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Deterministic Amplification of Space-Bounded Probabilistic Algorithms

Deterministic Amplification of Space-Bounded Probabilistic Algorithms. Ziv Bar-Yossef Oded Goldreich U.C. Berkeley Weizmann Institute of Science Avi Wigderson The Hebrew University. Amp- 1. Monte-Carlo Algorithms. Use random bits to determine whether x  L

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Deterministic Amplification of Space-Bounded Probabilistic Algorithms

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  1. Deterministic Amplification of Space-Bounded Probabilistic Algorithms Ziv Bar-Yossef Oded Goldreich U.C. Berkeley Weizmann Institute 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,e)-Monte-Carlo algorithm: • uses r random bits • has a constant error probability 0 < e < 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,e)-Monte-Carlo algorithm for L Yields: (l,d)-Monte-Carlo algorithm for L with: - d < e -l  r as small as possible • Amplifies the success of the algorithm • Tries to save random bits Amp-3

  4. Naive Amplification A1 y1 kindependent random strings y2 A2 Majority yk Ak • Error probability <eW(k)(Chernoff) • No. of random bits:kr Amp-4

  5. Black-Box Amplification y1 A1 y2 A2 Random seed of length l Weak Extractor Majority yk Ak Amp-5

  6. Weak Extractors (1) V1 V2 • bipartite graph • 2l nodes on the left • 2r nodes on the right • left degree k 2lNodes 2r Nodes k • How black-box amplifiers use weak extractors: • Choose a random node yV1 • Compute the k neighbors y1,…,yk of y • Usey1,…,yk as random strings for the k simulations of A Amp-6

  7. Weak Extractors (2) • Bad witness set of A: a subset W V2 of volume <e • Bad witness set of MA: the set of yV1 whose majority of neighbors belong to W • (d,e)-weak extractor: For any subset W V2of volume < e, the set of y  V1 whose majority of neighbors belong to W is of volume < d Amp-7

  8. Known Amplifiers MethodRandom Error Bits (l) Probability (d) Chor-Goldreich2r O(1/k) Karp-Pippenger-Sipserr O(1/k) Impagliazzo-ZuckermanO(r+k2) 2-W(k) NisanO(r log k) 2-W(k) Ajtai-KomlÓs-Szemerédir + O(k) 2-W(k) Amp-8

  9. Applicability of Black-Box Amplifiers • C-applicability: for all A  C, also MA  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 SA space, then MA uses O(S + pSA) space • BPL - logspace polynomial time Monte-Carlo algorithms • Fact: BPL-applicability  (O(log(n),O(1))-efficiency Amp-10

  11. BPL-ApplicableBlack-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 < d < e we obtain an amplifier which is: • BPL-applicable • reduces the error from e to d • 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 e error probability to less than eO(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 2r nodes (d is constant) • The AKS weak extractor: • V1 - all walks on G of length k • V2 - 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 e to eW(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: j1,…,jk,v0 • Initialization: read j1,…,jk from the random tape • Make r iterations. At the ith iteration: • read the ith bit of v0 • compute the ith bits of v1,…,vk • feed these bits into the simulations A1,…,Ak Amp-17

  18. Proof of Positive Result (3) j1 j2 jk v0 v1 v2 . . . vk R1 R2 Rk A1 A2 Ak Amp-18

  19. The Margulis-Gabber-Galil Expander • A graph on m2 nodes • Every node is a pair (x,y) where x,y Zm • (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 22w nodes is an online constant-space expander. • Proof: • Encoding for a node (x,y): x1,y1,x2,y2,…,xw,yw • To compute (x’,y’), a neighbor of (x,y), we need to calculate a few summations modulo 2w. • Calculation of x’i,y’i requires only xi,yi 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 e error probability to less than eO(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

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