# (67902) Topics in Theory and Complexity Nov 2, Lecture 7

Save this PDF as:

Size: px
Start display at page:

Download "(67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7"

## Transcription

1 (67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition of the zigzag product of graphs and the expansion property of these graph We will also see another operation on graphs which is the replacement graph In the second part we will introduce the notion of probabilistically checkable proofs, define the class PCP, and make some easy remarks 2 ZigZag Recall we have defined the zigzag product between two graphs as follows Definition 1 Let G 1 be a (n, d 1, α) graph and H 1 be a (d 1, d 2, β) graph Let {1,, d 1 } be the numbering of vertices in H, and for each vertex v G fix an ordering of v s neighbors We define the zigzag product G z H as follows: V (G z H) V (G) V (H) ((v, i), (u, j)) E(G z H) if exists i, j such that: (i, i ) E(H) and u is the i th neighbor of v in V (G) (j, j ) E(H) and v is the j th neighbor of u in V (G) this means that e i v e j u Remark A way to look at this graph is as taking the graph G and replacing each vertex v V (G) with a cloud that is the graph H (thus creating the vertices (v, i) i V (H)), we will refer to this cloud as H v Now, with respect to the edges, we can look at the edges of each instance of H and refer to them as in-cloud edges from each cloud there are between-clouds edges that are connected according to the edges of G and the ordering of the edges for each vertex in G (This is actually the replacement-product graph edges - see below) This way, the edges of the zigzag-product graph are edges that connect vertices that have a in-cloud between-clouds in-cloud path between them The ZigZag product allows us to create a family of expander graphs as we have seen in the previous lecture We will now see a proof of the theorem that states this Theorem 2 Let G 1 be a (n, d 1, α)-graph and H 1 be a (d 1, d 2, β)-graph Then G z H is a (n d 1, d 2 2, γ) graph where γ α + β + β 2 Proof The size and the degree of the graph are trivial and derive from the construction Proving the bound on γ Proof Idea: We will see that the normalized adjacency matrix of G z H can be expressed with the adjacency matices of G and H We will then use this in order to bound the eigenvalue of that matrix 7-1

2 Let A, B be the normalized adjacency matrices of G and H respectively A describes a step in a random walk on G and B describes a step in a random walk on H A single step in G z H can be thought of as three steps: Zig - a step in the cloud of H v - can be described as a step in H Zag - a step in the graph G (going from H v to some other cloud H v ) -can be described as a step in G Zig - a step in the cloud of H v - another step that can be described as a step in H Let Z n d1 n d 1 be the normalized adjacency matrix of G z H we would like to know what does Z look like with respect to A and B Observation 3 We can think of Z as a n n block matrix where each block is of size d 1 d 1 This way, the first and last steps are limited to steps within the respective blocks of the vertices in the diagonal blocks of the matrix Definition 4 The tensor product of two matices A and B is the matrix C A B This matrix can be thought of as a block matrix, in which the i, j block is A i,j B: A 1,1 B, A 1,2 B, A B A n,1 B, A n,2 B, We define B I n B, to be the matrix that describes the the steps within the clouds We define Ã to be the following nd 1 nd 1 matrix defined by A (u,i),(v,j) 1 iff u is the j-th neighbor of v and v is the i-th neighbor of u and A (u,i),(v,j) 0 otherwise It is immediate that Ã has a single 1 in each row, and since it is a symmetric matrix it holds that in each column there is a single 1 Thus, Ã is a permutaion matrix, meaninig that the second step (Zag) is deterministic We can now represent Z in the following manner Z Now, let γ be the second largest eigenvalue of Z Let x 1, and we can use the Rayleigh quotient to obtain: x, Zx γ max x 1,x 0 x, x We will represent x in the following manner: x x 0 + x 1 where x 0 1 and for each cloud its values are identical a v1 a v1 x 0 a v2 a v2 BÃ B 7-2

3 We can construct this vector by letting a vi to be the average value x on each cloud The vector x 1 is a vector that is orthogonal to 1, and such that on each cloud it is orthogonal to the uniform vector on that cloud We now examine the Rayleigh quotient: Zx, x BÃ Bx, x Since B is a symmetric and real matrix: Ã Bx, Bx Ã B(x0 + x 1 ), B(x 0 + x 1 ) Ã Bx0 + Ã Bx 1, Bx 0 + Bx 1 Since each segment of x 0 is orthogonal to 1 d1 then x 0 is invariant on B, ie, Bx0 x 0 Thus, Ã(x0 + Bx 1 ), x 0 + Bx 1 Ãx0, x 0 + Ã Bx1, x 0 + Ãx0, Bx 1 + Ã Bx1, Bx 1 ( ) According to the definition of x 0, there is a vector ω R n, ω 1 n such that x 0 ω 1 d1 (the vector 1 with d 1 entries This way: Ãx0, x 0 ω u ω v d 1 Aω, ω d 1 α ω, ω α x 0, x 0 (v,i) (u,j) x 0(v,i) x 0(u,j) u v For the remaining three terms, let us observe that Bx 1 β x1 This follows since in each cloud, the vector x 1 is orthogonal to the uniform distribution on that cloud Let y Bx 1 By the Cauchy-Schwartz inequality we get Ã Bx1, Bx 1 Ãy, y y 2 Bx 2 1 β 2 x 2 Next observe that Ã cannot increase norm, so Ã Bx 1 Bx1 β x1 Cauchy-Schwartz inequality it also holds that Ã Bx1, x 0 Ã Bx 1 x0 β x 1 x 0 Now, using the and similarly, Ãx0, Bx 1 β x 0 x 1 Now we can get back to where we left: ( ) α x β x 1 x 0 + β x 0 x 1 + β 2 x 1 2 α x β x 1 x 0 + β 2 x 1 2 ( ) Now, Since x 0 x 1 and ( x 0 + x 1 ) 2 > 0: 2β x 1 x 0 β(( x x 1 2 ) ( x 0 + x 1 ) 2 ) β( x 2 ( x 0 + x 1 ) 2 ) β x 2 7-3

4 And since we can always increase the norms in the the equation to be x we get that ( ) (α + β + β 2 ) x 2 This concludes the proof since we have shown that: meaning that γ Zx, x (α + β + β 2 ) x 2 x, Zx max x 1,x 0 x, x (α + β + β2 ) x 2 x 2 α + β + β 2 Corollary 5 If α and β are small then so is γ Meaning that if G and H are expanders so is G z H This theorem can be used to prove that the replacement graphs (metioned above) are also expanders We will start with the definition of the replacement product and then see that these are also expanders Definition 6 Let G 1 be a (n, d 1, α)-graph and H 1 be a (d 1, d 2, β)-graph We define the replacement product G r H as follows: V (G r H) V (G) V (H) E(G r H) E 1 E 2 where E 1 {((v, i), (v, j)) (i, j) E(H)} E 2 {((v, i), (u, j) u i v, v j u} The notation v j u states that v is the i th neighbour of u Claim 7 The replacement product graph is an expander Proof Let R G r H be a graph created by the replacement product Consider the graph G 3, which is the graph on the same vertex set but whose edges correspond to length-3 paths (the adjacency matrix of this graph is simply A 3 ) It is immediate that E(G 3 ) E(G z H) This means that G 3 has at least the same edge expansion as G z H, meaning that its second eigenvalue is also bounded away from 1 Since λ(g) 3 λ(g 3 ) we deduce that λ(g) is also bounded away from 1, and thus it is also an expander We conclude this section by reminding that the construction of a family of expander graphs using the zigzag product require some initial expander graph to begin with We have seen last week, that once we have this initial graph, we can create a whole family of expanders Since this expander is only on a constant number of vertices, it can be found in O(1) time 7-4

5 3 PCP - Probabilistically Checkable Proofs 31 Proof systems and the class NP Definition 8 Given a set of axioms and deduction rules A Proof is defined to be a series of formulas φ 1, φ 2,, φ n where the each of the formulas are either an axiom or was deduced by the deduction rules by previous formulas Definition 9 A system which holds the above is called a Proof System A proof system must have the two following properties: Completeness: if φ is true, then there exists a valid proof φ 1, φ 2,, φ m And it holds that φ φ m Soundness: if φ is false, then there is no proof φ 1, φ 2,, φ m, such that φ φ m Definition 10 The Class NP is the set of languages L for which there exists a polynomial algorithm A and a constant c such that: 1 x L y, y x c ; A(x, y) 1 2 x / L y, y x c ; A(x, y) 0 Remark 1 A is a verification algorithm 2 A can be viewed as a proof system for L 32 Randomized Verifiers Definition 11 It is said that an algorithm (receiving input x and a proof y ) has oracle access to the string y, if it can access a specific index within y with a cost of one step This is denoted by A y (x) We use this oracle access to define randomized algorithms which access only a limited number of bits in the string y Definition 12 An algorithm A is a (r, q)-verifier for a language L if for an input string x and a proof string y, A uses at most r random bits, reads at most q bits from y, runs in polynomial time, such that: 1 x L y, y x c ; Pr ρ R {0,1} r[ay (x, ρ) 1] 1 2 x / L y, y x c ; Pr ρ R {0,1} r[ay (x, ρ) 1] < 1 2 The notation A y (x, ρ) means that A has regular access to the input x and the random bits ρ, and oracle access to the proof string y Definition 13 We define the class P CP [r, q] to be the collection of all languages L for which there is an (r, q)-verifier Remark Note that by this definition, NP c P CP [0, n c ] Remark P P CP (0, O(1)) Remark NP c P CP (log n, n c ) The direction is immediate For the direction, observe that any verifier which uses O(log n) random bits can be made deterministic by enumerating over all possible random strings (there are at most 2 O(log n) of them) and outputting 1 unless we encounter any string for which the algorithm returns 0 7-5

6 Theorem 14 (PCP Theorem) P CP [O(log n), O(1)] NP This is a very surprising theorem since it states that every language in NP has a proof format that can be verified by reading only a constant number of bits! These must be randomly chosen of course 7-6

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

### Chapter 6. Orthogonality

6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

### Lecture 2: Universality

CS 710: Complexity Theory 1/21/2010 Lecture 2: Universality Instructor: Dieter van Melkebeek Scribe: Tyson Williams In this lecture, we introduce the notion of a universal machine, develop efficient universal

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### We shall turn our attention to solving linear systems of equations. Ax = b

59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### 1 Formulating The Low Degree Testing Problem

6.895 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 5: Linearity Testing Lecturer: Dana Moshkovitz Scribe: Gregory Minton and Dana Moshkovitz In the last lecture, we proved a weak PCP Theorem,

### princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

### Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013

Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous

### (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

### Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:

Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible

### 2.1 Complexity Classes

15-859(M): Randomized Algorithms Lecturer: Shuchi Chawla Topic: Complexity classes, Identity checking Date: September 15, 2004 Scribe: Andrew Gilpin 2.1 Complexity Classes In this lecture we will look

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various

### Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

### Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

### Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization

### 1 The Line vs Point Test

6.875 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 5: Low Degree Testing Lecturer: Dana Moshkovitz Scribe: Gregory Minton and Dana Moshkovitz Having seen a probabilistic verifier for linearity

### Lecture 4: AC 0 lower bounds and pseudorandomness

Lecture 4: AC 0 lower bounds and pseudorandomness Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: Jason Perry and Brian Garnett In this lecture,

### Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix

Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.

### Entropy Waves, The Zig-Zag Graph Product, and New Constant-Degree Expanders

Entropy Waves, The Zig-Zag Graph Product, and New Constant-Degree Expanders Omer Reingold Salil Vadhan Avi Wigderson. August 1, 2001 Abstract The main contribution of this work is a new type of graph product,

### 10. Graph Matrices Incidence Matrix

10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### Bindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8

Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e

### DETERMINANTS. b 2. x 2

DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

### Orthogonal Diagonalization of Symmetric Matrices

MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

### Lecture 7: NP-Complete Problems

IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 7: NP-Complete Problems David Mix Barrington and Alexis Maciel July 25, 2000 1. Circuit

### Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

### We know a formula for and some properties of the determinant. Now we see how the determinant can be used.

Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we

### 5.3 The Cross Product in R 3

53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

### ONLINE DEGREE-BOUNDED STEINER NETWORK DESIGN. Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015

ONLINE DEGREE-BOUNDED STEINER NETWORK DESIGN Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015 ONLINE STEINER FOREST PROBLEM An initially given graph G. s 1 s 2 A sequence of demands (s i, t i ) arriving

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

### Lecture Notes: Matrix Inverse. 1 Inverse Definition

Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,

### Matrix Inverse and Determinants

DM554 Linear and Integer Programming Lecture 5 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1 2 3 4 and Cramer s rule 2 Outline 1 2 3 4 and

### MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### A Sublinear Bipartiteness Tester for Bounded Degree Graphs

A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublinear-time algorithm for testing whether a bounded degree graph is bipartite

### Combinatorial PCPs with ecient veriers

Combinatorial PCPs with ecient veriers Or Meir Abstract The PCP theorem asserts the existence of proofs that can be veried by a verier that reads only a very small part of the proof. The theorem was originally

### Chapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6

Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a

### Derivatives of Matrix Functions and their Norms.

Derivatives of Matrix Functions and their Norms. Priyanka Grover Indian Statistical Institute, Delhi India February 24, 2011 1 / 41 Notations H : n-dimensional complex Hilbert space. L (H) : the space

### Notes on Symmetric Matrices

CPSC 536N: Randomized Algorithms 2011-12 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.

### MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

### Lecture 3: Finding integer solutions to systems of linear equations

Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

### 1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

### Private Approximation of Clustering and Vertex Cover

Private Approximation of Clustering and Vertex Cover Amos Beimel, Renen Hallak, and Kobbi Nissim Department of Computer Science, Ben-Gurion University of the Negev Abstract. Private approximation of search

### Conductance, the Normalized Laplacian, and Cheeger s Inequality

Spectral Graph Theory Lecture 6 Conductance, the Normalized Laplacian, and Cheeger s Inequality Daniel A. Spielman September 21, 2015 Disclaimer These notes are not necessarily an accurate representation

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### 4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

### MATH 240 Fall, Chapter 1: Linear Equations and Matrices

MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### Week 5 Integral Polyhedra

Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory

### A matrix over a field F is a rectangular array of elements from F. The symbol

Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

### 160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v

### On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

### Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### Definition of a Linear Program

Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

### Linear Algebra and TI 89

Linear Algebra and TI 89 Abdul Hassen and Jay Schiffman This short manual is a quick guide to the use of TI89 for Linear Algebra. We do this in two sections. In the first section, we will go over the editing

### On the Unique Games Conjecture

On the Unique Games Conjecture Antonios Angelakis National Technical University of Athens June 16, 2015 Antonios Angelakis (NTUA) Theory of Computation June 16, 2015 1 / 20 Overview 1 Introduction 2 Preliminary

### Factorization Theorems

Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization

### Orthogonal Projections and Orthonormal Bases

CS 3, HANDOUT -A, 3 November 04 (adjusted on 7 November 04) Orthogonal Projections and Orthonormal Bases (continuation of Handout 07 of 6 September 04) Definition (Orthogonality, length, unit vectors).

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### Lecture 4: BK inequality 27th August and 6th September, 2007

CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound

### 2.5 Elementary Row Operations and the Determinant

2.5 Elementary Row Operations and the Determinant Recall: Let A be a 2 2 matrtix : A = a b. The determinant of A, denoted by det(a) c d or A, is the number ad bc. So for example if A = 2 4, det(a) = 2(5)

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### On Rank vs. Communication Complexity

On Rank vs. Communication Complexity Noam Nisan Avi Wigderson Abstract This paper concerns the open problem of Lovász and Saks regarding the relationship between the communication complexity of a boolean

### Algebraic and Combinatorial Circuits

C H A P T E R Algebraic and Combinatorial Circuits Algebraic circuits combine operations drawn from an algebraic system. In this chapter we develop algebraic and combinatorial circuits for a variety of

### Reminder: Complexity (1) Parallel Complexity Theory. Reminder: Complexity (2) Complexity-new

Reminder: Complexity (1) Parallel Complexity Theory Lecture 6 Number of steps or memory units required to compute some result In terms of input size Using a single processor O(1) says that regardless of

### Reminder: Complexity (1) Parallel Complexity Theory. Reminder: Complexity (2) Complexity-new GAP (2) Graph Accessibility Problem (GAP) (1)

Reminder: Complexity (1) Parallel Complexity Theory Lecture 6 Number of steps or memory units required to compute some result In terms of input size Using a single processor O(1) says that regardless of

### A Linear Online 4-DSS Algorithm

A Linear Online 4-DSS Algorithm V. Kovalevsky 1990: algorithm K1990 We consider the recognition of 4-DSSs on the frontier of a region in the 2D cellular grid. Algorithm K1990 is one of the simplest (implementation)

GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

### Random graphs with a given degree sequence

Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

### Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

### MATH36001 Background Material 2015

MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be

### Why? A central concept in Computer Science. Algorithms are ubiquitous.

Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online

### October 3rd, 2012. Linear Algebra & Properties of the Covariance Matrix

Linear Algebra & Properties of the Covariance Matrix October 3rd, 2012 Estimation of r and C Let rn 1, rn, t..., rn T be the historical return rates on the n th asset. rn 1 rṇ 2 r n =. r T n n = 1, 2,...,

### Linear Programming I

Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

### Introduction to Logic in Computer Science: Autumn 2006

Introduction to Logic in Computer Science: Autumn 2006 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today Now that we have a basic understanding

### Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,

### Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

### Lecture 4: Partitioned Matrices and Determinants

Lecture 4: Partitioned Matrices and Determinants 1 Elementary row operations Recall the elementary operations on the rows of a matrix, equivalent to premultiplying by an elementary matrix E: (1) multiplying

### 15-750 Graduate Algorithms Spring 2010

15-750 Graduate Algorithms Spring 2010 1 Cheap Cut (25 pts.) Suppose we are given a directed network G = (V, E) with a root node r and a set of terminal nodes T V. We would like to remove the fewest number

### Inner products on R n, and more

Inner products on R n, and more Peyam Ryan Tabrizian Friday, April 12th, 2013 1 Introduction You might be wondering: Are there inner products on R n that are not the usual dot product x y = x 1 y 1 + +

### MATH 551 - APPLIED MATRIX THEORY

MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points

### 1 Eigenvalues and Eigenvectors

Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x