Download presentation

Presentation is loading. Please wait.

Published byMaria Rogers Modified over 2 years ago

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

Similar presentations

OK

Extracting Randomness From Few Independent Sources Boaz Barak, IAS Russell Impagliazzo, UCSD Avi Wigderson, IAS.

Extracting Randomness From Few Independent Sources Boaz Barak, IAS Russell Impagliazzo, UCSD Avi Wigderson, IAS.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on real numbers for class 9th physics Ppt on art of war author Ppt on job rotation disadvantages Multifunction display ppt online A ppt on loch ness monster Ppt on retail sales Ppt on cleanliness of surroundings Ppt on acid-base titration calculation Download ppt on training and development in hrm Ppt on vacuum circuit breaker