1. Deterministic Amplification of Space-Bounded Probabilistic Algorithms Ziv Bar-Yossef Oded Goldreich U.C. Berkeley Weizmann Institute U.C. Berkeley Weizmann Institute of Science of Science Avi Wigderson The Hebrew University Amp-1

2. Monte-Carlo Algorithms Use random bits to determine whether x  L Random tape is one-way Bad witness: a sequence of random bits that leads to an error Error probability: proportional volume of bad witness set (r,  -Monte-Carlo algorithm: –uses r random bits –has a constant error probability 0 <  1/2 Amp-2

3. Deterministic Amplification Goals: –to amplify the success of a Monte-Carlo algorithm –to save random bits Deterministic Amplifier: Given: (r,  )-Monte-Carlo algorithm for L Yields:(l,  )-Monte-Carlo algorithm for L with: -  - l  r as small as possible Amplifies the success of the algorithm Tries to save random bits Amp-3

4. Naive Amplification A1A1A1A1 A2A2A2A2 AkAkAkAk Majority y1y1y1y1 y2y2y2y2 ykykykyk k independent random strings   (k) Error probability <   (k) (Chernoff) kr No. of random bits: kr Amp-4

5. Black-Box Amplification Amp-5 A1A1A1A1 A2A2A2A2 AkAkAkAk y1y1y1y1 y2y2y2y2 ykykykyk Random seed of length l Weak Extractor Majority

6. Weak Extractors (1) 2 l Nodes 2 r Nodes k bipartite graph 2 l nodes on the left 2 r nodes on the right left degree k V1V1 V2V2 Amp-6 How black-box amplifiers use weak extractors: Choose a random node y  V 1 Compute the k neighbors y 1,…,y k of y Use y 1,…,y k as random strings for the k simulations of A

7. Weak Extractors (2) Amp-7 Bad witness set of A: a subset W  V 2 of volume <  Bad witness set of M A : the set of y  V 1 whose majority of neighbors belong to W (  )-weak extractor: For any subset W  V 2 of volume < , the set of y  V 1 whose majority of neighbors belong to W is of volume < 

8. Known Amplifiers Amp-8 Method Random Error Bits (l) Probability (  ) Chor-Goldreich 2r O(1/k) Karp-Pippenger-Sipser r O(1/k) Impagliazzo-Zuckerman O(r+k 2 ) 2 -  (k) Nisan O(r log k) 2 -  (k) Ajtai-Koml Ó s-Szemerédi r + O(k) 2 -  (k)

9. Applicability of Black-Box Amplifiers C-applicability: for all A  C, also M A  C Determined by the complexity of computing neighborhoods in the weak extractor All above amplifiers are BPP-applicable Amp-9

10. Space-Bounded Amplification An (S,p)-efficient black-box amplifier: –uses S space for computing the k neighbors in the weak extractor –runs at most p simulations simultaneously (p-parallel) If A uses S A space, then M A uses O(S + pS A ) space BPL - logspace polynomial time Monte-Carlo algorithms Fact: BPL-applicability  (O(log(n),O(1))-efficiency Amp-10

11. BPL-Applicable Black-Box Amplifiers Problems: –Naive amplifier is BPL-applicable but uses too many random bits –Straightforward implementations of other amplifiers need to store the random seed in their work space random seed may be of polynomial size Conclusion: No known BPL-applicable amplifier that uses a small number of random bits Amp-11

12. Positive Result Theorem: A new implementation of the AKS amplifier: –uses O(k) space for computing the k neighbors in the weak extractor –k-parallel Corollary: For any constant 0 <  we obtain an amplifier which is: –BPL-applicable –reduces the error from  to  –uses r + O(1) random bits Amp-12

13. Negative Result Theorem: Any black-box amplifier that: –is p-parallel –uses < r/4 space for computing the k neighbors –uses O(r) random bits cannot reduce an  error probability to less than  O(p). Corollary: BPL-applicable black-box amplifiers that use O(r) random bits can achieve only a constant amplification. Amp-13

14. The AKS Amplifier G: d-regular expander on 2 r nodes (d is constant) The AKS weak extractor: –V 1 - all walks on G of length k –V 2 - all nodes of G –every walk is connected to all the nodes that occur in it Theorem (AKS,CW,IZ): The AKS amplifier uses r + O(k) random bits and reduces the error probability from  to   k). Amp-14

15. Proof of Positive Result (1) We want to find a new implementation of the AKS amplifier which: –has a one-way access to the random seed (a random walk on G) –computes the k neighbors of the seed (the k nodes of the walk) in O(k) space –runs at most k simulations simultaneously Amp-15

16. Online Constant-Space Expanders Online constant space expander: Has a neighborhood algorithm R, which if given: –a node v  G –a neighbor index j outputs: –the j’th neighbor of v with: –one-way access to the bits of v –constant space Amp-16

17. Proof of Positive Result (2) Use an online constant-space expander G Run the k simulations simultaneously Encoding of a walk: j 1,…,j k,v 0 Initialization: read j 1,…,j k from the random tape Make r iterations. At the i th iteration: –read the i th bit of v 0 –compute the i th bits of v 1,…,v k –feed these bits into the simulations A 1,…,A k Amp-17

18. Proof of Positive Result (3) j1j1 R1R1R1R1 v1v1... A1A1A1A1 j2j2 R2R2R2R2 v2v2 A2A2A2A2 v0v0 jkjk RkRkRkRk vkvk AkAkAkAk Amp-18

19. The Margulis-Gabber-Galil Expander A graph on m 2 nodes Every node is a pair (x,y) where x,y  Z m (x,y) is connected to –(x+y,y), (x-y,y) –(x+y+1,y), (x-y-1,y) –(x,y+x), (x,y-x) –(x,y+x+1), (x,y-x-1) (all operations are modulo m) Amp-19

20. The MGG Expander is Online Constant-Space Theorem: Under a certain encoding, the MGG expander on 2 2w nodes is an online constant-space expander. Proof: –Encoding for a node (x,y): x 1,y 1,x 2,y 2,…,x w,y w –To compute (x’,y’), a neighbor of (x,y), we need to calculate a few summations modulo 2 w. –Calculation of x’ i,y’ i requires only x i,y i and a few carry bits –Can be carried out online and in constant space Amp-20

21. Idea of the Negative Result’s Proof Information-theoretic fact: A black-box amplifier that makes k simulations cannot reduce an  error probability to less than  O(k). We show that if a black-box amplifier: –cannot store the seed in its work space –uses cr random bits –is p-parallel then it works as if k = cp. Amp-21

22. Summary of Results First non-trivial BPL-applicable amplifier Proof of optimality with respect to BPL-applicable black- box amplifiers that use O(r) random bits First example of an online constant-space expander First example of an online constant-space weak extractor Amp-22

23. Open Problems Can non-black-box amplifiers do better than a constant amplification for BPL algorithms? Other applications of online constant-space weak extractors and expanders? Amp-23

24. Thank You Amp-24