Cache-Oblivious and Data-Oblivious Sorting and Applications

Transcription

1 Cache-Oblivious and Data-Oblivious Sorting and Applications T-H. Hubert Chan, Yue Guo, Wei-Kai Lin, and Elaine Shi Jan, 2018

2 External Memory Model Cache efficiency: # of blocks Time: # of words Memory Word (fast) Cache Block (slow) CPU

3 Data-Obliviousness Adversarial storages Encryption is not enough! Memory Cache CPU

4 Data-Obliviousness Access patterns leak sensitive information Memory Read, address Write, address Breast cancer Liver cancer

5 Definition of Data-Oblivious Access patterns should be simulatable [Goldreich, Ostrovsky 96] Memory Simulator Read, address Write, address Read, address Write, address Statistically close

6 Example: Data-Oblivious Sort Emulated sorting circuits Bitonic sort [Batcher 68] Merge sort (or Funnel sort) [Ajtai, Komlós, Szemerédi 83] AKS sort

7 Cache-Oblivious. and Cache-Efficient [Frigo, Leiserson, Prokop, Ramachandran 99] Unknown cache size and block size Implies other nice properties Memory Word Cache size M Cache Block size B Block CPU

8 Sorting Alorithms Data-oblivious Yes Cache-efficient No [Ajtai, Komlós, Szemerédi 83] [Frigo, Leiserson, Prokop, Ramachandran 99] No O N B log M B N B Lower bound [Aggarwal, Vitter 88] Can we construct comparison-based, cache-oblivious and data-oblivious sorting that is optimal in cache effcncy?

9 Sorting Alorithms Data-oblivious Yes Cache-efficient No [Ajtai, Komlós, Szemerédi 83] [Frigo, Leiserson, Prokop, Ramachandran 99] This work No O N B log M B Yes O N B log M B N B Lower bound [Aggarwal, Vitter 88] N B

10 Overview of Cache-Efficient Sort Data-Oblivious Random Permutation () Balls & Bins Trivial Concat. Funnel sort original permuted sorted bins Trivial Trivial Trivial Permuted bins

11 Access pattern only depends on ordering New permutation hides ordering N words data- Oblivious Random Permutation Cache efficient Sort Funnel sort [Frigo, Leiserson, Prokop, Ramachandran 99] original permuted sorted

12 Idea: Funnel Sort [Frigo, Leiserson, Prokop, Ramachandran 99] Merger Main body: Divide and Merge Merge-sort For cache efficiency: Solve subproblem in cache Challenge: unknown cache size Funnel sort ,4 1,2 5,6 7,8 9,,12 0,13 14,15 Sorted array

13 Idea: Funnel Sort [Frigo, Leiserson, Prokop, Ramachandran 99] Main body: Divide and Merge Merge-sort For cache efficiency: Solve subproblem in cache Solution: smart recursion O N. O N Funnel sort O N

14 Data-Oblivious Random Permutation () Randomness: uniformly random shuffle Input array Funnel sort Output: π(input), π: permutation

15 Data-Oblivious Random Permutation () Randomness: uniformly random shuffle Data-Oblivious: unknown π given access pattern Funnel sort Addr 1 Addr 2 Addr 3... Is π uniform?

16 Warmup: Trivial (Data-oblivious) Sort random keys Not cache-efficient! Sort Funnel sort

17 Cache-Efficient Balls & Bins + Trivial inside each bin Data-Oblivious Random Permutation () Balls & Bins Trivial Concat. Funnel sort original permuted sorted bins Trivial Trivial Trivial Permuted bins

18 Ball & Bins Input: N balls Output: N balls in n bins, each with capacity Z Uniform random Balls & Bins Trivial concat Funnel sort

19 Ball & Bins (Bucket Funnel Sort) ** ** ** ** 01 0* 1* 0* 1* Bitonic sort Merge two same-prefix bins 01 01

20 Ball & Bins: Data-Oblivious Bitonic sort: data-oblivious Merging bins: fixed pattern 0* 1* Data-Oblivious 01 0* Negligible overflow probability: Bin size Z = ω log λ λ: security parameter, ~1,0 1*

21 Ball & Bins: Cache Efficiency ** 0* ** 1* 01 ** 0* Borrow recursion from Funnel Sort ** 1*

22 Ball & Bins: Cache Efficiency ** O n ** 01 Assume M = Ω log 1.1 λ, Cache efficiency is O N B log N M B B ** ** O. n O n

23 Cache-Efficient and Data-Oblivious Sort Simple algo: Balls & Bins + Trivial + Funnel sort Balls & Bins Trivial Trivial Trivial Bin size Z = log 1.1 λ Trivial # of bins n = 4N bins Z Concat. Funnel sort original permuted sorted Permuted bins

24 Sorting Alorithms Data-oblivious Yes Cache-efficient No [Ajtai, Komlós, Szemerédi 83] [Frigo, Leiserson, Prokop, Ramachandran 99] This work No Comparison-based, cache-oblivious O N B log M B Yes O N B log M B N B N B Time: O N log N log 2 log λ

25 Applications Cache-efficient data-oblivious RAM compiler: O log N Other data-oblivious frameworks MapReduce GraphSC

26 Summary This work: Cache-efficient, data-oblivious sort Open: Cache-efficient, data-oblivious sort and optimal in time? Followup: Data-oblivious sorting in o N log N time (non-comparisonbased)

27 Thank you!