Sublinear Algorithms Lecture 3

Transcription

1 Sublinear Algorithms Lecture 3 Sofya Raskhodnikova Penn State University Thanks to Madhav Jha (Penn State) for help with creating these slides.

2 Tentative Plan Lecture. Background. Testing properties of images and lists. Lecture 2. Testing properties of lists. Sublinear-time approximation for graph problems. Lecture 3. Testing properties of functions. Linearity testing. Lecture 4. Techniques for proving hardness. Other models for sublinear computation.

3 Testing Linearity

4 Linear Functions Over Finite Field ABoolean function :, {,}islinearif,, = + + for some,, {,} no free term Work in finite field Other accepted notation for : and Z Addition and multiplication is mod 2 =,,,=,,,that is,,, += +,, + example + Based on Ryan O Donell s lecture notes: 4

5 Testing if a Boolean function is Linear Input:Boolean function :, {,} Question: Is the functionlinear or -far from linear ( 2 values need to be changed to make it linear)? Today: can answer in time 5

6 Motivation Linearity test is one of the most celebrated testing algorithms A special case of many important property tests Computations over finite fields are used in Cryptography Coding Theory Originally designed for program checkers and self-correctors Low-degree testing is needed in constructions of Probabilistically Checkable Proofs (PCPs) Used for proving inapproximability Main tool in the correctness proof: Fourier analysis of Boolean functions Powerful and widely used technique in understanding the structure of Boolean functions 6

7 Equivalent Definitions of Linear Functions Definition. islinearif,, = + + for some,,,, = for some. Definition. is linearif + = +()for all,,. [] is a shorthand for {, } Definition Definition + = + = + = +. Definition Definition Let =((,,,,,,)) Repeatedly apply Definition :,, = = =. Based on Ryan O Donell s lecture notes: 7

8 Linearity Test [Blum Luby Rubinfeld 9] BLR Test (f, ε). Pick and independently and uniformly at random from,. 2. Set =+and query on,, and. Acceptiff = +. Analysis If is linear, BLR always accepts. Correctness Theorem [Bellare Coppersmith Hastad Kiwi Sudan 95] If is -far from linear then>fraction of pairs and fail BLR test. Then, by Witness Lemma (Lecture ), 2/iterations suffice. 8

9 Analysis Technique: Fourier Expansion

10 Representing Functions as Vectors Stack the 2 values of ()and treat it as a vector in {,}. = () () () () () () () ()

11 Linear functions There are 2 linear functions: one for each subset []. =, =,, = Parity on the positions indexed by set is,, =

12 Great Notational Switch Idea: Change notation, so that we work over realsinstead of a finite field. Vectors in, Vectors in R. /False /True -. Addition (mod 2) Multiplication in R. Boolean function:, {,}. Linear function, {,}is given by =. 2

13 Benefit of New Notation The dot product of and as vectors in, : (# s such that =()) (# s such that ()) = 2 2 (# s such that ()) disagreements between and Inner product of functions,, {,}, = dot product of and as vectors 2 = avg = E,, [ ]., = 2(fraction of disagreements between and ) 3

14 Benefit 2 of New Notation Claim. The functions form an orthonormal basis for R. If then and are orthogonal:, =. Let be an element on which and differ (w.l.o.g. ) Pair up all -bit strings: (, ) where is with the th bit flipped. Each such pair contributes =to,. Since all s are paired up,, =. Recall that there are 2 linear functions., = In fact,, =for all,,. (The normof,denoted, is, )

15 Fourier Expansion Theorem Idea: Work in the basis, so it is easy to see how close a specific function is to each of the linear functions. Fourier Expansion Theorem Every function, Ris uniquely expressible as a linear combination (over R) of the 2 linear functions: =, where =, is the Fourier Coefficient of on set. Proof: can be written uniquely as a linear combination of basis vectors: It remains to prove that = for all. = =, = Definition of Fourier coefficients [], =, [] Linearity of, =, = if = otherwise 5

16 Examples: Fourier Expansion Fourier transform = = AND(, ) MAJORITY(,, )

17 Parseval Equality Parseval Equality Let :, R. Then, = Proof: By Fourier Expansion Theorem, =, By linearity of inner product =, By orthonormality of s = 7

18 Parseval Equality Parseval Equality for Boolean Functions Let :,,. Then Proof:, = = By definition of inner product, = E, [ ] = Since is Boolean 8

19 BLR Test in {-,} notation BLR Test (f, ε). Pick and independently and uniformly at random from,. 2. Set = and query on,, and. Acceptiff =. Vector product notation: =(,,, ) Sum-Of-Cubes Lemma. Proof: Indicator variable = if BLR accepts otherwise Pr BLR accepts =,, [] = +. Pr,, BLR accepts = E,, = E,, By linearity of expectation 9

20 Proof of Sum-Of-Cubes Lemma So far: Pr BLR accepts =,, + Next: E,, E,, By Fourier Expansion Theorem = E,, () [] () [] () [] Distributing out the product of sums = E,, () () (),, [] By linearity of expectation = E,, [ () () ()],, [] 2

21 Proof of Sum-Of-Cubes Lemma (Continued) Pr,, Claim. BLR accepts LetΔdenote symmetric difference of sets and a E,, [ () () ()] = 2 + 2,, [] E,, [ () () ()] E,, [ () () ()]is if ==and otherwise. a= E,, a= E,, a= E,, a= E, E, a= E, [ ] E, [ ] a= E [ ] E {,} [ ] {,} Since = Since = = Since and are independent Since and s coordinates are independent = when Δ= and Δ= == otherwise 2

22 Proof of Sum-Of-Cubes Lemma (Done) Pr,, BLR accepts = 2 + 2,, [] E,, [ () () ()] = [] Sum-Of-Cubes Lemma. Pr BLR accepts =,, [] 22

23 Proof of Correctness Theorem Correctness Theorem (restated) If is ε-far from linear then Pr BLR accepts. Proof: Suppose to the contrary that < Pr BLR accepts,, = [] max = max Then max > 2. That is, > 2for some. But =, = 2(fraction of disagreements between and ) disagrees with a linear function on <fraction of values. By Sum-Of-Cubes Lemma Since Parseval Equality 23

24 Summary BLR tests whether a function :, {,}is linear or -far from linear ( 2 values need to be changed to make it linear) in time. 24