Large-Scale Similarity Joins With Guarantees · NEW YORK, NY, April 9, 2013—ACM (the Association for Computing Machinery) today announced the winners of six prestigious awards […]

Large-Scale Similarity Joins With Guarantees

Rasmus Pagh IT University of Copenhagen

SISAPOctober 13, 2015

SCALABLESIMILARITYSEARCH

Licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. This images of book covers, movie posters, and research articles and the copyright for them are most likely owned

either by the publishers. It is believed that the use of low-resolution images qualifies as fair use. 1

http://creativecommons.org/licenses/by-sa/3.0/deed.en

http://sss.projects.itu.dk

Talk outline

• In theory…

• The (approximate) similarity join problem

• Techniques for candidate set generation

- Locality-sensitive hashing- Cache-efficiency via recursion- CoveringLSH: Achieving 100% recall

• In practice…

2

In theory…

3

• I was trained as an algorithm theorist

- Worst case assumptions on data

- Big-O notation

- Papers full of math, rarely experiments

4

Confession

• I was trained as an algorithm theorist

- Worst case assumptions on data

- Big-O notation

- Papers full of math, rarely experiments

• “In theory there is no difference between theory and practice… But in practice there is!”

4

Confession

5

Figure 3: Our map of computer science: The map was constructed by embedding the conference graphinto a 2-dimensional Euclidean space. Only top-tier conferences (according to Libra) are shown. Note thatthe map only represents pairwise distances, there is no notion of orientation, i.e. the axes can be chosenarbitrarily.

ACM SIGACT News 56 December 2007 Vol. 38, No. 4

Similarity map of CS conferences

Source: Kuhn & Wattenhofer. The Theoretic Center of Computer Science

SISAP (?)

5

Figure 3: Our map of computer science: The map was constructed by embedding the conference graphinto a 2-dimensional Euclidean space. Only top-tier conferences (according to Libra) are shown. Note thatthe map only represents pairwise distances, there is no notion of orientation, i.e. the axes can be chosenarbitrarily.

ACM SIGACT News 56 December 2007 Vol. 38, No. 4

Similarity map of CS conferences

Source: Kuhn & Wattenhofer. The Theoretic Center of Computer Science

interaction gap

SISAP (?)

Theory with impact• Almost always algorithms

that are easy to describe, implement, and adapt

6

Simpledescription



• Analysis often simple enough to be taught to undergraduates

6

Can be taught

Simpledescription



• Analysis often simple enough to be taught to undergraduates

• Solving a real problem

6

Can be taught

Simpledescription

Applicable

7

NEW YORK, NY, April 9, 2013—ACM (the Association for Computing Machinery) today announced the winners of six prestigious awards […]Andrei Broder, Moses Charikar, and Piotr Indyk, recipients of the Paris Kanellakis Theory and Practice Award for algorithms that allow for quickly finding similar entries in large databases, known as locality-sensitive hashing (LSH). […] The Kanellakis Award honors specific theoretical accomplishments that significantly affect the practice of computing.

Main contributions in the years 1997-2002

8

Now

8

Now

Soon?

8

Now

Soon?

Similarity joins

9

Similarity join example 1 (record linkage)

Country Name

USA IBM

USA Microsoft

Germany SAP

China Baidu

Token ID

Mircosoft 1

SAP SE 2

I.B.M. 3

baidu.com 4

10

Similarity join example 1 (record linkage)

Country Name

USA IBM

USA Microsoft

Germany SAP

China Baidu

Token ID

Mircosoft 1

SAP SE 2

I.B.M. 3

baidu.com 4

10

Class

Mammalia

Mammalia

Reptilia

Aves

Name

Cat

Dog

Snake

Parrot

ID

1

2

3

4

Image

Images by Bodlina and Marek Szczepanek

11

Similarity join example 2 (classification)

Class

Mammalia

Mammalia

Reptilia

Aves

Name

Cat

Dog

Snake

Parrot

ID

1

2

3

4

ImageFeatures

0101101

1001000

1101101

1101110


11


Class

Mammalia

Mammalia

Reptilia

Aves

Name

Cat

Dog

Snake

Parrot

Features

1100101

1101101

1011000

1010111

ID

1

2

3

4

ImageFeatures

0101101

1001000

1101101

1101110


11


Class

Mammalia

Mammalia

Reptilia

Aves

Name

Cat

Dog

Snake

Parrot

Features

1100101

1101101

1011000

1010111

ID

1

2

3

4

ImageFeatures

0101101

1001000

1101101

1101110


11


User Follows

@rasmuspagh1 {@ERC_SSS, @NateSilver538, @NatureNews, @sapinker,@techreview,…}

@Reza_Zadeh {@medialab,@techreview, @AndrewYNg,…}

… …

@simMachines {@neiltyson,@NatureNews, @RichardDawkins,…}

12

Similarity join example 3 (recommendation)

https://twitter.com/neiltyson

Algorithmic problem

• Given a tolerance r compute:

13

Q ./r S = {(q, x) 2 Q⇥ S | ||q � x|| r}

Algorithmic problem


13

Q ./r S = {(q, x) 2 Q⇥ S | ||q � x|| r}

Distance measure

Algorithmic problem


13

• This talk: Consider n vectors in {0,1}d and Hamming distance.

1100101

1101101

1100101

1101101

q

x

=

=

Q ./r S = {(q, x) 2 Q⇥ S | ||q � x|| r}

||q � x|| = 1

Distance measure

Variants

• kNN similarity join

• Batched similarity search

14

15

Many names…

15

2015 Many names…

Similarity join in a picture

16

Q ./r SQR=


16

Q ./r SQR=


16

Q ./r SQR=

Similarity join in a picture in high dimensional space (d=80)

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

17

Why is this hard?

18

Why is this hard?

18

Because of the CURSE of

dimensionality!

Why is this hard?

• [Williams ’04], [Alman & Williams ’15]: Hamming similarity search in time n0.99 2o(d) ⟹

k-SAT w. n variables can be solved in time cn, c < 2

18

Why is this hard?

• [Williams ’04], [Alman & Williams ’15]: Hamming similarity search in time n0.99 2o(d) ⟹

k-SAT w. n variables can be solved in time cn, c < 2

18

Strong ETH states that this is not possible

19

c-approximate similarity joinQ ./r SQ◆C

20

c-approximate similarity joinQ ./r SQ◆C

20

c-approximate similarity joinQ ./r SQ◆C False positive pairs,

filter away in time|Q ./cr S|

20

Effect of approximation

21

10−3 10−2 10−1 10010−5

10−4

10−3

10−2

10−1

100

Jaccard distance

(b) Enron email dataset

CD

F of

pai

rwis

e di

stan

ces

103 104 10510−5

10−4

10−3

10−2

10−1

100

L1 distance

(a) MNIST dataset

CD

F of

pai

rwis

e di

stan

ces

Frac

tion

of a

ll pa

irs

within Hamming distance

MNIST data set (unary encoding)

r


21

10−3 10−2 10−1 10010−5

10−4

10−3

10−2

10−1

100

Jaccard distance


CD

F of

pai

rwis

e di

stan

ces

103 104 10510−5

10−4

10−3

10−2

10−1

100

L1 distance

(a) MNIST dataset

CD

F of

pai

rwis

e di

stan

ces

Frac

tion

of a

ll pa

irs



r

Q ./r S


21

10−3 10−2 10−1 10010−5

10−4

10−3

10−2

10−1

100

Jaccard distance


CD

F of

pai

rwis

e di

stan

ces

103 104 10510−5

10−4

10−3

10−2

10−1

100

L1 distance

(a) MNIST dataset

CD

F of

pai

rwis

e di

stan

ces

Frac

tion

of a

ll pa

irs



r cr

Q ./r S

Q ./cr S


21

10−3 10−2 10−1 10010−5

10−4

10−3

10−2

10−1

100

Jaccard distance


CD

F of

pai

rwis

e di

stan

ces

103 104 10510−5

10−4

10−3

10−2

10−1

100

L1 distance

(a) MNIST dataset

CD

F of

pai

rwis

e di

stan

ces

Frac

tion

of a

ll pa

irs



Additional fraction of pairs that may

become candidates

r cr

Q ./r S

Q ./cr S


21

10−3 10−2 10−1 10010−5

10−4

10−3

10−2

10−1

100

Jaccard distance


CD

F of

pai

rwis

e di

stan

ces

103 104 10510−5

10−4

10−3

10−2

10−1

100

L1 distance

(a) MNIST dataset

CD

F of

pai

rwis

e di

stan

ces

Frac

tion

of a

ll pa

irs



Additional fraction of pairs that may

become candidates

r cr

In rest of the talk: Assumeis not much larger than

|Q ./cr S|

|Q ./r S|Q ./r S

Q ./cr S

Candidate set generation- Locality-sensitive hashing (LSH)

22

Locality-sensitive hashing

23

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]


23

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



23

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




23

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


h(x) = x ⋀ a



23

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


Candidate m

atch

h(x) = x ⋀ a



23

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


Candidate m

atch

Repeat enough times that vectors at distance r produce

at least one collision

h(x) = x ⋀ a



23

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


Candidate m

atch

Repeat enough times that vectors at distance r produce

at least one collision

h(x) = x ⋀ ahi(x) = x ⋀ ai


Probability of collision?

24

Collision probability for ith hash table:(1� p)||x�q|| ⇡ e�p||x�q||


Pr[x ⋀ ai = q ⋀ ai] =


24


With , probability ≈ at distance cr and ≈ at distance r1/n

1/n1/c

p = ln(n)cr


Pr[x ⋀ ai = q ⋀ ai] =


24

x ⋀ a1

x ⋀ a2

x ⋀ a3

⠇x ⋀ at

q ⋀ a1

q ⋀ a2

q ⋀ a3

⠇q ⋀ at

?=

?=

?=

?=



1/n1/c

p = ln(n)cr

repetitions ensure

constant success

probability

t = n1/c


Pr[x ⋀ ai = q ⋀ ai] =


24

x ⋀ a1

x ⋀ a2

x ⋀ a3

⠇x ⋀ at

q ⋀ a1

q ⋀ a2

q ⋀ a3

⠇q ⋀ at

?=

?=

?=

?=



1/n1/c

p = ln(n)cr

repetitions ensure

constant success

probability

t = n1/c

Possibility of false negatives: Saying ‘No’ when ‘Yes’ is

required


Pr[x ⋀ ai = q ⋀ ai] =

Summary of analysis

Analysis (GIM ’99): Each bit of ai is 1 with probability p.25

Summary of analysis


Summary of analysis



26


26

Number of operations is O(dn1+1/c), expected;

Newer theory developments*

27

Reference Exponent, search time Comment

Linear search 1Indyk & Motwani

STOC ’98 1/c Worst case upper bound

* Focus on subquadratic space; lower order terms ignored.


27




O’Donnell, Wu & ZhouITCS ’10 1/c Data indep. LSH, worst case lower bound



27





DubinerTrans. Inf. Theory ’10 1/(2c-1) Random data upper bound



27






Andoni & RazenshteynSTOC ‘15 1/(2c-1) Data dep. LSH, worst case upper bound



27








and lower bound


27







KapralovPODS ‘15 4/(c+1) Linear space upper bound


and lower bound


27








ValiantFOCS ‘12

Batched search [random data]


1

4�! +O⇣

1�1/clog d

⌘

and lower bound

ω < 2.38


27








ValiantFOCS ‘12


Karppa, Kaski & Kohonen SODA ‘16

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


1

4�! +O⇣

1�1/clog d

⌘

and lower bound

2!�33

ω < 2.38


27








ValiantFOCS ‘12


Karppa, Kaski & Kohonen SODA ‘16

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

Alman & WilliamsFOCS ‘15

Batched search, c=11� ˜

⌦(log(n)/d)


1

4�! +O⇣

1�1/clog d

⌘

and lower bound

2!�33

ω < 2.38

Can LSH work for large data?• Quote from a database paper:

“LSH needs large memory space and long processing time to achieve good performance when searching a massive dataset”.

28



28

For similarity join, space is linear:

Process one hash function at a time.



• Other issues:- LSH parameters are pessimistic;

chosen to work for worst-case data set.

28





• Other issues:- LSH parameters are pessimistic;

chosen to work for worst-case data set.- Poor use of internal memory: A simple nested loop join

often has better I/O complexity.

28



On pessimism

• LSH chosen to ensure few collisions at distance cr, even in worst-case scenarios:

29

On pessimism

• LSH chosen to ensure few collisions at distance cr, even in worst-case scenarios:

29

Price paid: Also small collision

probability at distance r, so need many repetitions.

Candidate set generation- Cache-efficiency via recursion

30

I/O model

• Data is stored on an external storage device, with B vectors / storage block

• Internal memory can hold M vectors

• Count the number of block transfers between internal memory and external storage (I/Os)

31

D

P

M

BlockI/O

Figure courtesy of Lars Arge

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

Cautious LSH

32

Joint work with Pham, Silvestri, and Stöckel

Idea: Apply a weak LSH with constant

collision probability.

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

Cautious LSH

32


Idea: Apply a weak LSH with constant

collision probability.

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 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 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 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 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 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 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 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 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 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 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 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 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

Cautious LSH

33


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 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 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 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 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 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 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 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 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 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 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 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 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

Cautious LSH

33

RECURSIVE SUBPROBLEM



Q0 ./r S0

Q1 ./r S1

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 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 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 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 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 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 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 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 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 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 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 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 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

Cautious LSH

33




Subproblems of size M or less require no further I/Os

Q0 ./r S0

Q1 ./r S1

Example, cautious LSH• Assume weak LSH collision prob. 0.9.

• Prob. that q and x collide i times is 0.9i.• If depth is log n, success probability is ≈ n-0.152.

34

Q ./r S

Q0 ./r S0

Q00 ./r S00 Q01 ./r S01 Q10 ./r S10 Q11 ./r S11

Q1 ./r S1

qx

Example, cautious LSH• Assume weak LSH collision prob. 0.9.

• Prob. that q and x collide i times is 0.9i.• If depth is log n, success probability is ≈ n-0.152.

34

Q ./r S

Q0 ./r S0

Q00 ./r S00 Q01 ./r S01 Q10 ./r S10 Q11 ./r S11

Q1 ./r S1

q x

Example 2, cautious LSH• Assume weak LSH collision prob. 0.5.

• Prob. that q and x collide i times is 0.5i.

35

Q ./r S

Q0 ./r S0 Q1 ./r S1

qx

Q00 ./r S00 Q01 ./r S01 Q10 ./r S10 Q11 ./r S11


• Prob. that q and x collide i times is 0.5i.• Idea: Recurse twice at each node.

35

Q ./r S

Q0 ./r S0 Q1 ./r S1

qx

Q00 ./r S00 Q01 ./r S01 Q10 ./r S10 Q11 ./r S11


• Prob. that q and x collide i times is 0.5i.• Idea: Recurse twice at each node.

35

Q ./r S

Q0 ./r S0 Q1 ./r S1

qx

#subproblems containing q and x: 1 at each level, expected!

Aside: Don’t be fooled by great expectations

36

Aside: Don’t be fooled by great expectations

36

What is the expected TNT equivalent of asteroid impacts during this talk?

Branching processes

• Abstract setting:

- One pair (q,x) at root problem.

- Generate t subproblems such that (q,x) is “reproduced” in each with probability ≥ 1/t.

37

Branching processes




• What is the probability that (q,x) is extinct at recursive level i?

37

Branching processes




• What is the probability that (q,x) is extinct at recursive level i?

- From theory of branching processes: Ω(1/√i)

37

I/O complexity, sketch

38

recursion tree


38

recursion tree

n

Size

of a

ll su

bpro

blem

s


38

recursion tree

n

nt

Size

of a

ll su

bpro

blem

s


38

recursion tree

n

nt

nt2

nt3

Size

of a

ll su

bpro

blem

s

⠇


38

recursion tree

n

nt

nt2

nt3

Size

of a

ll su

bpro

blem

s

Expe

cted

ave

rage

#p

oint

s at d

ista

nce

> cr

.n

⠇


38

recursion tree

n

nt

nt2

nt3

Size

of a

ll su

bpro

blem

s

Expe

cted

ave

rage

#p

oint

s at d

ista

nce

> cr

.nn/tc

⠇


38

recursion tree

n

nt

nt2

nt3

Size

of a

ll su

bpro

blem

s

Expe

cted

ave

rage

#p

oint

s at d

ista

nce

> cr

.nn/tc

n/t2c

n/t3c

⠇ ⠇


38

recursion tree

n

nt

nt2

nt3

Size

of a

ll su

bpro

blem

s

Expe

cted

ave

rage

#p

oint

s at d

ista

nce

> cr

.nn/tc

n/t2c

n/t3c

⠇ ⠇In-memory computation


38

recursion tree

n

nt

nt2

nt3

Size

of a

ll su

bpro

blem

s

Expe

cted

ave

rage

#p

oint

s at d

ista

nce

> cr

.nn/tc

n/t2c

n/t3c


log

tc(n/M

)


38

recursion tree

n

nt

nt2

nt3

Size

of a

ll su

bpro

blem

s

Expe

cted

ave

rage

#p

oint

s at d

ista

nce

> cr

.nn/tc

n/t2c

n/t3c


log

tc(n/M

)

Pessimistic

I/O complexity

39

O

0

@⇣ n

M

⌘1/c

0

@ n

B+

|Q ./r

S|

MB

1

A

1

A I/Os

• Simplified:

Assumingis not much larger than

|Q ./cr S|

|Q ./r S|

I/O complexity

39

O

0

@⇣ n

M

⌘1/c

0

@ n

B+

|Q ./r

S|

MB

1

A

1

A I/Os

• Simplified:


|Q ./cr S|

|Q ./r S|

Cost of reading input

I/O complexity

39

O

0

@⇣ n

M

⌘1/c

0

@ n

B+

|Q ./r

S|

MB

1

A

1

A I/Os

• Simplified:


|Q ./cr S|

|Q ./r S|


Cost of generating

output

I/O complexity

39

O

0

@⇣ n

M

⌘1/c

0

@ n

B+

|Q ./r

S|

MB

1

A

1

A I/Os

• Simplified:


|Q ./cr S|

|Q ./r S|


Cost of generating

outputOverhead

I/O complexity

• In general:

39

O

0

@⇣ n

M

⌘1/c

0

@ n

B+

|Q ./r

S|

MB

1

A+

|Q ./cr

S|

MB

1

A I/Os

O

0

@⇣ n

M

⌘1/c

0

@ n

B+

|Q ./r

S|

MB

1

A

1

A I/Os

• Simplified:

Candidate set generation- CoveringLSH: Achieving 100% recall

40

41

Approximation

Two kinds of approximation:

• Approximate distances

• Allow false positives and negatives (precision and recall below 100%)

42

Approximation




42

Inherent to LSH

Approximation




42

Inherent to LSH

Total recall possible!

Basic correlated LSH

43

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]


43

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



43

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



43

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



43

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



43

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



43

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



44

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



45

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



46

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



46

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


Result in LSH framework• The bit sampling LSH achieves:

47

Pr[h(q) = h(x) | ||x� q|| = cr] Pr[h(q) = h(x) | ||x� q|| = r]c


• Bound on PartEnum construction with partitioning + enumeration + permutation:

47

Pr[h(q) = h(x) | ||x� q|| = cr] Pr[h(q) = h(x) | ||x� q|| = r]0.36 c




47



Need ≈ 3× larger c



47



Need ≈ 3× larger c

Probabilistic argument suggests that it is possible to do much better. But how?

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.

48




48

Small collision

probability




48

Small collision

probability

Collision guarantee




• Question: How few rows can such a matrix have?

48

Small collision

probability

Collision guarantee





48

Small collision

probability

Collision guarantee

Number of hash functions

� � � � � � ��





48

p = 4/7r = 2

Small collision

probability

Collision guarantee


� � � � � � ��





48

p = 4/7r = 2

Small collision

probability

Collision guarantee


� � � � � � ��





48

p = 4/7r = 2

Small collision

probability

Collision guarantee


� � � � � � ��





48

p = 4/7r = 2

Small collision

probability

Collision guarantee


Mathematical question

49

Related to covering problems in extremal combinatorics, but good constructions have been

known only for p=O(1/d)

• A (p,r)-covering matrix of dim. d satisfies:




CoveringLSH• Next: Answer for d = 2r+1-1, pd = 2r+1 ≈ d/2

50

[P., SODA ’16]

� � � � � � ��


50

001

010

011

101

110

111

100

001010011

101110111

100

Inde

x (b

inar

y)

[P., SODA ’16]

� � � � � � ��


50

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

001

010

011

101

110

111

100

001010011

101110111

100

Inde

x (b

inar

y)

[P., SODA ’16]

� � � � � � ��


50


001

010

011

101

110

111

100

001010011

101110111

100

Inde

x (b

inar

y)

[P., SODA ’16]

(0,1,1)·(0,1,0)=1

� � � � � � ��


50


001

010

011

101

110

111

100

001010011

101110111

100

Inde

x (b

inar

y)

[P., SODA ’16]

(0,1,1)·(0,1,1)=0

� � � � � � ��


51


001

010

011

101

110

111

100

001010011

101110111

100

Inde

x (b

inar

y)

[P., SODA ’16]

� � � � � � ��


51


001

010

011

101

110

111

100

001010011

101110111

100

Inde

x (b

inar

y)

[P., SODA ’16]

� � � � � � ��


51

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

001010011

101110111

100

Inde

x (b

inar

y)

[P., SODA ’16]

� � � � � � ��


51

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

001010011

101110111

100

Inde

x (b

inar

y)

[P., SODA ’16]

Optimality?

• Could there be a smaller (4/7,2)-covering?

• We need to “cover” sets of two columns

52

✓7

2

◆= 21

� � � � � � ��

Optimality?



• Each vector can cover sets of two columns

52

✓7

2

◆= 21

✓3

2

◆= 3

� � � � � � ��

Optimality?




52

✓7

2

◆= 21

✓3

2

◆= 3 7 vectors is

optimal!

� � � � � � ��

Optimality?




52

✓7

2

◆= 21

✓3

2

◆= 3 7 vectors is

optimal!

� � � � � � ��

For r>2: Within factor 2 of optimal

Optimality?

• Could there be a smaller (1/2,r)-covering?

• We need to cover sets of r columns

53

✓d

r

◆

2r+1�

1

2r+1 � 1� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Optimality?



• Each vector can cover sets of r columns

53

✓d

r

◆

✓d/2

r

◆

2r+1�

1

2r+1 � 1� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Optimality?



• Each vector can cover sets of r columns

53

✓d

r

◆

✓d/2

r

◆

Within factor 2 of optimal!

�dr

�/�d/2

r

�> 2r

2r+1�

1

2r+1 � 1� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Use in similarity search• Combine with random permutation trick:

Each bit sampled w. prob. 1/2 in each hash value.

54



• “Sweet spot” is for r = log(n)/c:

- 2r+1 = 2 n1/c hash functions

- Collision probability 1/n at distance cr = log(n), so number of “far” collisions insignificant

54



• “Sweet spot” is for r = log(n)/c:

- 2r+1 = 2 n1/c hash functions

- Collision probability 1/n at distance cr = log(n), so number of “far” collisions insignificant

• Matches bound of Indyk and Motwani.

54

Smaller radius?

• 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

- Details: arXiv:1507.03225 [cs.DS]

55

1100101

1101101

1100101

1101101

1100101

1101101

1100101

1101101

1100101

1101101

qt=xt=

http://arxiv.org/abs/1507.03225

Example

56

106 1010 1014 1018n (size of set)

1

104

108

1012

1016

1020

TimeSimilarity search, radius 8, approx. factor 3

Linear searchExhaustive search in Hamming ballClassical LSH, error prob. 1/nClassical LSH, error prob. 1%CoveringLSH (1 partition)

106 1010 1014 1018n (size of set)

1

104

108

1012

1016

1020


Linear searchExhaustive search in Hamming ballClassical LSH, error prob. 1/nClassical LSH, error prob. 1%CoveringLSH (1 partition)

r=8, c=3

Larger radius?• Partition to reduce to pd < d/2 sampled bits:

- Iterate over 1/(2p) parts

57

Part 1 Part 2 Part 1/(2p)…

� � � � � � � � � � � � � ��


Larger radius?• Partition to reduce to pd < d/2 sampled bits:

- Iterate over 1/(2p) parts

• Distance in some part will be ≤ 2pr

- Use CoveringLSH with radius 2pr on each part

- Details: arXiv:1507.03225 [cs.DS]

57

Part 1 Part 2 Part 1/(2p)…

� � � � � � � � � � � � � ��


106 1010 1014 1018n (size of set)

105

109

1013

1017

1021


Linear searchClassical LSH, error prob. 1/nClassical LSH, error prob. 1%CoveringLSH (1 partition)

Example

58

r=256, c=2

106 1010 1014 1018n (size of set)

105

109

1013

1017

1021


Linear searchClassical LSH, error prob. 1/nClassical LSH, error prob. 1%CoveringLSH (multi-partition)

59

(d+1)r

Euclidean space - shifted grids

60

(d+1)r


61

(d+1)r


61

(d+1)r

In same cell ⇒ d+1-approximate

near neighbor


61

(d+1)r

In same cell ⇒ d+1-approximate

near neighbor

Time d+1.Approx. factor d+1.


JL dimension reduction

62

Euclidean vector x

random linear mapping

Length concentrated around

Projection

||x||2

JL dimension reduction

62

Euclidean vector x

random linear mapping

Length concentrated around

Projection

Lengths can increase ⇒false negatives possible

||x||2

Correlated dimension reduction

63

Euclidean vector x

Rotated vector 𝜋(x)

Projection 1 Projection 2 Projection t

random rotation

partitioning, scaling by t

…


63

Euclidean vector x



random rotation


…

Length concentrated around||x||2


63

Euclidean vector x



random rotation


…

Length concentrated around||x||2

Some projection will have length at most ||x||2

Correlated Euclidean LSH

• d nÕ(1/c) hash functions

• Collision guaranteed within distance r

• Collision probability 1/n at distance cr

64

(joint work with Matthew Skala)

In practice…

65

In practice…

66

?

• What are the good use cases for similarity join?

• When is 100% recall of particular value?

In practice…

66

Can be taught

Simpledescription

Applicable

?

• What are the good use cases for similarity join?

• When is 100% recall of particular value?

Linking new theory to practice?

67

Match performance of classical (or data dep.) LSH without false neg.?

(SODA ’16)


67


Indexing: When is

sublinear query time

possible with linear space

(PODS ’15)(SODA ’16)


67


Indexing: When is



Making new LSH

constructions truly

practical?

(PODS ’15)

(NIPS ’15)

(SODA ’16)


67


Indexing: When is



Data dep. LSH that works in theory and in practice?

Making new LSH

constructions truly

practical?

(PODS ’15)

(NIPS ’15) (STOC ’15)

(SODA ’16)

Thank you

68

To people with whom I have discussed material of this talk: Annalisa De Bonis, Francesco Silvestri, Ilya Razenshteyn, Johan von Tangen Sivertsen, Matthew Skala, Ninh Pham, Riko Jacob, Thomas Dybdahl Ahle, Tobias Christiani, Ugo Vaccaro, and more.

For economic support:

Thank you

69

for yourattention!

Documents

Large-Scale Similarity Joins With Guarantees · NEW YORK, NY, April 9, 2013—ACM (the Association for Computing Machinery) today announced the winners of six prestigious awards […]