Some Hash Functions

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Experiments with Hashing 15-451 Feb. 15, 2001

• Some Hash Functions
• Bucket Size Distribution
• Maximum Bucket Sizes

http//www.cs.cmu.edu/bryant
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Parameters

• Keys
• /usr/dict/words
• N 45,402 English words
• From Aarhus to Zurich
• 128 characters long
• antidisestablishmentarianism
• Hashing
• Into M buckets
• Load N/M
• 8 different hash functions

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Hash Functions

• Key x c1 c2 clen(K)
• Functions
• h1(x) c1 mod M
• This is really bad!
• Since only have 52 characters
• h2(x) ? ci mod M
• Hashes not and ton to same bucket
• h3(x) ? (ai ci) mod M
• ais random 22-bit numbers
• This should be a good function
• h4(x) ? (ai ci bi) mod M
• ais, bis random 22-bit numbers
• This should be even better function

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More Hash Functions

• h5(x) ? (ai ci) mod M
• ais random 22-bit numbers
• All sums products computed module p 524,287
• This should be a good function
• h6(x) ? (ai ci bi) mod M
• ais, bis random 22-bit numbers
• All sums products computed module p 524,287
• This should be the best function
• h7(x) h6(first 5 characters of K)
• hashes botch, botches, botching, and
  botched to same bucket
• h8(x) random(0..M-1)
• Not a real hash function
• Should represent ideal case

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Bucket Size Distribution

• Experiment
• Hash 45,402 keys into 128 buckets
• Load 354.7
• Average number of keys per bucket
• Measure
• Range of bucket sizes
• Normalize as count/load
• Average 1.0
• Determines how well hash function does at
  distributing keys

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Bucket Size Distribution Results
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Bucket Size Dist. Results (closeup)
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Distribution Observations

• Load 354.7
• h1 is really bad
• only uses 52 buckets
• Largest one has 4532 elements
• h7 is pretty bad too
• Good function, but only over first 5 characters
• Largest has 529 elements
• Rest look fairly decent
• h2 441 max. Ignores order of characters
• h4 428 max. Why not better than h3?
• h6 409 max. Why not better than h5?
• h8 403 max. Random
• Hey! This should be the best!
• h3 402 max. Mod p helps
• h5 400 max. Mod p helps

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Maximum Bucket Size

• Experiment
• Hash 45,402 keys into M buckets
• M powers of 2 from 128 to 65,536
• Load 354.7 to 0.69
• Measure
• Maximum bucket size
• Normalize as count/load
• Determines worst case access time

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Max. Bucket Size Results
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Max. Bucket Size Results (Closeup)
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Bucket Size Observations

• h1 is really bad
• only uses 52 buckets
• Largest one has 4532 elements, independent of M
• h7 is pretty bad too
• Good function, but only over first 5 characters
• Largest bucket with M65,536 has 197 elements
• h2 doesnt do very well
• Ignores order of characters
• Largest bucket with M65,536 has 163 elements
• Rest are comparable
• 67 elements in largest bucket for M 65,536
• Compare to theory
• When MN, Elargest bucket size log N / log
  log N
• For M65,536, this would be 16/4 4.