Hash Functions

1
Hash Functions

• Andy Wang
• Data Structures, Algorithms, and Generic
  Programming

2
Introduction

• Hash function
• Maps keys to integers (buckets)
• Hash(Key) Integer
• Ideally in a random-like manner
• Evenly distributed bucket values
• Even if the input data is not evenly distributed

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An Example

• ID Number Generation
• Key your name
• Hash(Key) a number
• Not a great hash function
• Two people with the same name will have the same
  number

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Simple Hash Functions

• Assumptions
• K an unsigned 32-bit integer
• M the number of buckets (the number of entries
  in a hash table)
• Goal
• If a bit is changed in K, all bits are equally
  likely to change for Hash(K)

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A Simple Hash Function

• What if K M?
• Hash(K) K
• What is wrong?
• Your student ID SSN
• I cant use your SSN to post your grades

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Another Simple Function

• If K gt M
• Hash(K) K M
• What is wrong?
• Suppose M 4, K 2, 4, 6, 8
• K M 2, 0, 2, 0

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Yet Another Simple Function

• If K gt P, P prime number
• Hash(K) K P
• Suppose P 3, K 2, 4, 6, 8
• K P 2, 1, 0, 3
• More uniform distributionbut still problematic
  for other cases

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More on Prime Numbers

• K gt P1 gt P2, P1 and P2 are prime numbers
• Hash(K) (K P1) P2
• Suppose P1 5, P2 3, K 2, 4, 6, 8, 10
• (K 5) 2, 4, 1, 3, 0
• (K 5) 3 2, 1, 1, 0, 0
• Still uniform distribution

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Polynomial Functions

• If K gt P, P prime number
• Hash(K) K(K 3) P
• Slightly better than pure modulo functions

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How About

• Hash(K) rand()
• What is wrong?
• Not repeatable

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How About

• K gt P, P prime number
• Hash(K) rand(K) P
• Better randomness
• Can be expensive to compute random numbers

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Pre-generated Randomness

• Two prime numbers P1 and P2
• K gt P1 and K gt P2
• A table RP1, with Ri pre-initialized to
  rand(i) P2
• Hash(K) RK P1
• Slight Problem Possible duplicate mapping

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To Avoid Duplicate Mapping

• Two prime numbers P1 and P2
• K gt P1 and K gt P2
• A table RP1, with Ri pre-initialized to
  unique random numbers
• Hash(K) RK P1

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An Example

• K 0232, P1 3, P2 5
• R3 0, 4, 1
• Hash(K) RK 3

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Hashing a Sequence of Keys

• K K1, K2, , Kn)
• E.g., Hash(test) 98157
• Design Principles
• Use the entire key
• Use the ordering information
• Use pre-generated randomness

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Use the Entire Key

• unsigned int Hash(const char Key)
• unsigned int hash 0
• for (unsigned int j 0 j lt K j)
• hash hash Keyj
• return hash
• Problem Hash(ab) Hash(ba)

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Use the Ordering Information

• unsigned int Hash(const char Key)
• unsigned int hash 0
• for (unsigned int j 0 j lt K j)
• hash hash Keyj
• hash / hash with some shiftings /
• return hash
• Problem H(short keys) will not perturb all
  32-bits (clustering)

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Use Pre-generated Randomness

• unsigned int Hash(const char Key)
• unsigned int hash 0
• for (unsigned int j 0 j lt K j)
• hash hash RKeyj
• hash / hash with some shiftings /
• return hash

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CRC Variant

• Do 5-bit circular shift of hash
• XOR hash and Kj
• for ()
• highorder hash 0xf8000000
• hash hash ltlt 5
• hash hash (highorder gtgt 27)
• hash hash Kj

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CRC Variant

• For long keys, all 32-bits are exercised
• More randomness toward lower bits
• - Not all bits are changed for short keys

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BUZ Hash

• Set up an array R to store precomputed random
  numbers
• for ()
• highorder hash 0x80000000
• hash hash ltlt 1
• hash hash (highorder gtgt 31)
• hash hash RKj

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References

• Aho, Sethi, and Ullman. Compilers Principles,
  Techniques, and Tools, 1986.
• Cormen, Leiserson, River. Introduction to
  Algorithms, 1990
• Knuth. The Art of Computer Programming, 1973
• Kuenning. Hash Functions, 2003.