{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

l26 - 6.896 Sublinear Time Algorithms Lecture 26 Lecturer...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
6.896 Sublinear Time Algorithms May 10, 2007 Lecture 26 Lecturer: Ronitt Rubinfeld Scribe: Chih-yu Chao 1R e c a p f r o m L a s t T i m e Defnition 1 (Fourier cofficient) ˆ f ( s )= h f, χ s i = 1 2 n x ∈{± 1 } n f ( x ) χ s ( x ) Theorem 1 f,x f ( x s [ n ] ˆ f ( s ) χ s ( x ) Last time, we learned that the magnitute of the Fourier coefficient is in agreement with the linear function. Based on this, we can use Fourier analysis to test linearity. 2U s e s o f F o u r i e r A n a l y s i s Now, we are going to use Fourier analysis to: 1. estimate sums of powers of degree 1 Fourier coefficient of f 2. estimate the maximum of degree 1 Fourier coefficient 2.1 Estimate sums of powers We would like to estimate sums of powers of degree 1 Fourier coefficients of f , given oracle access to f . (We are only inerested in sets of size 1, and thus “degree 1”.) X | s | =1 ˆ f ( s ) p (for p 2) The algorithm is a random process: Pick x (1) ,x (2) ,...,x ( p 1) R 1 } n uniformly Pick noise vector μ such that entry +1 1 with probability 1 2 + η 2 1 2 η 2 y f ( x (1) ) f ( x (2) ) ··· f ( x ( p 1) ) f ( x (1) ² x (2) x ( p 1) ² μ ) (note that for p =3,italmost looks like a linearity test with noise) We will show: E [ y ]= s [ n ] η | s | ˆ f ( s ) p , which will look a lot like the analysis of the linearity test last time.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 4

l26 - 6.896 Sublinear Time Algorithms Lecture 26 Lecturer...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online