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Sparse Prefix Sums Michael Shekelyan, Anton Digns, Johann Gamper 1 - PowerPoint PPT Presentation

Jan 01, 2024 •570 likes •1.2k views

1 7 8 6 7 6 5 16 5 6 77 8 2 5 54 Sparse Prefix Sums Michael Shekelyan, Anton Digns, Johann Gamper 1 0 0 0 1 7 7 15 0 0 0 6 6 0 0 0 Free University of Bozen-Bolzano, Italy 7 0 0 6 13 0 0 6 7 0 5 27 34

  1. 1 7 8 6 7 6 5 16 5 6 77 8 2 5 54 Sparse Prefix Sums Michael Shekelyan, Anton Dignös, Johann Gamper 1 0 0 0 1 7 7 15 0 0 0 6 6 0 0 0 Free University of Bozen-Bolzano, Italy 7 0 0 6 13 0 0 6 7 0 5 27 34 0 0 6 8 5 10 32 46 7 7 98 8 0 0 0 8 2 7 7 25.09.2017, Nicosia, Cyprus 8 0 0 54 62 2 7 7 8 0 0 54 62 2 7 7 Michael Shekelyan ! 1 ADBIS’17 - Sparse Prefix Sums ! 1

  2. Outline Introduction • range sums • prefix sums • technique to compute range sums in constant-time • relative prefix sums • technique to achieve faster updating • related work Contribution • sparse prefix sums • compression of relative prefix sums preserving constant query time Experiments • low-resolution grids • high-resolution grids • impact of dimensionality • population grids based on satellite imagery Michael Shekelyan ! 2 ADBIS’17 - Sparse Prefix Sums ! 2

  3. Range Sums original data table 1 7 8 6 7 6 5 16 5 6 77 8 2 5 54 • e.g. each cell counts the number of inhabitants in a city Michael Shekelyan ! 3 ADBIS’17 - Sparse Prefix Sums ! 3

  4. Range Sums original data table 1 7 8 6 7 6 5 16 5 6 77 8 2 5 54 • e.g. each cell counts the number of inhabitants in a city Michael Shekelyan ! 4 ADBIS’17 - Sparse Prefix Sums ! 4

  5. Range Sums original data table 1 7 8 6 S 7 6 5 16 5 6 77 8 2 5 T 54 • range sum: sum values in a sub-matrix/tensor • query: how many inhabitants live in range between S and T? • answer: 16+6+77 = 99 Michael Shekelyan ! 5 ADBIS’17 - Sparse Prefix Sums ! 5

  6. Prefix Sums original data table S = origin 1 7 8 6 7 6 5 16 5 6 77 T 8 2 5 54 “sum of preceding values” • prefix sum: range sum with one corner at origin • query: how many inhabitants in range between origin and T? • answer: 1+7+5 = 13 Michael Shekelyan ! 6 ADBIS’17 - Sparse Prefix Sums ! 6

  7. Prefix Sums • well-known concept across many disciplines probability theory computer graphics CDFs (1984) summed area integral tables images prefix sums computer vision database systems (2001) (1997) Michael Shekelyan ! 7 ADBIS’17 - Sparse Prefix Sums ! 7

  8. Prefix Sums original data table S = origin 1 7 8 6 7 6 5 16 5 6 77 T 8 2 5 54 • prefix sum: range sum with one corner at origin • query: how many inhabitants in range between origin and T? • answer: 1+7+5 = 13 Michael Shekelyan ! 8 ADBIS’17 - Sparse Prefix Sums ! 8

  9. Prefix Sums original data table prefix sums 1 7 8 1 1 1 1 1 8 8 16 6 1 1 1 7 7 14 14 22 7 6 8 8 8 14 14 21 21 35 5 16 8 8 13 35 35 42 42 56 5 6 77 8 13 18 40 46 53 53 144 8 2 5 16 21 26 48 54 63 68 159 54 16 21 26 54 108 117 122 213 16 21 26 102 108 117 122 213 • data table with N cells has N prefix sums! • idea: store result for each prefix sum • result: O(1) querying Michael Shekelyan ! 9 ADBIS’17 - Sparse Prefix Sums ! 9

  10. Prefix Sums original data table prefix sums 1 7 8 1 1 1 1 1 8 8 16 6 1 1 1 7 7 14 14 22 7 6 8 8 8 14 14 21 21 35 5 16 8 8 13 35 35 42 42 56 5 6 77 8 13 18 40 46 53 53 144 8 2 5 16 21 26 48 54 63 68 159 54 16 21 26 54 108 117 122 213 16 21 26 102 108 117 122 213 • data table with N cells has N prefix sums! • idea: store result for each prefix sum • result: O(1) querying • example: 13 = 1+7+5 Michael Shekelyan ! 10 ADBIS’17 - Sparse Prefix Sums ! 10

  11. Prefix Sums original data table prefix sums 1 7 8 1 1 1 1 1 8 8 16 6 1 1 1 7 7 14 14 22 8 35 7 6 8 8 14 14 21 21 5 16 8 8 13 35 35 42 42 56 18 53 144 5 6 77 8 13 40 46 53 8 2 5 16 21 26 48 54 63 68 159 54 16 21 26 54 108 117 122 213 16 21 26 102 108 117 122 213 • each range sum can be computed from 2 d prefix sums Michael Shekelyan ! 11 ADBIS’17 - Sparse Prefix Sums ! 11

  12. Prefix Sums original data table prefix sums 1 7 8 1 1 1 1 1 8 8 16 6 1 1 1 7 7 14 14 22 7 6 8 8 8 14 14 21 21 35 5 16 8 8 13 35 35 42 42 56 5 6 77 8 13 18 40 46 53 53 144 8 2 5 16 21 26 48 54 63 68 159 54 16 21 26 54 108 117 122 213 16 21 26 102 108 117 122 213 • add prefix sum (144) Michael Shekelyan ! 12 ADBIS’17 - Sparse Prefix Sums ! 12

  13. Prefix Sums original data table prefix sums 1 7 8 1 1 1 1 1 8 8 16 6 1 1 1 7 7 14 14 22 7 6 8 8 8 14 14 21 21 35 5 16 8 8 13 35 35 42 42 56 5 6 77 8 13 18 40 46 53 53 144 8 2 5 16 21 26 48 54 63 68 159 54 16 21 26 54 108 117 122 213 16 21 26 102 108 117 122 213 • prefix sum (144) adds too much Michael Shekelyan ! 13 ADBIS’17 - Sparse Prefix Sums ! 13

  14. Prefix Sums original data table prefix sums 1 7 8 1 1 1 1 1 8 8 16 6 1 1 1 7 7 14 14 22 7 6 8 8 8 14 14 21 21 35 5 16 8 8 13 35 35 42 42 56 5 6 77 8 13 18 40 46 53 53 144 8 2 5 16 21 26 48 54 63 68 159 54 16 21 26 54 108 117 122 213 16 21 26 102 108 117 122 213 • subtract prefix sum (35) Michael Shekelyan ! 14 ADBIS’17 - Sparse Prefix Sums ! 14

  15. Prefix Sums original data table prefix sums 1 7 8 1 1 1 1 1 8 8 16 6 1 1 1 7 7 14 14 22 7 6 8 8 8 14 14 21 21 35 5 16 8 8 13 35 35 42 42 56 5 6 77 8 13 18 40 46 53 53 144 8 2 5 16 21 26 48 54 63 68 159 54 16 21 26 54 108 117 122 213 16 21 26 102 108 117 122 213 • still too much added on the left (5+5) Michael Shekelyan ! 15 ADBIS’17 - Sparse Prefix Sums ! 15

  16. Prefix Sums original data table prefix sums 1 7 8 1 1 1 1 1 8 8 16 6 1 1 1 7 7 14 14 22 7 6 8 8 8 14 14 21 21 35 5 16 8 8 13 35 35 42 42 56 5 6 77 8 13 18 40 46 53 53 144 8 2 5 16 21 26 48 54 63 68 159 54 16 21 26 54 108 117 122 213 16 21 26 102 108 117 122 213 • subtract prefix sum (18) Michael Shekelyan ! 16 ADBIS’17 - Sparse Prefix Sums ! 16

  17. Prefix Sums original data table prefix sums 1 7 8 1 1 1 1 1 8 8 16 6 1 1 1 7 7 14 14 22 7 6 8 8 8 14 14 21 21 35 5 16 8 8 13 35 35 42 42 56 5 6 77 8 13 18 40 46 53 53 144 8 2 5 16 21 26 48 54 63 68 159 54 16 21 26 54 108 117 122 213 16 21 26 102 108 117 122 213 • top-left (1+7) was first added once and then subtracted twice Michael Shekelyan ! 17 ADBIS’17 - Sparse Prefix Sums ! 17

  18. Prefix Sums original data table prefix sums 1 7 8 1 1 1 1 1 8 8 16 6 1 1 1 7 7 14 14 22 7 6 8 8 8 14 14 21 21 35 5 16 8 8 13 35 35 42 42 56 5 6 77 8 13 18 40 46 53 53 144 8 2 5 16 21 26 48 54 63 68 159 54 16 21 26 54 108 117 122 213 16 21 26 102 108 117 122 213 • add top-left again to balance out additions/subtractions Michael Shekelyan ! 18 ADBIS’17 - Sparse Prefix Sums ! 18

  19. Prefix Sums original data table prefix sums 1 7 8 1 1 1 1 1 8 8 16 16+6+77 = 99 144-35-18+8 = 99 6 1 1 1 7 7 14 14 22 7 6 8 8 8 14 14 21 21 35 5 16 8 8 13 35 35 42 42 56 5 6 77 8 13 18 40 46 53 53 144 8 2 5 16 21 26 48 54 63 68 159 54 16 21 26 54 108 117 122 213 16 21 26 102 108 117 122 213 • range sum computed with prefix sums at corner cells of range Michael Shekelyan ! 19 ADBIS’17 - Sparse Prefix Sums ! 19

  20. Prefix Sum Updating original data table prefix sums 1 7 8 1 1 1 1 1 8 8 16 6 1 1 1 7 7 14 14 22 7 6 8 8 8 14 14 21 21 35 5 16 8 8 13 35 35 42 42 56 5 6 77 8 13 18 40 46 53 53 144 8 2 5 16 21 26 48 54 63 68 159 54 16 21 26 54 108 117 122 213 16 21 26 102 108 117 122 213 update complexity update complexity O(1) with dense storage O(N) where N is # of cells • • Michael Shekelyan ! 20 ADBIS’17 - Sparse Prefix Sums ! 20

  21. Prefix Sum Updating original data table prefix sums relative prefix sums 1 7 8 1 1 1 1 1 8 8 16 1 0 0 0 1 7 7 15 6 1 1 1 7 7 14 14 22 0 0 0 6 6 0 0 0 7 6 8 8 8 14 14 21 21 35 7 0 0 6 13 0 0 6 5 16 8 8 13 35 35 42 42 56 7 0 5 27 34 0 0 6 5 6 77 8 13 18 40 46 53 53 144 8 5 10 32 46 7 7 98 8 2 5 16 21 26 48 54 63 68 159 8 0 0 0 8 2 7 7 54 16 21 26 54 108 117 122 213 8 0 0 54 62 2 7 7 16 21 26 102 108 117 122 213 8 0 0 54 62 2 7 7 update complexity update complexity update complexity O(1) with dense storage O(N) where N is # of cells O(N 1/2 ) • • • Michael Shekelyan ! 21 ADBIS’17 - Sparse Prefix Sums ! 21

  22. Relative Prefix Sums original data table prefix sums relative prefix sums 1 7 8 1 1 1 1 1 8 8 16 1 0 0 0 1 7 7 15 6 1 1 1 7 7 14 14 22 0 0 0 6 6 0 0 0 7 6 8 8 8 14 14 21 21 35 7 0 0 6 13 0 0 6 5 16 8 8 13 35 35 42 42 56 7 0 5 27 34 0 0 6 5 6 77 8 13 18 40 46 53 53 144 8 5 10 32 46 7 7 98 8 2 5 16 21 26 48 54 63 68 159 8 0 0 0 8 2 7 7 54 16 21 26 54 108 117 122 213 8 0 0 54 62 2 7 7 16 21 26 102 108 117 122 213 8 0 0 54 62 2 7 7 relative prefix sums • split into around N 1/2 blocks • each block has an anchor cell storing the prefix sum • each block has border cells storing prefix sum minus anchor cell • each block has local cells storing local prefix sum Michael Shekelyan ! 22 ADBIS’17 - Sparse Prefix Sums ! 22

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