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Effective Web Graph Representations Giulio Ermanno Pibiri - PowerPoint PPT Presentation

Jul 11, 2023 •279 likes •1.33k views

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

  1. k 2 -trees k 2 -ary tree k 2 -trees for Compact Web Graph Representation , representation of the Brisaboa-Ladra-Navarro, SPIRE 2009 adjacency matrix. k = 2 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0111 0100 01 00 10 11 Credits to A. Gómez-Brandón

  2. k 2 -trees k 2 -ary tree k 2 -trees for Compact Web Graph Representation , representation of the Brisaboa-Ladra-Navarro, SPIRE 2009 adjacency matrix. k = 2 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0111 0100 01 00 10 01 00 11 Credits to A. Gómez-Brandón

  3. k 2 -trees k 2 -ary tree k 2 -trees for Compact Web Graph Representation , representation of the Brisaboa-Ladra-Navarro, SPIRE 2009 adjacency matrix. k = 2 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0111 0100 01 00 10 01 00 1110101010001001 00011101000100100100 T L 11 Credits to A. Gómez-Brandón

  4. k 2 -trees k 2 -ary tree k 2 -trees for Compact Web Graph Representation , representation of the Brisaboa-Ladra-Navarro, SPIRE 2009 adjacency matrix. k = 2 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0111 0100 01 00 10 01 00 Access 1110101010001001 00011101000100100100 T L 11 Credits to A. Gómez-Brandón

  5. k 2 -trees k 2 -ary tree k 2 -trees for Compact Web Graph Representation , representation of the Brisaboa-Ladra-Navarro, SPIRE 2009 adjacency matrix. k = 2 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0111 0100 01 00 10 01 00 Access 1110101010001001 00011101000100100100 1 T L 11 Credits to A. Gómez-Brandón

  6. k 2 -trees k 2 -ary tree k 2 -trees for Compact Web Graph Representation , representation of the Brisaboa-Ladra-Navarro, SPIRE 2009 adjacency matrix. k = 2 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0111 0100 01 00 10 01 00 Access 1110101010001001 00011101000100100100 1 0 T L 11 Credits to A. Gómez-Brandón

  7. k 2 -trees k 2 -ary tree k 2 -trees for Compact Web Graph Representation , representation of the Brisaboa-Ladra-Navarro, SPIRE 2009 adjacency matrix. k = 2 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0111 0100 01 00 10 01 00 Access 1110101010001001 00011101000100100100 1 0 1 T L 11 Credits to A. Gómez-Brandón

  8. k 2 -trees k 2 -ary tree k 2 -trees for Compact Web Graph Representation , representation of the Brisaboa-Ladra-Navarro, SPIRE 2009 adjacency matrix. k = 2 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0111 0100 01 00 10 01 00 Access 1110101010001001 00011101000100100100 1 0 1 1 0 T L 11 Credits to A. Gómez-Brandón

  9. k 2 -trees k 2 -ary tree k 2 -trees for Compact Web Graph Representation , representation of the Brisaboa-Ladra-Navarro, SPIRE 2009 adjacency matrix. k = 2 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0111 0100 01 00 10 1 01 00 Access 1110101010001001 00011101000100100100 1 0 1 1 0 1 T L 11 Credits to A. Gómez-Brandón

  10. k 2 -trees k 2 -ary tree k 2 -trees for Compact Web Graph Representation , representation of the Brisaboa-Ladra-Navarro, SPIRE 2009 adjacency matrix. k = 2 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0111 0100 01 00 10 1 01 00 Access 1110101010001001 00011101000100100100 1 0 1 1 0 1 T L rank 1 x k 2 on T in O(1) 11 Credits to A. Gómez-Brandón

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sha1_base64="NBm/54KWBqYjwdzq4/XS3LU5RmQ=">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</latexit> k 2 -trees pages links m n h = d log k 2 n 2 e mk 2 d log k 2 n 2 e So the total space is bits k 2 Õ b log k 2 m c� 1 k 2 i + mk 2 ( d log k 2 n e � b log k 2 m c ) i = 0 k 2 − 1 + mk 2 log k 2 n 2 = k 2 m − 1 m = mk 2 ⇣ ⌘ log k 2 n 2 m + O ( 1 ) bits k = 2 Information theoretic 2 m log n 2 m + O ( m ) lower bound � ≈ m log n 2 log � n 2 m + O ( m ) 2X away m 12

  16. k 2 -trees Space results in bits x link. 13

  17. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. 14 Credits to A. Gómez-Brandón

  18. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = 14 Credits to A. Gómez-Brandón

  19. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = AABAABBA ABAACBBA 14 Credits to A. Gómez-Brandón

  20. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = AABAABBA ABAACBBA AABA ABBA 14 Credits to A. Gómez-Brandón

  21. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = AABAABBA ABAACBBA AABA ABBA ABAA CBBA 14 Credits to A. Gómez-Brandón

  22. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = AABAABBA ABAACBBA AABA ABBA ABAA CBBA offset=1 14 Credits to A. Gómez-Brandón

  23. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = AABAABBA ABAACBBA AABA ABBA ABAA CBBA offset=1 AA BA 14 Credits to A. Gómez-Brandón

  24. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = AABAABBA ABAACBBA AABA ABBA ABAA CBBA offset=1 AA BA AB BA 14 Credits to A. Gómez-Brandón

  25. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = AABAABBA ABAACBBA AABA ABBA ABAA CBBA offset=1 AA BA AB BA offset=1 14 Credits to A. Gómez-Brandón

  26. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = AABAABBA ABAACBBA AABA ABBA ABAA CBBA offset=1 AA BA AB BA offset=1 offset=0 14 Credits to A. Gómez-Brandón

  27. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = AABAABBA ABAACBBA AABA ABBA ABAA CBBA offset=1 AA BA AB BA CB BA offset=1 offset=0 14 Credits to A. Gómez-Brandón

  28. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = AABAABBA ABAACBBA AABA ABBA ABAA CBBA offset=1 AA BA AB BA CB BA offset=1 offset=0 offset=0 14 Credits to A. Gómez-Brandón

  29. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = offset=1 AA BA CB offset=0 offset=1 offset=0 15 Credits to A. Gómez-Brandón

  30. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = Access(4) = ? offset=1 AA BA CB offset=0 offset=1 offset=0 15 Credits to A. Gómez-Brandón

  31. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = Access(4) = ? offset=1 AA BA CB offset=0 offset=1 offset=0 15 Credits to A. Gómez-Brandón

  32. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = Access(4) = ? offset=1 AA BA CB offset=0 offset=1 offset=0 15 Credits to A. Gómez-Brandón

  33. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA input = Access(4) = ? offset=1 AA BA CB offset=0 offset=1 offset=0 15 Credits to A. Gómez-Brandón

  34. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA A input = A Access(4) = ? offset=1 AA BA CB offset=0 offset=1 offset=0 15 Credits to A. Gómez-Brandón

  35. Block-trees A block-tree divides a string into fixed- Queries on LZ-Bounded Encodings , size blocks and those appearing Belazzougui et al., DCC 2014 earlier are represented with pointers. AABAABBAABAACBBA A input = A Access(4) = ? offset=1 AA BA CB offset=0 offset=1 offset=0 Compression ratios close to Lempel-Ziv in practice and access to any subsequence. 15 Credits to A. Gómez-Brandón

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  37. 2D Block-trees A hybrid between the k 2 -tree to exploit the clustering of the 0s Two-Dimensional Block Trees , and the block tree to exploit the Brisaboa, Gagie, Gómez-Brandón, Navarro , DCC 2018 repetitiveness of the adjacency matrix. A source block may overlap up to 4 adjacent blocks. 1 0 Target 1 0 1 0 Source 1 0 Credits to A. Gómez-Brandón 17

  38. 2D Block-trees A hybrid between the k 2 -tree to exploit the clustering of the 0s Two-Dimensional Block Trees , and the block tree to exploit the Brisaboa, Gagie, Gómez-Brandón, Navarro , DCC 2018 repetitiveness of the adjacency matrix. A source block may overlap up to 4 adjacent blocks. 1 0 Target 1 0 1 0 Source 1 0 Credits to A. Gómez-Brandón 17

  39. 2D Block-trees A hybrid between the k 2 -tree to exploit the clustering of the 0s Two-Dimensional Block Trees , and the block tree to exploit the Brisaboa, Gagie, Gómez-Brandón, Navarro , DCC 2018 repetitiveness of the adjacency matrix. A source block may overlap up to 4 adjacent blocks. Pointer (0,0) (1,0) Offset 1 0 Target 1 0 1 0 Source 1 0 (0,1) (1,1) Credits to A. Gómez-Brandón 17

  40. 2D Block-trees 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 18

  41. 2D Block-trees 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 18

  42. 2D Block-trees 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 (0,0) 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 18

  43. 2D Block-trees 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 (0,0) 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 18

  44. 2D Block-trees 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 (0,0) 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 (0,0) 0 1 0 0 0 1 0 0 18

  45. 2D Block-trees 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 (0,0) 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 (0,0) 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1 18

  46. 2D Block-trees 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 (0,0) 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 (0,0) (0,0) 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1 18

  47. 2D Block-trees 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 (0,0) 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 (0,0) (0,0) 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1 18

  48. 2D Block-trees 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 (0,0) 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 (0,0) (0,0) 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1 18

  49. 2D Block-trees 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 (0,0) 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 (0,0) (0,0) 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1 18

  50. 2D Block-trees 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 (0,0) 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 (0,0) (0,0) (0,1) 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1 18

  51. 2D Block-trees 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 (0,0) 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 (0,0) (0,0) (0,1) 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 18

  52. 2D Block-trees 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 (0,0) 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 (0,0) (0,0) (0,0) (0,0) (0,1) (0,0) (0,0) 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 18

  53. 2D Block-trees 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 On Web graphs, we 1 1 0 0 1 1 0 0 are not exploiting the 0 1 0 0 0 1 0 0 “empty” zones 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 of zeroes as k 2 -trees. 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 (0,0) 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 (0,0) (0,0) (0,0) (0,0) (0,1) (0,0) (0,0) 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 18

  54. 2D Block-trees on Web graphs 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 (0,0) 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,1) 0 1 0 0 0 1 1 0 1 0 19

  55. 2D Block-trees on Web graphs 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 Introduce a new 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 type of node to 0 0 0 0 0 0 0 0 mark empty zones. 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 (0,0) 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 (0,0) (0,0) (0,0) (0,0) (0,0) (0,0) (0,1) 0 1 0 0 0 1 1 0 1 0 19

  56. 2D Block-trees on Web graphs 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 (0,0) 1 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 (0,1) 0 1 0 1 1 0 1 0 20

  57. 2D Block-trees on Web graphs 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 (0,0) 1 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 (0,1) 0 1 0 1 1 0 1 0 21

  58. 2D Block-trees on Web graphs 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 T 1010 1010 1010 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 (0,0) 1 1 0 0 1 0 0 1 L 0001 1101 0010 1010 0 1 0 0 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 (0,1) 0 1 0 1 1 0 1 0 21

  59. 2D Block-trees on Web graphs 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 T 1010 1010 1010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 (0,0) 1 1 0 0 1 0 0 1 L 0001 1101 0010 1010 0 1 0 0 1 0 1 1 N 100001 0 0 0 0 0 1 1 0 0 1 0 (0,1) 0 1 0 1 1 0 1 0 21

  60. 2D Block-trees on Web graphs 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 T 1010 1010 1010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 (0,0) 1 1 0 0 1 0 0 1 L 0001 1101 0010 1010 0 1 0 0 1 0 1 1 N 100001 P 1 1 O 1 00 0 0 0 0 0 1 1 0 0 1 0 (0,1) 0 1 0 1 1 0 1 0 21

  61. 2D Block-trees on Web graphs 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 T 1010 1010 1010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 (0,0) 1 1 0 0 1 0 0 1 L 0001 1101 0010 1010 0 1 0 0 1 0 1 1 N 100001 P 1 1 O 1 00 0 0 0 0 0 1 1 0 0 1 0 (0,1) 0 1 0 1 1 0 1 0 P 2 7 O 2 01 21

  62. 2D Block-trees on Web graphs 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 Access 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 T 1010 1010 1010 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 (0,0) 1 1 0 0 1 0 0 1 L 0001 1101 0010 1010 0 1 0 0 1 0 1 1 N 100001 P 1 1 O 1 00 0 0 0 0 0 1 1 0 0 1 0 (0,1) 0 1 0 1 1 0 1 0 P 2 7 O 2 01 21

  63. 2D Block-trees on Web graphs 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 Access 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 T 1010 1010 1010 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 (0,0) 1 1 0 0 1 0 0 1 L 0001 1101 0010 1010 0 1 0 0 1 0 1 1 N 100001 P 1 1 O 1 00 0 0 0 0 0 1 1 0 0 1 0 (0,1) 0 1 0 1 1 0 1 0 P 2 7 O 2 01 21

  64. 2D Block-trees on Web graphs 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 Access 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 T 1010 1010 1010 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 (0,0) 1 1 0 0 1 0 0 1 L 0001 1101 0010 1010 0 1 0 0 1 0 1 1 N 100001 P 1 1 O 1 00 0 0 0 0 0 1 1 0 0 1 0 (0,1) 0 1 0 1 1 0 1 0 P 2 7 O 2 01 21

  65. 2D Block-trees on Web graphs 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 Access 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 T 1010 1010 1010 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 (0,0) 1 1 0 0 1 0 0 1 L 0001 1101 0010 1010 0 1 0 0 1 0 1 1 N 100001 P 1 1 O 1 00 0 0 0 0 0 1 1 0 0 1 0 (0,1) 0 1 0 1 1 0 1 0 P 2 7 O 2 01 21

  66. 2D Block-trees on Web graphs 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 Access 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 T 1010 1010 1010 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 (0,0) 1 1 0 0 1 0 0 1 L 0001 1101 0010 1010 0 1 0 0 1 0 1 1 N 100001 P 1 1 O 1 00 0 0 0 0 0 1 1 0 0 1 0 (0,1) 0 1 0 1 1 0 1 0 P 2 7 O 2 01 21

  67. 2D Block-trees on Web graphs 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 Access 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 T 1010 1010 1010 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 (0,0) 1 1 0 0 1 0 0 1 L 0001 1101 0010 1010 0 1 0 0 1 0 1 1 N 100001 1 P 1 1 O 1 00 0 0 0 0 0 1 1 0 0 1 0 (0,1) 0 1 0 1 1 0 1 0 P 2 7 O 2 01 21

  68. 2D Block-trees on Web graphs 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 Access 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 T 1010 1010 1010 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 (0,0) 1 1 0 0 1 0 0 1 L 0001 1101 0010 1010 0 1 0 0 1 0 1 1 N 100001 1 P 1 1 O 1 00 0 0 0 0 0 1 1 0 0 1 0 (0,1) 0 1 0 1 1 0 1 0 P 2 7 O 2 01 21

  69. 2D Block-trees on Web graphs 22

  70. 2D Block-trees on Web graphs Up to 50% reduction wrt the k 2 -tree. 22

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