Talk outline Hamming similarity search Approximate similarity - - PowerPoint PPT Presentation

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Talk outline Hamming similarity search Approximate similarity - - PowerPoint PPT Presentation

Dec 20, 2023 419 likes •1.08k views

Locality-sensitive Hashing without False Negatives S CALABLE S IMILARITY Rasmus Pagh IT University of Copenhagen S EARCH SODA, January 10, 2016 1-2 post-doc positions available starting fall 2016 - contact me for details! 1 Talk outline

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SLIDE 1

Locality-sensitive Hashing without False Negatives

Rasmus Pagh
 IT University of Copenhagen SODA, January 10, 2016

SCALABLE SIMILARITY SEARCH

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1-2 post-doc positions available starting fall 2016 - contact me for details!

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SLIDE 2

Talk outline

  • Hamming similarity search
  • Approximate similarity search using LSH
  • Recent developments
  • New result: Avoiding false negatives

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SLIDE 3

Hamming similarity search

  • Build data structure for set S ⊆ {0,1}

d s.t. given query

vector q and radius r, can decide
 
 
 where = Hamming distance between x and q.

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v a n i l l a v e r s i

  • n

∃x ∈ S : ||x − q|| ≤ r ||x − q||

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SLIDE 4
  • [Williams ’04], [Alman & Williams ’15]:


Hamming similarity search in time n0.99 2o(d) ⟹
 k-SAT w. n variables can be solved in time αn, α < 2

Hamming similarity search

  • Build data structure for set S ⊆ {0,1}

d s.t. given query

vector q and radius r, can decide
 
 
 where = Hamming distance between x and q.

3

v a n i l l a v e r s i

  • n

∃x ∈ S : ||x − q|| ≤ r ||x − q||

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SLIDE 5
  • [Williams ’04], [Alman & Williams ’15]:


Hamming similarity search in time n0.99 2o(d) ⟹
 k-SAT w. n variables can be solved in time αn, α < 2

Hamming similarity search

  • Build data structure for set S ⊆ {0,1}

d s.t. given query

vector q and radius r, can decide
 
 
 where = Hamming distance between x and q.

3

v a n i l l a v e r s i

  • n

∃x ∈ S : ||x − q|| ≤ r ||x − q||

S t r

  • n

g E T H s t a t e s : N

  • t

p

  • s

s i b l e !

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SLIDE 6

Approximate similarity search

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radius cr

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SLIDE 7

Approximate similarity search

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radius cr

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SLIDE 8

Approximate similarity search

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radius cr

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SLIDE 9

Approximate similarity search

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radius cr

[Indyk & Motwani, STOC ’98]:

Time O(dn1/c), space O(n1+1/c + dn)

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SLIDE 10

Locality-sensitive hashing

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1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1

[Indyk & Motwani ’98]

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SLIDE 11

Locality-sensitive hashing

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1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1

Idea: Consider projection onto a random subset of dimensions, each chosen with probability p

[Indyk & Motwani ’98]

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SLIDE 12

Locality-sensitive hashing

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1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1

Idea: Consider projection onto a random subset of dimensions, each chosen with probability p

[Indyk & Motwani ’98]

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SLIDE 13

Locality-sensitive hashing

5

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1

Idea: Consider projection onto a random subset of dimensions, each chosen with probability p

h(x) = x ⋀ a

[Indyk & Motwani ’98]

slide-14
SLIDE 14

Locality-sensitive hashing

5

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1

Idea: Consider projection onto a random subset of dimensions, each chosen with probability p

Candidate match

h(x) = x ⋀ a

[Indyk & Motwani ’98]

slide-15
SLIDE 15

Locality-sensitive hashing

5

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1

Idea: Consider projection onto a random subset of dimensions, each chosen with probability p

Candidate match

h(x) = x ⋀ a hi(x) = x ⋀ ai

[Indyk & Motwani ’98]

slide-16
SLIDE 16

Locality-sensitive hashing

5

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1

Idea: Consider projection onto a random subset of dimensions, each chosen with probability p

Candidate match

h(x) = x ⋀ a hi(x) = x ⋀ ai

[Indyk & Motwani ’98]

repetitions ensure constant success probability

t = n1/c

slide-17
SLIDE 17

Locality-sensitive hashing

5

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1

Idea: Consider projection onto a random subset of dimensions, each chosen with probability p

Candidate match

h(x) = x ⋀ a hi(x) = x ⋀ ai

[Indyk & Motwani ’98]

repetitions ensure constant success probability

t = n1/c

Possibility of false negatives

slide-18
SLIDE 18

Some recent developments*

6

Reference

Exponent, search time Comment

Linear search

1

Indyk & Motwani STOC ’98

1/c

* Focus on subquadratic space; lower order terms ignored.

slide-19
SLIDE 19

Some recent developments*

6

Reference

Exponent, search time Comment

Linear search

1

Indyk & Motwani STOC ’98

1/c

Andoni & Razenshteyn STOC ‘15

1/(2c-1)

Data dependent LSH

* Focus on subquadratic space; lower order terms ignored.

slide-20
SLIDE 20

Some recent developments*

6

Reference

Exponent, search time Comment

Linear search

1

Indyk & Motwani STOC ’98

1/c

Andoni & Razenshteyn STOC ‘15

1/(2c-1)

Data dependent LSH

Laarhoven arXiv ‘15

(2c-1)/c2

Linear space

* Focus on subquadratic space; lower order terms ignored.

slide-21
SLIDE 21

Some recent developments*

6

Reference

Exponent, search time Comment

Linear search

1

Indyk & Motwani STOC ’98

1/c

Andoni & Razenshteyn STOC ‘15

1/(2c-1)

Data dependent LSH

Laarhoven arXiv ‘15

(2c-1)/c2

Linear space

Alman & Williams FOCS ‘15

Batched, c=1

Karppa, Kaski & Kohonen 
 SODA ‘16

Batched 
 [c>1, random data, ω > 2.25]

1 − ˜ Ω(log(n)/d)

* Focus on subquadratic space; lower order terms ignored.

2ω−3 3

slide-22
SLIDE 22

Recent developments*

7

Reference

Exponent, search time Comment

Linear search

1

Indyk & Motwani STOC ’98

1/c

Andoni & Razenshteyn STOC ‘15

1/(2c-1)

Data dependent LSH

Laarhoven arXiv ‘15

(2c-1)/c2

Linear space

Alman & Williams FOCS ‘15

Batched, c=1

Karppa, Kaski & Kohonen 
 SODA ‘16

Batched 
 [c>1, random data, ω > 2.25]

1 − ˜ Ω(log(n)/d)

* Focus on subquadratic space; lower order terms ignored.

2ω−3 3

P

  • s

s i b i l i t y

  • f

f a l s e n e g a t i v e s

slide-23
SLIDE 23

Similarity search without false negatives

8

Reference

Exponent, search time Comment

Arasu, Ganti & Kaushik
 VLDB ‘06

≈ 3/c

(analysis not in their paper)

This paper

ln(4)/c

This paper

1/c

For cr = log (n)

slide-24
SLIDE 24

Basic correlated LSH

9

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0

partitioning

[Arasu, Ganti & Kaushik ’06]

slide-25
SLIDE 25

Basic correlated LSH

9

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0

h1 partitioning

[Arasu, Ganti & Kaushik ’06]

slide-26
SLIDE 26

Basic correlated LSH

9

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0

h2 partitioning

[Arasu, Ganti & Kaushik ’06]

slide-27
SLIDE 27

Basic correlated LSH

9

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0

h3 partitioning

[Arasu, Ganti & Kaushik ’06]

slide-28
SLIDE 28

Basic correlated LSH

9

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0

h4 partitioning

[Arasu, Ganti & Kaushik ’06]

slide-29
SLIDE 29

Basic correlated LSH

9

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0

For Hamming distance ≤ 3, a collision is guaranteed!

partitioning

[Arasu, Ganti & Kaushik ’06]

slide-30
SLIDE 30

Basic correlated LSH

9

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0

For Hamming distance ≤ 3, a collision is guaranteed! [Arasu et al. ’06]:
 To bound probability of collision for distance > 3 randomly permute the dimensions

partitioning

[Arasu, Ganti & Kaushik ’06]

slide-31
SLIDE 31

Basic correlated LSH

10

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0

h12 enumeration

[Arasu, Ganti & Kaushik ’06]

slide-32
SLIDE 32

Basic correlated LSH

11

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0

h13 enumeration

[Arasu, Ganti & Kaushik ’06]

slide-33
SLIDE 33

Basic correlated LSH

12

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0

h14 enumeration

[Arasu, Ganti & Kaushik ’06]

slide-34
SLIDE 34

Basic correlated LSH

12

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0

h14 enumeration

For Hamming distance ≤ 2, a collision is guaranteed in h12, h13, h14, h23, h24, h34

[Arasu, Ganti & Kaushik ’06]

slide-35
SLIDE 35

Basic correlated LSH

12

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0

h14 enumeration

For Hamming distance ≤ 2, a collision is guaranteed in h12, h13, h14, h23, h24, h34

[Arasu, Ganti & Kaushik ’06] Probabilistic argument suggests that it is possible to do much better. But how?

slide-36
SLIDE 36

Mathematical question

  • A (p,r)-covering matrix of dim. d satisfies:
  • Has (1-p)d zeros and pd ones in each row;
  • for every set of r columns there exists a row

with 0s in all of them.

13

slide-37
SLIDE 37

Mathematical question

  • A (p,r)-covering matrix of dim. d satisfies:
  • Has (1-p)d zeros and pd ones in each row;
  • for every set of r columns there exists a row

with 0s in all of them.

13

S m a l l c

  • l

l i s i

  • n

p r

  • b

a b i l i t y

slide-38
SLIDE 38

Mathematical question

  • A (p,r)-covering matrix of dim. d satisfies:
  • Has (1-p)d zeros and pd ones in each row;
  • for every set of r columns there exists a row

with 0s in all of them.

13

S m a l l c

  • l

l i s i

  • n

p r

  • b

a b i l i t y C

  • l

l i s i

  • n

g u a r a n t e e

slide-39
SLIDE 39

Mathematical question

  • A (p,r)-covering matrix of dim. d satisfies:
  • Has (1-p)d zeros and pd ones in each row;
  • for every set of r columns there exists a row

with 0s in all of them.

  • Question: 


How few rows can 
 such a matrix have?

13

S m a l l c

  • l

l i s i

  • n

p r

  • b

a b i l i t y C

  • l

l i s i

  • n

g u a r a n t e e N u m b e r

  • f


 h a s h f u n c t i

  • n

s

slide-40
SLIDE 40

Mathematical question

  • A (p,r)-covering matrix of dim. d satisfies:
  • Has (1-p)d zeros and pd ones in each row;
  • for every set of r columns there exists a row

with 0s in all of them.

  • Question: 


How few rows can 
 such a matrix have?

13

p = 4/7
 r = 2

S m a l l c

  • l

l i s i

  • n

p r

  • b

a b i l i t y C

  • l

l i s i

  • n

g u a r a n t e e N u m b e r

  • f


 h a s h f u n c t i

  • n

s

slide-41
SLIDE 41

Mathematical question

  • A (p,r)-covering matrix of dim. d satisfies:
  • Has (1-p)d zeros and pd ones in each row;
  • for every set of r columns there exists a row

with 0s in all of them.

  • Question: 


How few rows can 
 such a matrix have?

13

p = 4/7
 r = 2

S m a l l c

  • l

l i s i

  • n

p r

  • b

a b i l i t y C

  • l

l i s i

  • n

g u a r a n t e e N u m b e r

  • f


 h a s h f u n c t i

  • n

s

slide-42
SLIDE 42

Mathematical question

  • A (p,r)-covering matrix of dim. d satisfies:
  • Has (1-p)d zeros and pd ones in each row;
  • for every set of r columns there exists a row

with 0s in all of them.

  • Question: 


How few rows can 
 such a matrix have?

13

p = 4/7
 r = 2

S m a l l c

  • l

l i s i

  • n

p r

  • b

a b i l i t y C

  • l

l i s i

  • n

g u a r a n t e e N u m b e r

  • f


 h a s h f u n c t i

  • n

s

slide-43
SLIDE 43

Mathematical question

  • A (p,r)-covering matrix of dim. d satisfies:
  • Has (1-p)d zeros and pd ones in each row;
  • for every set of r columns there exists a row

with 0s in all of them.

  • Question: 


How few rows can 
 such a matrix have?

13

p = 4/7
 r = 2

S m a l l c

  • l

l i s i

  • n

p r

  • b

a b i l i t y C

  • l

l i s i

  • n

g u a r a n t e e N u m b e r

  • f


 h a s h f u n c t i

  • n

s

slide-44
SLIDE 44

Mathematical question

  • A (p,r)-covering matrix of dim. d satisfies:
  • Has (1-p)d zeros and pd ones in each row;
  • for every set of r columns there exists a row

with 0s in all of them.

  • Question: 


How few rows can 
 such a matrix have?

13

p = 4/7
 r = 2

S m a l l c

  • l

l i s i

  • n

p r

  • b

a b i l i t y C

  • l

l i s i

  • n

g u a r a n t e e N u m b e r

  • f


 h a s h f u n c t i

  • n

s

In the mathematics literature such a construction is referred to as a covering design

slide-45
SLIDE 45

A covering design

  • Parameters: d = 2r+1-1, pd = 2r+1 ≈ d/2

14

[Gordon, Patashnik, Kuperberg ’95]

slide-46
SLIDE 46

A covering design

  • Parameters: d = 2r+1-1, pd = 2r+1 ≈ d/2

14

001 010 011 101 110 111 100 001 010 011 101 110 111 100

Index (binary) [Gordon, Patashnik, Kuperberg ’95]

slide-47
SLIDE 47

A covering design

  • Parameters: d = 2r+1-1, pd = 2r+1 ≈ d/2

14

Idea: Entry is dot product (mod 2) of row/column ID vectors (Hadamard code)

001 010 011 101 110 111 100 001 010 011 101 110 111 100

Index (binary) [Gordon, Patashnik, Kuperberg ’95]

slide-48
SLIDE 48

A covering design

  • Parameters: d = 2r+1-1, pd = 2r+1 ≈ d/2

14

Idea: Entry is dot product (mod 2) of row/column ID vectors (Hadamard code)

001 010 011 101 110 111 100 001 010 011 101 110 111 100

Index (binary)

(0,1,1)·(0,1,0)=1

[Gordon, Patashnik, Kuperberg ’95]

slide-49
SLIDE 49

A covering design

  • Parameters: d = 2r+1-1, pd = 2r+1 ≈ d/2

14

Idea: Entry is dot product (mod 2) of row/column ID vectors (Hadamard code)

001 010 011 101 110 111 100 001 010 011 101 110 111 100

Index (binary)

(0,1,1)·(0,1,1)=0

[Gordon, Patashnik, Kuperberg ’95]

slide-50
SLIDE 50

15

001 010 011 101 110 111 100 001 010 011 101 110 111 100

Index (binary)

A covering design

  • Parameters: d = 2r+1-1, pd = 2r+1 ≈ d/2

Idea: Entry is dot product (mod 2) of row/column ID vectors (Hadamard code)

[Gordon, Patashnik, Kuperberg ’95]

slide-51
SLIDE 51

15

Lemma:
 For every set of r vectors in {0,1}r+1 there exists a nonzero vector that is orthogonal to all

001 010 011 101 110 111 100 001 010 011 101 110 111 100

Index (binary)

A covering design

  • Parameters: d = 2r+1-1, pd = 2r+1 ≈ d/2

Idea: Entry is dot product (mod 2) of row/column ID vectors (Hadamard code)

[Gordon, Patashnik, Kuperberg ’95]

slide-52
SLIDE 52

16

m(1)

001 010 011 101 110 111 100

Index (binary)

Sampling the covering design

  • Parameters: d = 2r+1-1, pd = 2r+1 ≈ d/2

Idea: Choose columns of covering design using a hash function m: {1,…,d}⟶{0,1}r+1.

m(2) m(3) m(4) m(5) m(d) … … … …

slide-53
SLIDE 53

16

m(1)

001 010 011 101 110 111 100

Index (binary)

Sampling the covering design

  • Parameters: d = 2r+1-1, pd = 2r+1 ≈ d/2

Idea: Choose columns of covering design using a hash function m: {1,…,d}⟶{0,1}r+1.

m(2) m(3) m(4) m(5) m(d) …

Lemma:
 Pr[x ⋀ ai = y ⋀ ai ]
 ≤ 
 2−||x−y||

… … …

slide-54
SLIDE 54

17

Implementation in Python

slide-55
SLIDE 55

Too many collisions?

  • Can map vectors from {0,1}d to {0,1}td, increasing

all distances by an integer factor t.

  • Try to “hit” sweet spot tr = log(n)/c

18

1100101 1101101 1100101 1101101 1100101 1101101 1100101 1101101 1100101 1101101

qt= xt=

slide-56
SLIDE 56

Too many collisions?

  • Can map vectors from {0,1}d to {0,1}td, increasing

all distances by an integer factor t.

  • Try to “hit” sweet spot tr = log(n)/c

18

1100101 1101101 1100101 1101101 1100101 1101101 1100101 1101101 1100101 1101101

qt= xt=

  • For cr=o(log n/log log n), not in SODA version:
  • Use mod p (rather than mod 2) linear algebra to

produce denser coverings.

  • Matches Indyk-Matwani up to polylog(n) factor.
slide-57
SLIDE 57

19

Example

r=6, c=2

1000 106 109 1012 n (size of set) 1000 104 105 106 107 Time

Similarity search, radius 6, approx. factor 2

set)

CoveringLSH (1 partition) Classical LSH, error prob. 1%

slide-58
SLIDE 58

Time 2r too much?

  • Partition dimensions (randomly):





20

Part 1 Part 2 Part s

…

slide-59
SLIDE 59

Time 2r too much?

  • Partition dimensions (randomly):





  • Distance in some part will be ≤ r/s
  • Use CoveringLSH with 


radius r/s on each part

20

Part 1 Part 2 Part s

…

s = 2
 r = 5

slide-60
SLIDE 60

106 1010 1014 1018 n (size of set) 105 109 1013 1017 1021 Time

Similarity search, radius 256, approx. factor 2

Example

21

r=256, c=2

set)

Linear search Classical LSH, error prob. 1/n Classical LSH, error prob. 1% CoveringLSH (multi-partition)

slide-61
SLIDE 61

Some open questions

22

M a t c h p e r f

  • r

m a n c e

  • f

c l a s s i c a l (

  • r

d a t a d e p . ) L S H w i t h

  • u

t f a l s e n e g . ?

slide-62
SLIDE 62

Some open questions

22

M a t c h p e r f

  • r

m a n c e

  • f

c l a s s i c a l (

  • r

d a t a d e p . ) L S H w i t h

  • u

t f a l s e n e g . ? L i n e a r s p a c e n e a r n e i g h b

  • r

s e a r c h w i t h

  • u

t f a l s e n e g a t i v e s : M a t c h i n g k n

  • w

n b

  • u

n d s ?

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SLIDE 63

C

  • n

d i t i

  • n

a l l

  • w

e r b

  • u

n d s f

  • r

a p p r

  • x

i m a t e s i m i l a r i t y s e a r c h ? ( B a s e d

  • n

S E T H , 3 S U M , … )

Some open questions

22

M a t c h p e r f

  • r

m a n c e

  • f

c l a s s i c a l (

  • r

d a t a d e p . ) L S H w i t h

  • u

t f a l s e n e g . ? L i n e a r s p a c e n e a r n e i g h b

  • r

s e a r c h w i t h

  • u

t f a l s e n e g a t i v e s : M a t c h i n g k n

  • w

n b

  • u

n d s ?

slide-64
SLIDE 64

your attention! Thank you for

23

Acknowledgement of economic support: