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. . . . . . . . . . . . . . . A linear lower bound for incrementing a space-optimal integer representation in the bit-probe model Michael Raskin, raskin@mccme.ru LaBRI, Universit de Bordeaux July 13, 2017 Michael Raskin,

  1. . . . . . . . . . . . . . . . A linear lower bound for incrementing a space-optimal integer representation in the bit-probe model Michael Raskin, raskin@mccme.ru LaBRI, Université de Bordeaux July 13, 2017 Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . 1 / 18

  2. . . . . . . . . . . . . . . A linear lower bound for incrementing a space-optimal integer representation in the bit-probe model Michael Raskin, raskin@mccme.ru LaBRI, Université de Bordeaux This work was mostly done in Aarhus University July 13, 2017 Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 / 18

  3. . . . . . . . . . . . . . . . . 000, 001, 010, … Worst case (1111→0000): n reads, n writes Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . . 2 / 18 How to count from 0 to 2 n − 1 using n bits?

  4. . . . . . . . . . . . . . . . . . [Gray1953] Frank Gray. Pulse code communication Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 3 / 18 How to count from 0 to 2 n − 1 using n bits? Worst case: n reads, 1 write

  5. . . . . . . . . . . . . . . . . [BGPS2014] Gerth Stølting Brodal, Mark Greve, Vineet Pandey, Srinivasa Rao Satti. Integer Representations towards Effjcient Counting in the Bit Probe Model Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . . 4 / 18 How to count from 0 to 2 n − 1 using n bits? Worst case: n − 1 reads, 3 writes

  6. . . . . . . . . . . . . . . . . . Can we do even better? Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 5 / 18 How to count from 0 to 2 n − 1 using n bits? [Gray1953] n reads, 1 write [BGPS2014] n − 1 reads, 3 writes

  7. . . . . . . . . . . . . . . Redundant case [RM2010] M. Ziaur Rahman, J. Ian Munro. Integer Representation and Counting in the Bit Probe Model. writes (one possible idea: increment n -bit normal binary counter over n steps with Gray-coded pointer and carry bit) Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . 6 / 18 One extra bit for representation, worst case: log 2 n + O (1) reads, O (1)

  8. . . . . . . . . . . . . . . Previous lower bound [FMS1997] Gudmund Skovbjerg Frandsen and Peter Bro Miltersen and Sven Skyum Dynamic word problems Cannot write unread bits Every bit needs to be written at some point Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . 7 / 18 Reading at most k bits → reading 2 k − 1 distinct bits in total upper bound n − 1 Lower bound log 2 n

  9. . . . . . . . . . . . . . . . . . The main claim Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 8 / 18 Lower bound log 2 n , upper bound n − 1 Redundant case: log 2 n + O (1) This talk: non-redundant counter needs to read at least ⌊ n 2 ⌋ bits

  10. . . . . . . . . . . . . . . . . . The main claim Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 8 / 18 Lower bound log 2 n , upper bound n − 1 Redundant case: log 2 n + O (1) This talk: non-redundant counter needs to read at least ⌊ n 2 ⌋ bits

  11. . . . . . . . . . . . . . . . Hypercube shuffming Code: a vertex of the n -dimensional hypercube Fixing some bits: a block (a hypercube of lower dimension) Changing fjxed bits: parallel translation Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . 9 / 18

  12. . . . . . . . . . . . . . . . . The [BGPS2014] example Initial representation: a decision tree Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . 10 / 18 . . . . . x 0 x 1 x 2 x 3 x 3 x 1 x 3 x 0 ← 1; x 1 ← 1; x 0 ← 1; x 3 ← 0; x 0 ← 0 , x 3 ← 1; x 2 ← 0; x 0 ← 0 , x 2 ← 1; x 1 ← 0;

  13. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  14. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  15. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  16. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  17. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  18. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  19. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  20. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  21. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  22. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  23. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  24. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  25. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  26. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  27. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  28. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  29. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

  30. . . . . . . . . . . . . . . . . . Hypercube shuffming and [BGPS2014] Michael Raskin, raskin@mccme.ru (LaBRI) Linear lower bound for increment July 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . 11 / 18

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