Computer Science 340 Reasoning about Computation Review Session 1 Thursday, January 10, 2008 Problem 1 Scientists at the Princeton Genomics Institute have discovered the following process: Given a test tube filled with DNA strands, it is possible to

Computer Science 340 Reasoning about Computation Homework 1 Due at the beginning of class on Wed, September 26, 2007 Problem 1 Prove that if x -1, then for any integer n 0, (1 + x)n > nx. Solution: We will strengthen the inductive hypothesis: we wi

Computer Science 340 Reasoning about Computation Homework 6 (take home midterm) Due at the beginning of class on Wednesday, November 7, 2007 There are six problems on the midterm. You must work on them on your own. No collaboration is allowed. Proble

Computer Science 340 Reasoning about Computation Homework 11 Due on Friday, December 14, 2007 For this homework, read Section 5.3 on Mapping Reducibility in the notes distributed in class. Problem 1 Recall that ET M = { M |M is a TM and L(M ) = }. Pr

Computer Science 340 Reasoning about Computation Homework 6 (take home midterm) Due at the beginning of class on Wednesday, November 7, 2007 There are six problems on the midterm. You must work on them on your own. No collaboration is allowed.
Probl

Computer Science 340 Reasoning about Computation Review Session 3 Monday, January 14, 2008
Problem 1 An algebraic number is a complex number that is a root of a polynomial with integer coefficients. Is the set of algebraic numbers countable ? Soluti

Computer Science 340 Reasoning about Computation Review Session 1 Thursday, January 10, 2008
Problem 1 Scientists at the Princeton Genomics Institute have discovered the following process: Given a test tube filled with DNA strands, it is possible to

Computer Science 340 Reasoning about Computation Homework 1 Due at the beginning of class on Wed, September 26, 2007
Problem 1 Prove that if x -1, then for any integer n 0, (1 + x)n > nx. Problem 2 Show that at a party of n people, there are two p

Computer Science 340 Reasoning about Computation Review Session 2 Friday, January 11, 2008
Problem 1 Recall the cow grazing problem from Homework 5. The cow is on a path that leads to food, but the cow doesn't know what direction the food lies in. T

Computer Science 340 Reasoning about Computation Homework 4 Due at the beginning of class on Wednesday, October 17, 2007
Problem 1 (10 points) We analyzed the process of throwing n balls into n bins independently and at random log n and showed that

Computer Science 340 Reasoning about Computation Homework 2 Due at the beginning of class on Wednesday, October 3, 2007 Problem 1 Consider a convex n-gon (a polygon with n sides). A chord is a line segment between two non-adjacent vertices. If all ch

Computer Science 340 Reasoning about Computation
Homework 8 Due at the beginning of class on Wednesday, November 21, 2007 Problem 1 The ciphertext below QFL HCVPS PX V ANSWLCEZK NCJVS; PQ XQVCQX QFL BPSZQL RNZ JLQ ZT PS QFL BNCSPSJ VSW WNLX SNQ XQNT

Computer Science 340 Reasoning about Computation Homework 2 Due at the beginning of class on Wednesday, October 3, 2007 Problem 1 Consider a convex n-gon (a polygon with n sides). A chord is a line segment between two non-adjacent vertices. If all ch

Computer Science 340
Reasoning about Computation
Homework 3
Due 03/02/2016
Problem 1
Throw n balls at random into n bins. Prove the existence of a constant c > such that,
with probability at least 1 1/n, no bin has more than c(log n)/ log log n balls.
Hin

6.042/18.062J Mathematics for Computer Science
Tom Leighton and Ronitt Rubinfeld
October 19, 2004
The Asymptotic Cheat Sheet
Asymptotic notation consists of six funny symbols used to describe the relative growth
rates of functions. These six symbols are d

Computer Science 340
Reasoning about Computation
Precept 7
Problem 1
Let x1 , . . ., xN be sequential elements of a stream S, with each xj [1, n], i.e., the
elements are integers between 1 and nit can be showed that in this context1 this is without
loss o

Computer Science 340
Reasoning about Computation
Precept 5
Problem 1
We have a factory that makes widgets and the expected number of widgets per week is
500.
1. What is probability at least 1000 widgets are produced?
2. If variance is 100 what is the prob

Computer Science 340
Reasoning about Computation
Precept 4
Problem 1
The goal of this problem is to show that given n independent random bits (fair coin flips)
x1 , x2 , . . . , xn one can generate a much larger number of bits which are 2-wise (=pairwise)

COS 340: Reasoning About Computation
Online Algorithms 2
Moses Charikar
March 9, 2014
1
Introduction
In the previous lecture, we introduced the notion of online algorithms and competitive analysis. We studied the Rent or Buy problem, and then discussed th

Computer Science 340 Reasoning about Computation Homework 11 Due on Friday, December 14, 2007 For this homework, read Section 5.3 on Mapping Reducibility in the notes distributed in class. Problem 1 Recall that ET M = { M |M is a TM and L(M ) = }. Pr

Computer Science 340 Reasoning about Computation Homework 10 Due at the beginning of class on Wednesday, December 5, 2007
Problem 1 Give an implementation level description of a Turing machine that decides the following language over alphabet {0, 1}

Computer Science 340 Reasoning about Computation Homework 10 Due at the beginning of class on Wednesday, December 5, 2007
Problem 1 Give an implementation level description of a Turing machine that decides the following language over alphabet {0, 1}

Computer Science 340 Reasoning about Computation
Homework 9 Due at the beginning of class on Wednesday, November 28, 2007
Problem 1 Consider the following computational problem: Given N, a, b, x, y, where N 1 is prime, a, b, x, y are integers betw

Computer Science 340 Reasoning about Computation
Homework 9 Due at the beginning of class on Wednesday, November 28, 2007
Problem 1 Consider the following computational problem: Given N, a, b, x, y, where N 1 is prime, a, b, x, y are integers betw

Computer Science 340 Reasoning about Computation
Homework 8 Due at the beginning of class on Wednesday, November 21, 2007
Problem 1 The ciphertext below QFL HCVPS PX V ANSWLCEZK NCJVS; PQ XQVCQX QFL BPSZQL RNZ JLQ ZT PS QFL BNCSPSJ VSW WNLX SNQ XQN

Computer Science 340 Reasoning about Computation Homework 7 Due at the beginning of class on Wednesday, November 14, 2007
Problem 1 Let G be a graph and u and v be two vertices in G. If u and v are connected by a walk in G, then show that G contains

Computer Science 340 Reasoning about Computation Homework 5 Due at the beginning of class on Wednesday, October 24, 2007 Problem 1 Consider the three-element list with the following initial configuration: (x1 , x2 , x3 ), (ie, x1 is at the front). Wh

Computer Science 340 Reasoning about Computation Homework 5 Due at the beginning of class on Wednesday, October 24, 2007 This homework consists of 5 problems Problem 1 Consider the three-element list with the following initial configuration: (x1 , x2

Computer Science 340 Reasoning about Computation Homework 4 Due at the beginning of class on Wednesday, October 17, 2007
Problem 1 (10 points) We analyzed the process of throwing n balls into n bins independently and at random log n and showed that

Computer Science 340 Reasoning about Computation Homework 3 Due at the beginning of class on Wednesday, October 10, 2007
Problem 1 Suppose you toss three fair, mutually independent coins. Define the following events: A - the event that the first co