Effective Web Graph Representations Giulio Ermanno Pibiri - - PowerPoint PPT Presentation

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Effective Web Graph Representations Giulio Ermanno Pibiri - - PowerPoint PPT Presentation

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Effective Web Graph Representations Giulio Ermanno Pibiri University of Pisa and ISTI-CNR Pisa, Italy giulio.pibiri@di.unipi.it Pisa, 29/10/2018 1 Context - Web Graphs Web graphs are directed graphs of pages pointing to other pages on the

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SLIDE 1

Effective Web Graph Representations

Pisa, 29/10/2018

1

Giulio Ermanno Pibiri

University of Pisa and ISTI-CNR Pisa, Italy giulio.pibiri@di.unipi.it

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SLIDE 2

Context - Web Graphs

2

We focus on compression effectiveness

  • n large real-world Web graphs.

Web graphs are directed graphs of pages pointing to other pages on the Web.

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SLIDE 3

1: 2,3 2: - 3: 4 4: 7 5: 1,4,6,7 6: 7 7: -

3

Context - Web Graphs

1 1 1 1 1 1 1 1 1

1 6 7 3 2 5 4

Conceptual graph Adjacency matrix Adjacency lists

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SLIDE 4

1: 2,3 2: - 3: 4 4: 7 5: 1,4,6,7 6: 7 7: -

3

Context - Web Graphs

1 1 1 1 1 1 1 1 1

1 6 7 3 2 5 4

Conceptual graph Adjacency matrix Adjacency lists

Many results are known for compressing integer sequences.

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SLIDE 5

4

Outline

  • 1. The WebGraph framework
  • 2. k2-trees
  • 3. Block-trees
  • 4. 2D-Block-trees
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SLIDE 6

4

Outline

  • 1. The WebGraph framework
  • 2. k2-trees
  • 3. Block-trees
  • 4. 2D-Block-trees

2004 2009 2014 2018

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SLIDE 7

5

The WebGraph Framework

The WebGraph Framework I: Compression Techniques, Boldi-Vigna, WWW 2004

Java/C++ framework consisting in algorithms and compression codes for managing large Web Graphs. http://webgraph.di.unimi.it/

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SLIDE 8

5

The WebGraph Framework

The WebGraph Framework I: Compression Techniques, Boldi-Vigna, WWW 2004

Java/C++ framework consisting in algorithms and compression codes for managing large Web Graphs. http://webgraph.di.unimi.it/

Locality - pages links to pages whose URL is lexicographically similar. URLs
 share long common prefixes.

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SLIDE 9

5

The WebGraph Framework

The WebGraph Framework I: Compression Techniques, Boldi-Vigna, WWW 2004

Java/C++ framework consisting in algorithms and compression codes for managing large Web Graphs. http://webgraph.di.unimi.it/

Locality - pages links to pages whose URL is lexicographically similar. URLs
 share long common prefixes.

Use d-gap compression.

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SLIDE 10

5

The WebGraph Framework

The WebGraph Framework I: Compression Techniques, Boldi-Vigna, WWW 2004

Java/C++ framework consisting in algorithms and compression codes for managing large Web Graphs. http://webgraph.di.unimi.it/

Locality - pages links to pages whose URL is lexicographically similar. URLs
 share long common prefixes. Similarity - pages that are close together in lexicographic order, tend to have many common successors.

Use d-gap compression.

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SLIDE 11

5

The WebGraph Framework

The WebGraph Framework I: Compression Techniques, Boldi-Vigna, WWW 2004

Java/C++ framework consisting in algorithms and compression codes for managing large Web Graphs. http://webgraph.di.unimi.it/

Locality - pages links to pages whose URL is lexicographically similar. URLs
 share long common prefixes. Similarity - pages that are close together in lexicographic order, tend to have many common successors.

Use d-gap compression. Use reference compression.

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SLIDE 12

Adjacency lists d-gapped adjacency lists If we have: x: [y1,…,yk], then we represent [y1 - x, y2 - y1 - 1, y3 -y2 -1,…,yk - yk-1 -1]

First gap d=y1 - x is represented as 2d if d ≥ 0 or 2|d|-1 if d < 0

6

The WebGraph Framework

Exploiting locality.

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SLIDE 13

7

The WebGraph Framework

Exploiting similarity.

Idea: use reference compression, i.e., represent a list with respect to another one called its reference list.

1 2 3 4 Copy lists Adjacency lists

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SLIDE 14

8

The WebGraph Framework

Copy lists Copy blocks

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SLIDE 15

8

The WebGraph Framework

Copy lists Copy blocks Intervals An interval is a run, of size ≥ L, of consecutive integers. Example for L = 2.

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SLIDE 16

9

The WebGraph Framework

Reference List

W

List

W - window size R - maximum reference chain

Tradeoff between compression and decoding time.

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SLIDE 17

10

The WebGraph Framework

.uk 18.5 million pages 300 million links WebBase 118 million pages 1 billion links

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SLIDE 18

10

The WebGraph Framework

.uk 18.5 million pages 300 million links WebBase 118 million pages 1 billion links

Compression rates down to approximately 3 bits per link.

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SLIDE 19

10

The WebGraph Framework

.uk 18.5 million pages 300 million links WebBase 118 million pages 1 billion links

Compression rates down to approximately 3 bits per link.

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SLIDE 20

1 1 1 1 1 1 1

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2

Credits to A. Gómez-Brandón

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SLIDE 21

1 1 1 1 1 1 1

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

Credits to A. Gómez-Brandón

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SLIDE 22

1 1 1 1 1 1 1

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1

Credits to A. Gómez-Brandón

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SLIDE 23

1 1 1 1 1 1 1

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1

Credits to A. Gómez-Brandón

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SLIDE 24

1 1 1 1 1 1 1

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1

Credits to A. Gómez-Brandón

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SLIDE 25

1 1 1 1 1 1 1

1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1

Credits to A. Gómez-Brandón

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SLIDE 26

1 1 1 1 1 1 1

1 0 0 0

1 1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1

Credits to A. Gómez-Brandón

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SLIDE 27

1 1 1 1 1 1 1

1 0 0 1

1 1

1 0 0 0

1 1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1

Credits to A. Gómez-Brandón

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SLIDE 28

1 1 1 1 1 1 1

1 0 0 1

1 1

1 0 0 0

1 1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1 00 01

Credits to A. Gómez-Brandón

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SLIDE 29

1 1 1 1 1 1 1

1 0 0 1

1 1

1 0 0 0

1 1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1 00 0111 01

Credits to A. Gómez-Brandón

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SLIDE 30

1 1 1 1 1 1 1

1 0 0 1

1 1

1 0 0 0

1 1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1 00 0111 0100 01

Credits to A. Gómez-Brandón

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SLIDE 31

1 1 1 1 1 1 1

1 0 0 1

1 1

1 0 0 0

1 1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1 00 0111 0100 01 00 10

Credits to A. Gómez-Brandón

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SLIDE 32

1 1 1 1 1 1 1

1 0 0 1

1 1

1 0 0 0

1 1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1 00 0111 0100 01 00 10 01 00

Credits to A. Gómez-Brandón

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SLIDE 33

1 1 1 1 1 1 1

1 0 0 1

1 1

1 0 0 0

1 1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1 00 0111 0100 01 00 10 01 00

1110101010001001 00011101000100100100 T L

Credits to A. Gómez-Brandón

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SLIDE 34

1 1 1 1 1 1 1

1 0 0 1

1 1

1 0 0 0

1 1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1 00 0111 0100 01 00 10 01 00

1110101010001001 00011101000100100100 T L

Access

Credits to A. Gómez-Brandón

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SLIDE 35

1 1 1 1 1 1 1

1 0 0 1

1 1

1 0 0 0

1 1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1 00 0111 0100 01 00 10 01 00

1110101010001001 00011101000100100100 T L 1

1 Access

Credits to A. Gómez-Brandón

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SLIDE 36

1 1 1 1 1 1 1

1 0 0 1

1 1

1 0 0 0

1 1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1 00 0111 0100 01 00 10 01 00

1110101010001001 00011101000100100100 T L 1

1 Access

Credits to A. Gómez-Brandón

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SLIDE 37

1 1 1 1 1 1 1

1 0 0 1

1 1

1 0 0 0

1 1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1 00 0111 0100 01 00 10 01 00

1110101010001001 00011101000100100100 T L 1

1

1

1 Access

Credits to A. Gómez-Brandón

slide-38
SLIDE 38

1 1 1 1 1 1 1

1 0 0 1

1 1

1 0 0 0

1 1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1 00 0111 0100 01 00 10 01 00

1110101010001001 00011101000100100100 T L 1

1

1

1

1

Access

Credits to A. Gómez-Brandón

slide-39
SLIDE 39

1 1 1 1 1 1 1

1 0 0 1

1 1

1 0 0 0

1 1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1 00 0111 0100 01 00 10 01 00

1110101010001001 00011101000100100100 T L 1

1

1

1

1

1

1

Access

Credits to A. Gómez-Brandón

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SLIDE 40

1 1 1 1 1 1 1

1 0 0 1

1 1

1 0 0 0

1 1 1

1 0 1 0

11

k2-trees

k2-trees for Compact Web Graph Representation, Brisaboa-Ladra-Navarro, SPIRE 2009

k2-ary tree representation of the adjacency matrix. k = 2 1

1

1

1 1

1 00 0111 0100 01 00 10 01 00

1110101010001001 00011101000100100100 T L 1

1

1

1

1

1 rank1 x k2 on T in O(1)

1

Access

Credits to A. Gómez-Brandón

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SLIDE 41

12

k2-trees

h = dlogk2 n2e

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n

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pages links

slide-42
SLIDE 42

12

k2-trees

h = dlogk2 n2e

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m

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n

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pages links

mk2dlogk2 n2e

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So the total space is bits

slide-43
SLIDE 43

12

k2-trees

h = dlogk2 n2e

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m

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n

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pages links

mk2dlogk2 n2e

<latexit sha1_base64="ZcH8CFXU8ogNkvQgJVXU7TE92p8=">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</latexit><latexit sha1_base64="ZcH8CFXU8ogNkvQgJVXU7TE92p8=">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</latexit><latexit sha1_base64="ZcH8CFXU8ogNkvQgJVXU7TE92p8=">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</latexit><latexit sha1_base64="ZcH8CFXU8ogNkvQgJVXU7TE92p8=">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</latexit>

So the total space is bits = k2 m−1

k2−1 + mk2 logk2 n2 m

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= mk2⇣ logk2 n2

m + O(1)

⌘

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bits

k2 Õblogk2 mc1

i=0

k2i + mk2(dlogk2 ne blogk2 mc)

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slide-44
SLIDE 44

12

k2-trees

h = dlogk2 n2e

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m

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n

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pages links

mk2dlogk2 n2e

<latexit sha1_base64="ZcH8CFXU8ogNkvQgJVXU7TE92p8=">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</latexit><latexit sha1_base64="ZcH8CFXU8ogNkvQgJVXU7TE92p8=">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</latexit><latexit sha1_base64="ZcH8CFXU8ogNkvQgJVXU7TE92p8=">ACjnicbVFNTxsxEHW2QGn4CvTYi0WC1AOKdiMEvURF9MIREAGkZBt5ndlgxetd7FlKauUP8Gu4wj/h3+BdIkQCI1l+evPGnkTZVIY9P3nivdlYXHp6/K36srq2vpGbXPrwqS5tDhqUz1VcQMSKGgwIlXGUaWBJuIxGf4r85S1oI1J1juMwoQNlYgFZ+iofq3RSOjob4v2JAch3ZUO+9YRE6oKVhdso1+r+02/DPoRBFNQJ9M46W9WznqDlOcJKOSGdMN/AxDyzQKLmFS7eUGMsZHbAhdBxVLwIS2HGdCdxwzoHGq3VFIS/Z9hWJMeMkcsqE4bWZzxXkZ7lujvGv0AqV5QiKv34U5JiSgtv6EBo4CjHDjCuheuV8mumGUfn4MxLiO2YSQO7ZWHQRp1DaKWIwE2o5ga8ZfomD+1/sTfLv8lDq+Af3pV9zyjOg9AWHhTdVt0WgnP4KLVjPwm8Fpq354N3HMvlBtslPEpADckiOyQnpE7uyQN5JE9ezdv32t7vV6lXmdZ8JzPhHb8AMQvJZw=</latexit><latexit sha1_base64="ZcH8CFXU8ogNkvQgJVXU7TE92p8=">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</latexit>

So the total space is bits = k2 m−1

k2−1 + mk2 logk2 n2 m

<latexit sha1_base64="IaMT6pNFtZSw6pMZqG5LsTnJFM=">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</latexit><latexit sha1_base64="IaMT6pNFtZSw6pMZqG5LsTnJFM=">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</latexit><latexit sha1_base64="IaMT6pNFtZSw6pMZqG5LsTnJFM=">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</latexit><latexit sha1_base64="IaMT6pNFtZSw6pMZqG5LsTnJFM=">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</latexit>

= mk2⇣ logk2 n2

m + O(1)

⌘

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bits

k2 Õblogk2 mc1

i=0

k2i + mk2(dlogk2 ne blogk2 mc)

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log n2

m

≈ m log n2

m + O(m)

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Information theoretic lower bound

slide-45
SLIDE 45

12

k2-trees

h = dlogk2 n2e

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m

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n

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pages links

mk2dlogk2 n2e

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So the total space is bits = k2 m−1

k2−1 + mk2 logk2 n2 m

<latexit sha1_base64="IaMT6pNFtZSw6pMZqG5LsTnJFM=">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</latexit><latexit sha1_base64="IaMT6pNFtZSw6pMZqG5LsTnJFM=">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</latexit><latexit sha1_base64="IaMT6pNFtZSw6pMZqG5LsTnJFM=">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</latexit><latexit sha1_base64="IaMT6pNFtZSw6pMZqG5LsTnJFM=">ACo3icbVFNbxMxEHWj5bwlcKRi0VaCQmI1hES9BCpgsSl1IlbaVkibzObGrF9m7t2Zg7W/h13CFM/8G7yZCJGUky89v3tgz2mhpM4/t2Kbt2+c3dn917/oOHjx539p6cury0AkYiV7k9T7kDJQ2MUKC8IC16mCs3Txoc6fXYF1MjdDXBaQaD43MpOCY6CmncN9OqCL306ySwXr9mlQ/HsNGXVDcJlc+nNVetJCYgr6v9acb9+Im6E3A1qBL1nE83WudTGa5KDUYFIo7N2ZxgYnFqVQULUnpYOCiwWfwzhAwzW4xDczVvQgMDOa5TYsg7Rh/63wXDu31GlQao4XbjtXk/LjUvM3iVemqJEMGL1UFYqijmtDaMzaUGgWgbAhZWhVyoueHACg60bNyEOMq4cvGoK2QBtCYlXMoUwodka8IrbyzLx3+SbTf6vPEGrvFr0/eGYsgSX3tQd9sOv8C2Pb8JTvs9FvfY53736P36P3bJM/KcvCMvCVH5CM5JiMiyHfyg/wkv6KD6FN0Eg1X0qi1rnlKNiJK/gC8M9FN</latexit>

= mk2⇣ logk2 n2

m + O(1)

⌘

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bits

k2 Õblogk2 mc1

i=0

k2i + mk2(dlogk2 ne blogk2 mc)

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log n2

m

≈ m log n2

m + O(m)

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Information theoretic lower bound

2m log n2

m + O(m)

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k = 2

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2X away

slide-46
SLIDE 46

13

k2-trees

Space results in bits x link.

slide-47
SLIDE 47

14

Block-trees

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

Credits to A. Gómez-Brandón

slide-48
SLIDE 48

14

Block-trees

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-49
SLIDE 49

14

Block-trees

AABAABBA ABAACBBA

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-50
SLIDE 50

14

Block-trees

AABAABBA ABAACBBA AABA ABBA

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-51
SLIDE 51

14

Block-trees

AABAABBA ABAACBBA AABA ABBA ABAA CBBA

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-52
SLIDE 52

14

Block-trees

AABAABBA ABAACBBA AABA ABBA ABAA CBBA

  • ffset=1

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-53
SLIDE 53

14

Block-trees

AABAABBA ABAACBBA AABA ABBA ABAA CBBA

  • ffset=1

AA BA

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-54
SLIDE 54

14

Block-trees

AABAABBA ABAACBBA AABA ABBA ABAA CBBA

  • ffset=1

AA BA AB BA

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-55
SLIDE 55

14

Block-trees

AABAABBA ABAACBBA AABA ABBA ABAA CBBA

  • ffset=1

AA BA AB BA

  • ffset=1

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-56
SLIDE 56

14

Block-trees

AABAABBA ABAACBBA AABA ABBA ABAA CBBA

  • ffset=1

AA BA AB BA

  • ffset=1
  • ffset=0

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-57
SLIDE 57

14

Block-trees

AABAABBA ABAACBBA AABA ABBA ABAA CBBA

  • ffset=1

AA BA AB BA

  • ffset=1
  • ffset=0

CB BA

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-58
SLIDE 58

14

Block-trees

AABAABBA ABAACBBA AABA ABBA ABAA CBBA

  • ffset=1

AA BA AB BA

  • ffset=1
  • ffset=0

CB BA

  • ffset=0

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-59
SLIDE 59

15

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

Block-trees

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

  • ffset=1

AA BA

  • ffset=1
  • ffset=0

CB

  • ffset=0

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-60
SLIDE 60

15

Access(4) = ?

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

Block-trees

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

  • ffset=1

AA BA

  • ffset=1
  • ffset=0

CB

  • ffset=0

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-61
SLIDE 61

15

Access(4) = ?

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

Block-trees

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

  • ffset=1

AA BA

  • ffset=1
  • ffset=0

CB

  • ffset=0

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-62
SLIDE 62

15

Access(4) = ?

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

Block-trees

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

  • ffset=1

AA BA

  • ffset=1
  • ffset=0

CB

  • ffset=0

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-63
SLIDE 63

15

Access(4) = ?

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

Block-trees

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

  • ffset=1

AA BA

  • ffset=1
  • ffset=0

CB

  • ffset=0

AABAABBAABAACBBA

input =

Credits to A. Gómez-Brandón

slide-64
SLIDE 64

15

Access(4) = ?

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

Block-trees

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

  • ffset=1

AA BA

  • ffset=1
  • ffset=0

CB

  • ffset=0

AABAABBAABAACBBA

input =

A A

Credits to A. Gómez-Brandón

slide-65
SLIDE 65

15

Access(4) = ?

Queries on LZ-Bounded Encodings, Belazzougui et al., DCC 2014

Block-trees

A block-tree divides a string into fixed- size blocks and those appearing earlier are represented with pointers.

  • ffset=1

AA BA

  • ffset=1
  • ffset=0

CB

  • ffset=0

Compression ratios close to Lempel-Ziv in practice and access to any subsequence.

AABAABBAABAACBBA

input =

A A

Credits to A. Gómez-Brandón

slide-66
SLIDE 66

16

Block-trees

Bdlogσe > dlogne

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B = O(logn/logσ)

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h = log n log σ

log n

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O(ti logn)

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ti

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logz

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z logn

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log n log σ

z log n

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O(z logn)

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O(z logn log n log σ

z log n )

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n

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σ

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size of the string alphabet size Stop recursion when

z

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At level i nodes take bits

log n log σ

z log n

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+ =

bits

nodes

slide-67
SLIDE 67

17

Two-Dimensional Block Trees, Brisaboa, Gagie, Gómez-Brandón, Navarro, DCC 2018

2D Block-trees

A hybrid between the k2-tree to exploit the clustering of the 0s and the block tree to exploit the repetitiveness of the adjacency matrix.

1 1 1 1

Target Source A source block may overlap up to 4 adjacent blocks.

Credits to A. Gómez-Brandón

slide-68
SLIDE 68

17

Two-Dimensional Block Trees, Brisaboa, Gagie, Gómez-Brandón, Navarro, DCC 2018

2D Block-trees

A hybrid between the k2-tree to exploit the clustering of the 0s and the block tree to exploit the repetitiveness of the adjacency matrix.

1 1 1 1

Target Source A source block may overlap up to 4 adjacent blocks.

Credits to A. Gómez-Brandón

slide-69
SLIDE 69

17

Two-Dimensional Block Trees, Brisaboa, Gagie, Gómez-Brandón, Navarro, DCC 2018

2D Block-trees

A hybrid between the k2-tree to exploit the clustering of the 0s and the block tree to exploit the repetitiveness of the adjacency matrix.

1 1 1 1

Target Source A source block may overlap up to 4 adjacent blocks. Pointer Offset

(0,0) (1,0) (0,1) (1,1)

Credits to A. Gómez-Brandón

slide-70
SLIDE 70

18

2D Block-trees

1 1 1 1 1 1 1 1 1 1 1 1 1 1

slide-71
SLIDE 71

18

2D Block-trees

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

slide-72
SLIDE 72

18

2D Block-trees

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

slide-73
SLIDE 73

18

2D Block-trees

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1

slide-74
SLIDE 74

18

2D Block-trees

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1

(0,0)

slide-75
SLIDE 75

18

2D Block-trees

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1 1 1 1

(0,0)

slide-76
SLIDE 76

18

2D Block-trees

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1 1 1 1

(0,0) (0,0)

slide-77
SLIDE 77

18

2D Block-trees

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1 1 1 1

(0,0) (0,0)

slide-78
SLIDE 78

18

2D Block-trees

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1 1 1 1

(0,0) (0,0)

slide-79
SLIDE 79

18

2D Block-trees

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1 1 1 1

(0,0) (0,0)

slide-80
SLIDE 80

18

2D Block-trees

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1 1 1 1

(0,0) (0,0) (0,1)

slide-81
SLIDE 81

18

2D Block-trees

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1 1 1 1

(0,0) (0,0) (0,1)

slide-82
SLIDE 82

18

2D Block-trees

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1 1 1 1

(0,0) (0,0) (0,1) (0,0) (0,0) (0,0) (0,0)

slide-83
SLIDE 83

18

2D Block-trees

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1 1 1 1

(0,0) (0,0) (0,1) (0,0) (0,0) (0,0) (0,0)

On Web graphs, we are not exploiting the “empty” zones

  • f zeroes as k2-trees.
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SLIDE 84

19

2D Block-trees on Web graphs

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1

(0,0) (0,0) (0,1) (0,0) (0,0) (0,0) (0,0)

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SLIDE 85

19

2D Block-trees on Web graphs

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1

(0,0) (0,0) (0,1) (0,0) (0,0) (0,0) (0,0)

Introduce a new type of node to mark empty zones.

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SLIDE 86

20

2D Block-trees on Web graphs

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1

(0,1)

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SLIDE 87

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1

(0,1)

21

2D Block-trees on Web graphs

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SLIDE 88

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1

(0,1)

21

2D Block-trees on Web graphs

1010 1010 1010 T 0001 1101 0010 1010 L

slide-89
SLIDE 89

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1

(0,1)

21

2D Block-trees on Web graphs

1010 1010 1010 T 0001 1101 0010 1010 L 0 0 0 0 0 0 N 100001

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SLIDE 90

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1

(0,1)

21

2D Block-trees on Web graphs

1010 1010 1010 T 0001 1101 0010 1010 L 0 0 0 0 0 0 N 100001 P1 1 O1 00

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SLIDE 91

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1

(0,1)

21

2D Block-trees on Web graphs

1010 1010 1010 T 0001 1101 0010 1010 L 0 0 0 0 0 0 N 100001 P1 1 O1 00 P2 7 O2 01

slide-92
SLIDE 92

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1

(0,1)

21

2D Block-trees on Web graphs

1010 1010 1010 T 0001 1101 0010 1010 L 0 0 0 0 0 0 N 100001 P1 1 O1 00 P2 7 O2 01

Access

slide-93
SLIDE 93

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1

(0,1)

21

2D Block-trees on Web graphs

1010 1010 1010 T 0001 1101 0010 1010 L 0 0 0 0 0 0 N 100001 P1 1 O1 00 P2 7 O2 01 1

Access

slide-94
SLIDE 94

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1

(0,1)

21

2D Block-trees on Web graphs

1010 1010 1010 T 0001 1101 0010 1010 L 0 0 0 0 0 0 N 100001 P1 1 O1 00 P2 7 O2 01 1

Access

slide-95
SLIDE 95

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1

(0,1)

21

2D Block-trees on Web graphs

1010 1010 1010 T 0001 1101 0010 1010 L 0 0 0 0 0 0 N 100001 P1 1 O1 00 P2 7 O2 01 1

Access

slide-96
SLIDE 96

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1

(0,1)

21

2D Block-trees on Web graphs

1010 1010 1010 T 0001 1101 0010 1010 L 0 0 0 0 0 0 N 100001 P1 1 O1 00 P2 7 O2 01 1

Access

slide-97
SLIDE 97

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1

(0,1)

21

2D Block-trees on Web graphs

1010 1010 1010 T 0001 1101 0010 1010 L 0 0 0 0 0 0 N 100001 P1 1 O1 00 P2 7 O2 01 1 1

Access

slide-98
SLIDE 98

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(0,0)

1 1 1 1 1 1 1

(0,1)

21

2D Block-trees on Web graphs

1010 1010 1010 T 0001 1101 0010 1010 L 0 0 0 0 0 0 N 100001 P1 1 O1 00 P2 7 O2 01 1 1

Access

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SLIDE 99

22

2D Block-trees on Web graphs

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SLIDE 100

22

2D Block-trees on Web graphs

Up to 50% reduction wrt the k2-tree.

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SLIDE 101

23

2D Block-trees on Web graphs

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SLIDE 102

23

2D Block-trees on Web graphs

3-6X slower wrt the k2-tree.

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SLIDE 103

24

Summing up

k2-trees support direct and reverse navigation;
 good trade-off between space and time; do not exploit repetitiveness. 2D-block-trees combines the capturing-spareness behaviour

  • f k2-trees with the capturing-repetitiveness of block-trees.

Up to 50% reduction in space, but 3-6X slower than k2-trees. Block-trees compress (1D) strings to compression ratios close to LZ and support efficient random access to any substring. WebGraph supports efficient extraction of direct neighbours in excellent compressed space.

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SLIDE 104

25

Take home messages

k2-trees achieve a good trade-off between space and time when both direct and reverse navigation is needed. 2D-block-trees are even smaller than k2-trees but much slower. WebGraph is still the most compact representation if only direct navigation is allowed.

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SLIDE 105

26

Thanks for your attention, time, patience!

Any questions?