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Good Predictions Are Worth a Few Comparisons Carine Pivoteau with Nicolas Auger and Cyril Nicaud LIGM - Universit e Paris-Est-Marne-la-Vall ee April 2016 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 1/16 A case

  1. Good Predictions Are Worth a Few Comparisons Carine Pivoteau with Nicolas Auger and Cyril Nicaud LIGM - Universit´ e Paris-Est-Marne-la-Vall´ ee April 2016 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 1/16

  2. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 5 1 4 3 6 0 2 8 7 9 min = 5 max = 5 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  3. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 5 1 4 3 6 0 2 8 7 9 1 < min ? 1 > max ? min = 5 max = 5 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  4. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 5 1 4 3 6 0 2 8 7 9 1 < min ? 1 > max ? min = 1 max = 5 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  5. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 5 1 4 3 6 0 2 8 7 9 4 < min ? 4 > max ? min = 1 max = 5 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  6. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 5 1 4 3 6 0 2 8 7 9 3 < min ? 3 > max ? min = 1 max = 5 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  7. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 5 1 4 3 6 0 2 8 7 9 6 < min ? 6 > max ? min = 1 max = 5 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  8. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 5 1 4 3 6 0 2 8 7 9 6 < min ? 6 > max ? min = 1 max = 6 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  9. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 5 1 4 3 6 0 2 8 7 9 0 < min ? 0 > max ? min = 1 max = 6 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  10. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 5 1 4 3 6 0 2 8 7 9 0 < min ? 0 > max ? min = 0 max = 6 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  11. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 5 1 4 3 6 0 2 8 7 9 2 < min ? 2 > max ? min = 0 max = 6 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  12. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 5 1 4 3 6 0 2 8 7 9 8 < min ? 8 > max ? min = 0 max = 6 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  13. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 5 1 4 3 6 0 2 8 7 9 8 < min ? 8 > max ? min = 0 max = 8 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  14. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 5 1 4 3 6 0 2 8 7 9 7 < min ? 7 > max ? min = 0 max = 8 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  15. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 5 1 4 3 6 0 2 8 7 9 9 < min ? 9 > max ? min = 0 max = 8 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  16. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 5 1 4 3 6 0 2 8 7 9 9 < min ? 9 > max ? min = 0 max = 9 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  17. A case study Find both the min. and the max. of an array of size n . Naive Algorithm: 2 n comparisons 5 1 4 3 6 0 2 8 7 9 9 < min ? 9 > max ? min = 0 max = 9 Can we do better? N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  18. A case study Find both the min. and the max. of an array of size n . Optimized Algorithm: 5 1 4 3 6 0 2 8 7 9 min = 5 max = 5 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  19. A case study Find both the min. and the max. of an array of size n . Optimized Algorithm: 5 1 4 3 6 0 2 8 7 9 5 < 1 ? 1 < min ? min = 5 5 > max ? max = 5 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  20. A case study Find both the min. and the max. of an array of size n . Optimized Algorithm: 5 1 4 3 6 0 2 8 7 9 5 < 1 ? 1 < min ? min = 1 5 > max ? max = 5 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  21. A case study Find both the min. and the max. of an array of size n . Optimized Algorithm: 5 1 4 3 6 0 2 8 7 9 4 < 3 ? 3 < min ? min = 1 4 > max ? max = 5 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  22. A case study Find both the min. and the max. of an array of size n . Optimized Algorithm: 5 1 4 3 6 0 2 8 7 9 6 < 0 ? 0 < min ? min = 1 6 > max ? max = 5 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  23. A case study Find both the min. and the max. of an array of size n . Optimized Algorithm: 5 1 4 3 6 0 2 8 7 9 6 < 0 ? 0 < min ? min = 0 6 > max ? max = 6 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  24. A case study Find both the min. and the max. of an array of size n . Optimized Algorithm: 5 1 4 3 6 0 2 8 7 9 2 < 8 ? 2 < min ? min = 0 8 > max ? max = 6 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  25. A case study Find both the min. and the max. of an array of size n . Optimized Algorithm: 5 1 4 3 6 0 2 8 7 9 2 < 8 ? 2 < min ? min = 0 8 > max ? max = 8 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  26. A case study Find both the min. and the max. of an array of size n . Optimized Algorithm: 5 1 4 3 6 0 2 8 7 9 7 < 9 ? 7 < min ? min = 0 9 > max ? max = 8 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  27. A case study Find both the min. and the max. of an array of size n . Optimized Algorithm: 5 1 4 3 6 0 2 8 7 9 7 < 9 ? 7 < min ? min = 0 9 > max ? max = 9 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  28. A case study Find both the min. and the max. of an array of size n . Optimized Algorithm: 3 n/ 2 comparisons (optimal) 5 1 4 3 6 0 2 8 7 9 7 < 9 ? 7 < min ? min = 0 9 > max ? max = 9 N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  29. A case study Find both the min. and the max. of an array of size n . Optimized Algorithm: 3 n/ 2 comparisons (optimal) Naive Algorithm: 2 n comparisons N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  30. A case study Find both the min. and the max. of an array of size n . Optimized Algorithm: 3 n/ 2 comparisons (optimal) Naive Algorithm: 2 n comparisons In practice, on uniform random data? N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  31. A case study Find both the min. and the max. of an array of size n . in C, using gcc -O0 , random integers N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  32. A case study Find both the min. and the max. of an array of size n . in C, using gcc -O0 , random integers N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 2/16

  33. What “really” happens in the processor... optimized min/max search // RAND_ARRAY: an array of length N // filled with random integers min = RAND_ARRAY[0]; max = RAND_ARRAY[0]; for(i=0; i<N; i+=2){ //assume N is even a1 = RAND_ARRAY[i]; a2 = RAND_ARRAY[i+1]; if (a1 < a2) { if (a1 < min) min = a1; if (a2 > max) max = a2; } else { if (a2 < min) min = a2; if (a1 > max) max = a1; } } N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 3/16

  34. What “really” happens in the processor... sample of assembly code ( gcc -O0 ) mov esi, dword ptr [rbp - 60] cmp esi, dword ptr [rbp - 64] jge LBB2_8 mov eax, dword ptr [rbp - 60] cmp eax, dword ptr [rbp - 12] jge LBB2_5 mov eax, dword ptr [rbp - 60] mov dword ptr [rbp - 12], eax LBB2_5: mov eax, dword ptr [rbp - 64] cmp eax, dword ptr [rbp - 16] jle LBB2_7 ... N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 3/16

  35. What “really” happens in the processor... sample of assembly code ( gcc -O0 ) ◮ Each instruction mov esi, dword ptr [rbp - 60] cmp esi, dword ptr [rbp - 64] can be decomposed: jge LBB2_8 mov eax, dword ptr [rbp - 60] cmp eax, dword ptr [rbp - 12] ◮ Most modern jge LBB2_5 processors are pipelined mov eax, dword ptr [rbp - 60] mov dword ptr [rbp - 12], eax ◮ Instructions LBB2_5: mov eax, dword ptr [rbp - 64] are parallelized cmp eax, dword ptr [rbp - 16] jle LBB2_7 ... simple 5 stages pipeline: N. Auger , C. Nicaud , C. Pivoteau Good predictions are worth... 3/16

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