ITEC 2620M Introduction to Data Structures

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ITEC 2620MIntroduction to Data Structures

• Instructor Prof. Z. Yang
• Course Website http//people.math.yorku.ca/zyang
  /itec2620m.htm
• Office TEL 3049

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HASHING
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Key Points of this Lecture

• Hash tables
• Hash functions
• Collision resolution and clustering
• Deletions

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Indices vs. Keys

• Each key/record is associated with an array slot
• We could map each key to each slot
• e.g. last name to apartment number
• We could then search either the array (unsorted?)
  or a look-up table (sorted?)
• However, what if the look-up is actually a
  calculated function?
• eliminate look-up!

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Hash Functions

• A hash function h() converts a key (integer,
  string, float, etc) into a table index
• Example

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Hash Tables

• Records are stored in slots specified by a hash
  function
• Look-up/store
• Convert key into a table index with hash function
  h()
• h(key) index
• Find record/empty slot starting at index
  h(key)(use resolution policy if necessary)

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Comments

• Hash function should evenly distribute keys
  across table
• not easy given unspecified input data
  distribution
• Hash table should be about half full
• note time-space tradeoff
• more space -gt less time(and already twice as
  much space as a sorted array)
• if half full, 50 chance of one collision
• 25 chance of two collisions
• etc...
• 2 accesses on average(approaches n as table
  fills)

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How to do better

• What to do with collisions?
• linear probing (classic hashing)
• if collision, search spaces sequentially
• To eliminate clustering, we would like each
  remaining slot to have equal probability
• Cant use random needs to be reproducable
• Pseudo-random probing (see text)
• Goal of random probing? --gt cause divergence
• Probe sequences should not all follow same path

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Quadratic Probing

• Simple divergence method
• Linear probing ith probe is i slots away
• Quadratic probing

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Secondary Clustering

• If multiple keys are hashed to the same
  index/home position, quadratic probing still
  follows the same path each time
• This is secondary clustering
• Use second hash function to determine probe
  sequence