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Course: CS 70, Fall 2008
School: Berkeley
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70 CS Spring 2008 Lesson Plan Discrete Mathematics for CS David Wagner Note 2 In order to be uent in mathematical statements, you need to understand the basic framework of the language of mathematics. This rst week, we will start by learning about what logical forms mathematical theorems may take, and how to manipulate those forms to make them easier to prove. In the next few lectures, we will learn several...

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70 CS Spring 2008 Lesson Plan Discrete Mathematics for CS David Wagner Note 2 In order to be uent in mathematical statements, you need to understand the basic framework of the language of mathematics. This rst week, we will start by learning about what logical forms mathematical theorems may take, and how to manipulate those forms to make them easier to prove. In the next few lectures, we will learn several different methods of proving things. Propositions A proposition is a statement which is either true or false. These statements are all propositions: (1) 3 is irrational. (2) 1 + 1 = 5. (3) Julius Caesar had 2 eggs for breakfast on his 10th birthday. These statements are clearly not propositions: (4) 2 + 2. (5) x2 + 3x = 5. These statements arent propositions either (although some books say they are). Propositions should not include fuzzy terms. (6) Arnold Schwarzenegger often eats broccoli. (What is often?) (7) George W. Bush is popular. (What is popular?) Propositions may be joined together to form more complex statements. Let P, Q, and R be variables representing propositions (for example, P could stand for 3 is odd). The simplest way of joining these propositions together is to use the connectives...
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Berkeley - CS - 70
CS 70 Fall 2000Discrete Mathematics for CS WagnerMT1 SolSolutions to Midterm 11. (16 pts.) Theorems and proofs (a) (4 pts) Prove that if a and b are rational, then ab is rational. Since a and b are rational they can be written as the ratio of
Berkeley - CS - 70
CS 70 Spring 2008Discrete Mathematics for CS David WagnerNote 11Error Correcting CodesErasure ErrorsWe will consider two situations in which we wish to transmit information on an unreliable channel. The first is exemplified by the internet, w
Berkeley - CS - 170
UC BerkeleyCS 170 Lecturer: David WagnerProblem Set 9 Due on April 17 at 3:30 p.m.Problem Set 9 for CS 170Formatting Please use the following format for the top of the solution you turn in, with one line per item below (in the order shown below)
Berkeley - CS - 70
CS 70 Spring 2008Discrete Mathematics for CS David WagnerNote 6Well Ordering PrincipleHow can the induction axiom fail to be true? Recall that the axiom says the following: [P(0) (n . P(n) = P(n + 1)] = n . P(n). What would it take for n N .
Berkeley - CS - 70
CS 70 Spring 2008Discrete Mathematics for CS David WagnerNote 8Cake Cutting and Fair Division AlgorithmsThe cake-cutting problem is as follows. We have a cake, to be shared among n of us, and we want to split it amongst themselves fairly. Howe
Berkeley - CS - 276
U.C. Berkeley CS276: Cryptography Professor David WagnerLecture 30 May 9, 2006Lecture 30 Fun Topics 11.1Secret SharingSecret sharing backgroundConsider a bank vault locked with a combination lock, and n bank vice presidents that need to b
Berkeley - CS - 70
CS 70 Spring 2008P RINT your name:Discrete Mathematics for CS David Wagner,(last)Final Exam(rst)S IGN your name: P RINT your Unix account login: Your section time (e.g., Tue 3pm): Name of the person sitting to your left: Name of the person
Berkeley - MATH - 185
File: DSpr2p43Solution for Ex. 2 p.43 of Sarasons NotesOctober 16, 2006 10:18 amExercise 2 p. 43: Let be a complex number of unit modulus and an irrational real number. Prove that the values of form a dense subset of the unit circle. Solutio
Berkeley - CS - 262
Advanced Topics in Computer Systems, CS262B Prof Eric A. BrewerPractical Byzantine Fault ToleranceMarch 11, 2004[updated 3/12/04]I. MotivationWe need to make systems work without having trust all of the components. We call faults with arbitrar
Berkeley - CS - 174
CS174 Lecture 18Byzantine AgreementThis lecture, we describe the Byzantine agreement problem. Given n processors, the goal is to have all processors agree on a binary decision. This is a basic and deceptively difficult task in distributed computing
Berkeley - CS - 18
CS174 Lecture 18Byzantine AgreementThis lecture, we describe the Byzantine agreement problem. Given n processors, the goal is to have all processors agree on a binary decision. This is a basic and deceptively difficult task in distributed computing
Berkeley - CS - 174
CS174Formula SheetX u; Y=John CannyY v1 X= PrIndependence: Random variables X , Y are independent iff for all values u and v ,Pr =v= PrX u=Pr=Expected Value: The expected value EEX=X of X :EXX=kkPrX k=
Berkeley - CS - 174
Solutions for CS174 Homework 1Pr Y = 1 = 1=3; Pr Y = 1jX = 1 = 2=3; so Pr Y = 1jX = 1 6= Pr Y = 1 . Therefore X and Y are not independent. E XY = 1 Pr X = 1; Y = 1 + 0 Pr X = 1; Y = 0 + Pr X = 0; Y = 1 + Pr X = 0; Y = 0 = 1=3:Solution 1. Solutio
Berkeley - CS - 12
CS174 Lecture 12Program CheckingMany application programs (e.g. air traffic control, financial management) have become so complicated that its very difficult to discover and correct errors and produce a correct (or sufficiently correct) program. Pr
Berkeley - CS - 174
CS174 Lecture 12Program CheckingMany application programs (e.g. air traffic control, financial management) have become so complicated that its very difficult to discover and correct errors and produce a correct (or sufficiently correct) program. Pr
Berkeley - CS - 174
CS174 Lecture Note 4Based on notes by Alistair Sinclair, September 1998; based on earlier notes by Manuel Blum/Douglas Young. More on random permutations We might ask more detailed questions, such as: Q3: What is the probability that contains at le
Berkeley - CS - 4
CS174 Lecture Note 4Based on notes by Alistair Sinclair, September 1998; based on earlier notes by Manuel Blum/Douglas Young. More on random permutations We might ask more detailed questions, such as: Q3: What is the probability that contains at le
Berkeley - CS - 174
CS174Lecture 3John CannyRandomized Quicksort and BSTsA couple more needed results about permutations: Q1: Whats the probability that 4 comes before 5 in a random permutation? Tempting to say 1/2, but why? For every permutation where 4 is befor
Berkeley - CS - 174
Solutions for CS174 Homework 4Consider a nal stable marriage. Because all the males have the same preference ordering, then we can assign each female a unique number k representing her ranking on the lists. Female k has a spouse, and she must be th
Berkeley - CS - 4
Solutions for CS174 Homework 4Consider a nal stable marriage. Because all the males have the same preference ordering, then we can assign each female a unique number k representing her ranking on the lists. Female k has a spouse, and she must be th
Berkeley - CS - 174
CS174 CryptographyLecture 21John CannyThe idea of cryptography is to protect data by transforming into a representation from which the original is hard to recover. These days many networking technologies (internet, wireless) allow many agents t
Berkeley - CS - 21
CS174 CryptographyLecture 21John CannyThe idea of cryptography is to protect data by transforming into a representation from which the original is hard to recover. These days many networking technologies (internet, wireless) allow many agents t
Berkeley - CS - 17
CS174 Lecture 17Minimum Spanning TreesRemember the minimum spanning tree problem from CS170 you are given a graph G with weighted edges (real values on each edge) and the goal is to find a spanning tree T whose total weight is minimal. The minimum
Berkeley - CS - 174
CS174 Lecture 17Minimum Spanning TreesRemember the minimum spanning tree problem from CS170 you are given a graph G with weighted edges (real values on each edge) and the goal is to find a spanning tree T whose total weight is minimal. The minimum
Berkeley - CS - 174
CS174Lecture 25John CannySecret Sharing and Threshold DecryptionThe goal of secret-sharing is to divide a secret S into n pieces S1 , . . . , Sn such that any m + 1 pieces are sufcient to reconstruct S, but any m pieces give no information abo
Berkeley - CS - 25
CS174Lecture 25John CannySecret Sharing and Threshold DecryptionThe goal of secret-sharing is to divide a secret S into n pieces S1 , . . . , Sn such that any m + 1 pieces are sufcient to reconstruct S, but any m pieces give no information abo
Berkeley - CS - 174
CS174Lecture 24John CannyZero-Knowledge Proofs for discrete logsSuppose you want to prove your identity to someone, in order to cash a check or pick up a package. Most forms of ID can be copied or forged, but there is a zero-knowledge method t
Berkeley - CS - 24
CS174Lecture 24John CannyZero-Knowledge Proofs for discrete logsSuppose you want to prove your identity to someone, in order to cash a check or pick up a package. Most forms of ID can be copied or forged, but there is a zero-knowledge method t
Berkeley - CS - 174
CS174Monte-Carlo vs. Las VegasLecture 2John CannyA random algorithm is Las Vegas if it always produces the correct answer. The running time depends on the random choices made in the algorithm. Random Quicksort (where pivot elements are chosen
Berkeley - CS - 174
CS174Lecture 8John CannyMore on Coupon CollectingRecall that coupon collecting is equivalent to placing m balls in n bins so that no bin is empty. Last time we derived an upper bound for the probability that some bin is empty which is Pr[some
Berkeley - CS - 174
CS174Lecture 22John CannySecure Hash AlgorithmsAnother basic tool for cryptography is a secure hash algorithm. Unlike encryption, given a variablelength message x, a secure hash algorithm computes a function h(x) which has a xed and often smal
Berkeley - CS - 22
CS174Lecture 22John CannySecure Hash AlgorithmsAnother basic tool for cryptography is a secure hash algorithm. Unlike encryption, given a variablelength message x, a secure hash algorithm computes a function h(x) which has a xed and often smal
Berkeley - CS - 12
CS174 Sp2001Homework 12 Solutionsout: May 3, 20011. Each secret share si of a secret s is a pair xi ; yi where yi1= pxi andpx = rtxt + + r x + smod p is a polynomial whose coefcients r ; : : : ; rt are chosen independently and uniformly
Berkeley - CS - 174
CS174 Sp2001Homework 12 Solutionsout: May 3, 20011. Each secret share si of a secret s is a pair xi ; yi where yi1= pxi andpx = rtxt + + r x + smod p is a polynomial whose coefcients r ; : : : ; rt are chosen independently and uniformly
Berkeley - CS - 174
CS174Tail BoundsLecture 6John CannyLast time we looked at occupancy problems and derived some results on the distribution of some random variables. We derived bounds on the probability of a bin containing more than k balls, and the expected nu
Berkeley - CS - 6
CS174Tail BoundsLecture 6John CannyLast time we looked at occupancy problems and derived some results on the distribution of some random variables. We derived bounds on the probability of a bin containing more than k balls, and the expected nu
Berkeley - CS - 174
CS174 J. CannyMidterm 1Spring 2000 Feb 29This is a closed-book exam with 4 questions. You have 80 minutes. All questions are worth equal points, so be sure to budget 20 minutes per question. You are allowed to use the formula sheet that will be
Berkeley - CS - 174
CS174 J. CannyMidterm 2Spring 2000 April 6This is a closed-book exam with 4 questions. You are allowed to use the 4 sides of notes that you brought with you. The marks for each question are shown in parentheses, and the total is 80 points. Make
Berkeley - CS - 14
CS174 Lecture 14Data-Punctuated Token Trees (Berlekamp)Fingerprints provide a fast and communication-efficient way to check whether two strings are identical or not. Rather than sending S and T over a network, you can send the fingerprint f(S) from
Berkeley - CS - 174
CS174 Lecture 14Data-Punctuated Token Trees (Berlekamp)Fingerprints provide a fast and communication-efficient way to check whether two strings are identical or not. Rather than sending S and T over a network, you can send the fingerprint f(S) from
Berkeley - CS - 11
CS174 Lecture 11Routing in a Parallel ComputerWe study the problem of moving packets around in a parallel computer. In this lecture we will consider parallel computers with hypercube connection networks. The methods we describe are easy to adapt to
Berkeley - CS - 174
CS174 Lecture 11Routing in a Parallel ComputerWe study the problem of moving packets around in a parallel computer. In this lecture we will consider parallel computers with hypercube connection networks. The methods we describe are easy to adapt to
Berkeley - CS - 174
CS174 J. CannyMidterm 1Spring 99 Mar 2This is a closed-book exam with 4 questions. You have 80 minutes. All questions are worth equal points, so be sure to budget 20 minutes per question. You are allowed to use the formula sheet that will be ha
Berkeley - CS - 10
CS174 Chernoff BoundsLecture 10John CannyChernoff bounds are another kind of tail bound. Like Markoff and Chebyshev, they bound the total amount of probability of some random variable Y that is in the tail, i.e. far from the mean.Recall that
Berkeley - CS - 174
CS174 Chernoff BoundsLecture 10John CannyChernoff bounds are another kind of tail bound. Like Markoff and Chebyshev, they bound the total amount of probability of some random variable Y that is in the tail, i.e. far from the mean.Recall that
Berkeley - CS - 174
CS174 Lecture 19Paging, Online Algorithms and AdversariesYou should be familiar with the paging problem. A computer has a cache memory which can hold k pages. Then there is a much larger slow memory (or disk) which can hold an arbitrary number of p
Berkeley - CS - 19
CS174 Lecture 19Paging, Online Algorithms and AdversariesYou should be familiar with the paging problem. A computer has a cache memory which can hold k pages. Then there is a much larger slow memory (or disk) which can hold an arbitrary number of p
Berkeley - CS - 174
CS174 Occupancy ProblemsLecture 5John CannyOccupancy problems deal with pairings of objects. The basic occupancy problem is about placing m balls into n bins. This seemingly ordinary problem has a vast number of applications. Let Xi be the rand
Berkeley - CS - 5
CS174 Occupancy ProblemsLecture 5John CannyOccupancy problems deal with pairings of objects. The basic occupancy problem is about placing m balls into n bins. This seemingly ordinary problem has a vast number of applications. Let Xi be the rand
Berkeley - CS - 174
Solutions for CS174 Homework 6P1. From the lecture notes, E Hij n=2: If the probability of a given packet is delayed more than T n steps is bounded by 2,2n, then we can guarantee that all packets reach their destination in time T n with probabili
Berkeley - CS - 6
Solutions for CS174 Homework 6P1. From the lecture notes, E Hij n=2: If the probability of a given packet is delayed more than T n steps is bounded by 2,2n, then we can guarantee that all packets reach their destination in time T n with probabili
Berkeley - CS - 174
CS174 J. CannyFinal Exam SolutionsSpring 2001 May 161. Give a short answer for each of the following questions: (a) (4 points) Suppose two fair coins are tossed. Let X be 1 if the first coin is heads, 0 if the first coin is a tail. Let Y be 1 i
Berkeley - CS - 174
CS174Randomized Birthday SearchLecture 1John CannyFrom the table below, copy the number under the month of your birthday onto a piece of paper. Jan 323 Feb 106 Mar Apr May 261 13 75 Jun 137 July 354 Aug 292 Sept 230 Oct 168 Nov 44 Dec 199Now
Berkeley - CS - 174
CS174 Sp2001 J. CannyHomework 2out: Jan 25, 2001 due: Feb 1, 2001This homework is due by 5pm on Thursday Feb 1st. Please hand it to the CS174 homework box on the second oor of Soda Hall. 1. Suppose you need a biased coin which has probability k
Berkeley - CS - 174
CS174Lecture 29John CannyTraceable Anonymous CashTraceable anonymous cash sounds like an oxymoron. But the secret of good cryptography is to reveal only the information necessary, and only when appropriate. In this case, the cash is anonymous
Berkeley - CS - 29
CS174Lecture 29John CannyTraceable Anonymous CashTraceable anonymous cash sounds like an oxymoron. But the secret of good cryptography is to reveal only the information necessary, and only when appropriate. In this case, the cash is anonymous
Berkeley - CS - 174
CS174 Sp2001Quiz 1Feb 8, 2001Please write your name and SID number in the spaces below, and wait for the signal to start:NameSID11. Let X be a random variable which is 1 iff the number on a toss of a fair die is even, 0 otherwise. Let Y
Berkeley - CS - 174
CS174 Sp2001Homework 5due: Feb 22, 2001This homework is due by 5pm on Thursday Feb 22th. Please hand it to the CS174 homework box on the second oor of Soda Hall. 1. Let G be a random graph with n vertices and m edges generated using the rst ran
Berkeley - CS - 5
CS174 Sp2001Homework 5due: Feb 22, 2001This homework is due by 5pm on Thursday Feb 22th. Please hand it to the CS174 homework box on the second oor of Soda Hall. 1. Let G be a random graph with n vertices and m edges generated using the rst ran
Berkeley - HISTORY - 90
Solutions to homework #2. 1. Answer: 3 2/(3 + 2). Consider the cross-section through the vertex A of the cone and a diagonal BC of the top of the cube. We get two similar triangles ABC and AB C where B C is a diameter of the base the cone. If the s
Berkeley - CS - 294
The problemPractical Byzantine Fault ToleranceMiguel Castro and Barbara Liskov MITPresented to cs294-4 by Owen CooperProvide a reliable answer to a computation even in the presence of Byzantine faults. A client would like to Transmit a r