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# ParallelRandom-mannyko

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### ParallelRandom-mannyko

1. 1. Parallel Random Generator Manny Ko Principal Engineer Activision
2. 2. Outline ●Serial RNG ●Background ●LCG, LFG, crypto-hash ●Parallel RNG ●Leapfrog, splitting, crypto-hash
3. 3. RNG - desiderata ● White noise like ● Repeatable for any # of cores ● Fast ● Small storage
4. 4. RNG Quality ● DIEHARD ● Spectral test ● SmallCrush ● BigCrush GPUBBS
6. 6. Serial RNG: LCG ● Linear-congruential (LCG) ● 𝑋𝑖 = 𝑎 ∗ 𝑋𝑖−1 + 𝑐 𝑚𝑜𝑑 𝑀, ● a, c and M must be chosen carefully! ● Never choose 𝑀 = 231 ! Should be a prime ● Park & Miller: 𝑎 = 16807, 𝑚 = 214748647 = 231 − 1. 𝑚 is a Mersenne prime! ● Most likely in your C runtime
7. 7. LCG: the good and bad ● Good: ● Simple and efficient even if we use mod ● Single word of state ● Bad: ● Short period – at most m ● Low-bits are correlated especially if 𝑚 = 2 𝑛 ● Pure serial
8. 8. LCG - bad ● 𝑋 𝑘_+1 = (3 ∗ 𝑋 𝑘+4) 𝑚𝑜𝑑 8 ● {1,7,1,7, … }
9. 9. Mersenne Prime modulo ● IDIV can be 40~80 cycles for 32b/32b ● 𝑘 𝑚𝑜𝑑 𝑝 where 𝑝 = 2 𝑠 − 1: ● 𝑖 = 𝑘 & 𝑝 + 𝑘 ≫ 𝑠 ; ● 𝑟𝑒𝑡 𝑖 ≥ 𝑝 ? 𝑖 − 𝑝 ∶ 𝑖;
10. 10. Lagged-Fibonacci Generator ● 𝑋𝑖 = 𝑋𝑖−𝑝 ∗ 𝑋𝑖−𝑞; p and q are the lags ● ∗ is =-* mod M (or XOR); ● ALFG: 𝑋 𝑛 = 𝑋 𝑛−𝑗 + 𝑋 𝑛−𝑘(𝑚𝑜𝑑 2 𝑚) ● * give best quality ● Period = 2 𝑝 − 1 2 𝑏−3; 𝑀 = 2 𝑏
11. 11. LFG ● The good: ●Very efficient: 2 ops + power-of-2 mod ●Much Long period than LCG; ●Directly works in floats ●Higher quality than LCG ●ALFG can skip ahead
12. 12. LFG – the bad ● Need to store max(p,q) floats ● Pure sequential – ● multiplicative LFG can’t jump ahead.
13. 13. Mersenne Twister ● Gold standard ? ● Large state (624 ints) ● Lots of flops ● Hard to leapfrog ● Limited parallelism power spectrum
14. 14. ● End of Basic RNG Overview
15. 15. Parallel RNG ● Maintain the RNG’s quality ● Same result regardless of the # of cores ● Minimal state especially for gpu. ● Minimal correlation among the streams.
16. 16. Random Tree • 2 LCGs with different 𝑎 • L used to generate a seed for R • No need to know how many generators or # of values #s per-thread • GG
17. 17. Leapfrog with 3 cores • Each thread leaps ahead by 𝑁 using L • Each thread use its own R to generate its own sequence • 𝑁 = 𝑐𝑜𝑟𝑒𝑠 ∗ 𝑠𝑒𝑞𝑝𝑒𝑟𝑐𝑜𝑟𝑒
18. 18. Leapfrog ● basic LCG without c: ● 𝐿 𝑘+1 = 𝑎𝐿 𝑘 𝑚𝑜𝑑 𝑚 ● 𝑅 𝑘+1 = 𝑎 𝑛 𝑅 𝑘 𝑚𝑜𝑑 𝑚 ● LCG: 𝐴 = 𝑎 𝑛and 𝐶 = 𝑐(𝑎 𝑛 − 1)/(𝑎 − 1) – each core jumps ahead by n (# of cores)
19. 19. Leapfrog with 3 cores • Each sequence will not overlap • Final sequence is the same as the serial code
20. 20. Leapfrog – the good ● Same sequence as serial code ● Limited choice of RNG (e.g. no MLFG) ● No need to fix the # of random values used per core (need to fix ‘n’)
21. 21. Leapfrog – the bad ● 𝑎 𝑝no longer have the good qualities of 𝑎 ● power-of-2 N produce correlated sub- sequences ● Need to fix ‘n’ - # of generators/sequences ● the period of the original RNG is shorten by a factor of ‘n’. 32 bit LCG has a short period to start with.
22. 22. Sequence Splitting • If we know the # of values per thread 𝑛 • 𝐿 𝑘+1 = 𝑎 𝑛 𝐿 𝑘 𝑚𝑜𝑑 𝑚 • 𝑅 𝑘+1 = 𝑎𝑅 𝑘 𝑚𝑜𝑑 𝑚 • the sequence is a subset of the serial code
23. 23. Leapfrog and Splitting ● Only guarantees the sequences are non- overlap; nothing about its quality ● Not invariant to degree of parallelism ● Result change when # cores change ● Serial and parallel code does not match
24. 24. Lagged-Fibonacci Leapfrog ● LFG has very long period ● Period = 2 𝑝 − 1 2 𝑏−3; 𝑀 = 2 𝑏 ● 𝑀 can be power-of-two! ● Much better quality than LCG ● No leapfrog for the best variant – ‘*’ ● Luckily the ALFG supports leapfrogging
25. 25. Issues with Leapfrog & Splitting ● LCG’s period get even shorter ● Questionable quality ● ALFG is much better but have to store more state – for the ‘lag’.
26. 26. Crypto Hash ● MD5 ● TEA: tiny encryption algorithm
27. 27. Core Idea 1. input trivially prepared in parallel, e.g. linear ramp 2. feed input value into hash, independently and in parallel 3. output white noise hash input output
28. 28. TEA ● A Feistel coder ● Input is split into L and R ● 128B key ● F: shift and XORs or adds
29. 29. TEA
30. 30. Magic ‘delta’ ● 𝑑𝑒𝑙𝑡𝑎 = 5 − 1 231 ● Avalanche in 6 cycles (often in 4) ● * mixes better than ^ but makes TEA twice as slow