
• Array index is computed from the key

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Direct Address Tables

• Universe U
• a set that contains all the possible keys
• Table has U slots.
• Each key in U is mapped into one unique entry in
  the Table
• Insert, delete and search takes O(1) time
• Works well when U is small
• If U is large, impractical

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Hash Function

• Assume Hash Table has m slots
• Hash function h is used to compute slot from the
  key k
• h maps U into the slots of a hash table
• h U ? 0, 1, , m-1
• U gt m, at least 2 keys will have the same hash
  value, collision
• Good hash function can minimize the number of
  collision

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• If the keys are not natural number
• Interpret them as natural number using suitable
  radix notation
• Example character string into radix-128 integer
• Division method
• h(k) k mod m
• m is usually a prime
• Avoid m too close to an exact power of 2
• Ex 11.3-3
• Choose m 2p-1 and k is a character string
  interpreted in radix 2p. Show that if x can be
  derived from y by permuting its characters, then
  h(x) h(y).

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• Multiplication method
• h(k) ?m ( k A mod 1)? , 0 lt A lt 1
• Value m is not critical
• Usually choose m to be power of 2
• It works better with some values of A
• Eg. (v5 1 ) / 2

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Separate Chaining

• Put all the elements that hash to the same slot
  in a link list
• Element is inserted into the head of the link
  list
• Worst case insertion O(1)
• Worst case search O(n)
• Worst case deletion O(1)

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• Given a hash table has m slots that stores n
  elements, we define load factor a
• a n/m
• Simple uniform hashing
• Element is equally likely to hash into any of the
  m slots, independently of where any other element
  has hashed to
• Under Simple uniform hashing
• Average time for search T(1a)

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Open Addressing

• Each table slot contains either an element or NIL
• When collision happens, we successively examine,
  or probe, the hash table until we find an empty
  slot to put the key
• Deletion is done by marking the slot as Deleted
  but not NIL
• Hash function h now takes two values
• The key value and the probe number
• h(k, i)

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• Linear Probing
• h(k, i) ( h(k) i ) mod m
• Initial probe determine the entire probe
  sequence, there are only m distinct probe
  sequence
• Primary clustering
• Quadratic Probing
• h(k, i) ( h(k) c1i c2i ) mod m
• there are only m distinct probe sequence
• Secondary clustering

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• Double hashing
• Make use of 2 different hash function
• h(k, i) ( h1(k) ih2(k) ) mod m
• ih2(k) should be co-prime with m
• Usually take m as a prime number
• Probe sequence depends on both has function, so
  there are m2 probe sequences
• Double hashing is better then linear or quadratic
  probing

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Trie

• Assumption
• Digital data / radix
• Tree structure is used
• Insertion is done by creating a path of nodes
  from the root to the data
• Deletion is done by removing the pointer that
  points to that element
• Time Complexity O(L)
• Max of keys for given L 128L1 - 1

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• Memory Usage
• Node size Number of node
• ((N1)pointer size) (L n)
• N radix
• L maximum length of the keys
• n number of keys
• Improvement 1
• Put all nodes into an array of nodes
• Replace pointer by array index
• Array index ? lg (L n) ?

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• Improvement 2
• Eliminate nodes with a single child
• Do Skipping
• Label each internal node with its position
• Improvement 3
• De La Briandais Tree
• Eliminate null pointer in the internal node
• Save memory when array are sparsely populated