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### hw6

Course: CS 273, Spring 2008
School: University of Illinois,...
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Word Count: 273

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273: CS Intro to Theory of Computation, Spring 2008 Problem Set 6 (due Monday, February 25th, 4pm) This homework contains four problems, one of which is bonus. Please submit each on a separate sheet of paper. This will help us grade your homeworks more quickly. Turn in your homework at Elaine Wilsons oce (3229 Siebel). 1. Pumping lemma problems: (a) Use pumping lemma to prove that L is not regular, where: L = {ak...

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273: CS Intro to Theory of Computation, Spring 2008 Problem Set 6 (due Monday, February 25th, 4pm) This homework contains four problems, one of which is bonus. Please submit each on a separate sheet of paper. This will help us grade your homeworks more quickly. Turn in your homework at Elaine Wilsons oce (3229 Siebel). 1. Pumping lemma problems: (a) Use pumping lemma to prove that L is not regular, where: L = {ak bm : k m or m 2k} (b) Prove that the following language satises pumping lemma: L = {00, 11} From the fact that it satises the pumping lemma, can we deduce that L is regular? Why or why not? 2. Decide of the following languages are regular. If they are regular, give a DFA,NFA or regular expression the for language. If it is not regular, then give a proof using either closure properties or the pumping lemma. (a) L1 = {xy | x has the same number of as as y} (b) L2 = {w | w has three times the number of as than bs} (c) L3 = {xy | x has the same number of as as y and |x| = |y|} 3. Let T be the language {0n 1n : n 0}. Use closure properties to show that the following languages are not regular, using a proof by con...

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University of Illinois, Urbana Champaign - CS - 273
CS 273: Intro to Theory of Computation, Spring 2008 Problem Set 4 (due Monday, February 11th, 4pm)This homework contains four problems. As usual, please submit each problem on a separate sheet of paper. Turn in your homework at Elaine Wilsons oce (3
University of Illinois, Urbana Champaign - CS - 273
CS 273: Intro to Theory of Computation, Spring 2008 Problem Set 10 Due Tuesday, April 1st, 10amThis homework contains one problem and one bonus problem. Please submit each on a separate sheet of paper. Turn in your homework at Elaine Wilsons oce (32
University of Illinois, Urbana Champaign - CS - 273
CS 273: Intro to Theory of Computation, Spring 2008 Problem Set 7 Due Monday, March 3rd, 4pm.This homework contains four problems. Please submit each on a separate sheet of paper. This will help us grade your homeworks more quickly. Turn in your hom
University of Illinois, Urbana Champaign - CS - 273
CS 273: Intro to Theory of Computation, Spring 2008 Problem Set 9 Due Tuesday, March 25th, 10amThis homework contains ve problems. Please submit each on a separate sheet of paper. Turn in your homework at Elaine Wilsons oce (3229 Siebel). 1. Chomsky
University of Illinois, Urbana Champaign - CS - 473
CS 473Head-Banging Session 1Spring 20081. Problem 1: Decorating the DFS TreeDraw the DFS tree rooted at A for the above graph. Use alphabetic ordering to break ties. Label the vertices of the tree with their pre(v) : post(v) time. Add in the
University of Illinois, Urbana Champaign - CS - 473
CS 473ug: AlgorithmsMahesh Viswanathan vmahesh@cs.uiuc.edu 3232 Siebel CenterUniversity of Illinois, Urbana-ChampaignSpring 2008ViswanathanCS473ugWeighted Interval Scheduling Dynamic Programming: An Overview Segmented Least SquaresPart I
University of Illinois, Urbana Champaign - CS - 473
CS 473ug: AlgorithmsMahesh Viswanathan vmahesh@cs.uiuc.edu 3232 Siebel CenterUniversity of Illinois, Urbana-ChampaignSpring 2007ViswanathanCS473ugSta, Oce Hours, and Resources Grading Scheme Course Goals and Overview How to Perform BadlyP
University of Illinois, Urbana Champaign - CS - 473
CS 473ug: AlgorithmsMahesh Viswanathan vmahesh@cs.uiuc.edu 3232 Siebel CenterUniversity of Illinois, Urbana-ChampaignSpring 2008ViswanathanCS473ugComplementation Self ReductionPart I Complementation and Self-ReductionViswanathanCS473u
University of Illinois, Urbana Champaign - CS - 473
CS 473: AlgorithmsAssigned: Friday January 18, 2008 Homework Problems Before Starting! 1. Read the webpage http:/www.cs.uiuc.edu/class/sp08/cs473/hwfaq.html for homework instructions! 2. This Homework is to be done individually. 3. Each problem MUST
University of Illinois, Urbana Champaign - CS - 473
Head-Banging Session 6CS 473: AlgorithmsProblem 1. Overlap alignment for two sequences x = x1 . . . xn and y = y1 . . . ym is an alignment between a sux of x and a prex of y. This type of alignment is useful when you are attempting to assemble the
University of Illinois, Urbana Champaign - CS - 473
Head-Banging Session 2CS 473ug: AlgorithmsProblem 1.[Algebraic Manipulation] 1. Prove the following identity:nm=im i=n+1 i+12. Using the previous identity, evaluate 3. Provek c t=1 tk t=1t andk 2 t=1 t .= (k c+1 ), for any c N
University of Illinois, Urbana Champaign - CS - 473
CS 473: AlgorithmsAssigned: Friday February 22, 2008 Due on: February 29, 2008Homework 4Instructions: Solve the homework in groups of at most 3. Solution to each problem must start on a separate sheet. Please staple the pages for each problem bu
University of Illinois, Urbana Champaign - CS - 473
Head-Banging Session 3CS 473ug: AlgorithmsProblem 1.[Preliminaries] 1. Fast Exponentiation: (a) Given a constant c we are interested in computing n = c2 . Give an algorithm to compute n in time polynomial in k. (b) Given a constant c we are interes
University of Illinois, Urbana Champaign - CS - 473
CS 473: AlgorithmsAssigned: Friday February 8, 2008 Due on: February 15, 2008Homework 3Instructions: Solve the homework in groups of at most 3. Solution to each problem must start on a separate sheet. Please staple the pages for each problem but
University of Illinois, Urbana Champaign - CS - 473
Head-Banging Session 5CS 473: AlgorithmsProblem 1. [Maximum Weight Independent Set] Given an undirected graph G = (V, E), an independent set in G is a subset S V of nodes such that for any u, v S, the edge (u, v) E. Assume that each node v has
University of Illinois, Urbana Champaign - CS - 473
CS 473ug: AlgorithmsAssigned: Friday February 1, 2008 Due on: February 8, 2008Homework 2Instructions: Solve the homework in groups of at most 3. Solution to each problem must start on a separate sheet. Please staple the pages for each problem bu
University of Illinois, Urbana Champaign - CS - 473
CS 473: AlgorithmsAssigned: Friday February 29, 2008 Due on: March 7, 2008Homework 5Instructions: Solve the homework in groups of at most 3. Solution to each problem must start on a separate sheet. Please staple the pages for each problem but do
University of Illinois, Urbana Champaign - CS - 473
Sample Inductive ProofCS 473ug: AlgorithmsProblem 1. Consider a 2n 2n chess board with one (arbitrarily chosen) square removed. Prove that any such chessboard can be tiled without gaps or overlaps by L-shaped pieces consisting of 3 squares each. So
University of Illinois, Urbana Champaign - CS - 473
Practice HomeworkCS 473: AlgorithmsAssigned: Friday March 14, 2008 Due on: NeverInstructions: Please do not turn in solutions to this homework. Recommended Reading: Chapter 7 of the textbook and lecture notes 13 through 15 on the web. Homework Pr
University of Illinois, Urbana Champaign - CS - 473
How to Present a Paper in Theoretical Computer Science: A Speaker's Guide for StudentsIan Parberryy Department of Computer Sciences University of North Texas July 29, 1993There are many points in your career at which you will be called upon to pres
University of Illinois, Urbana Champaign - CS - 473
CS 473: AlgorithmsAssigned: Friday April 4, 2008 Due on: April 11, 2008Homework 8Instructions: Solve the homework in groups of at most 3. Solution to each problem must start on a separate sheet. Please staple the pages for each problem but do no
University of Illinois, Urbana Champaign - CS - 473
CS 473: AlgorithmsAssigned: Friday March 28, 2008 Due on: April 4, 2008Homework 7Instructions: Solve the homework in groups of at most 3. Solution to each problem must start on a separate sheet. Please staple the pages for each problem but do no
University of Illinois, Urbana Champaign - CS - 473
CS 473ug: AlgorithmsMahesh Viswanathan vmahesh@cs.uiuc.edu 3232 Siebel CenterUniversity of Illinois, Urbana-ChampaignSpring 2008ViswanathanCS473ugSet CoverPart I Set CoverViswanathanCS473ugSet CoverThe Problem Greedy Heuristic Ana
University of Illinois, Urbana Champaign - CS - 473
CS 473ug: AlgorithmsMahesh Viswanathan vmahesh@cs.uiuc.edu 3232 Siebel CenterUniversity of Illinois, Urbana-ChampaignSpring 2008ViswanathanCS473ugBipartite Matching Edge Disjoint PathsPart I Network Flow Applications IViswanathanCS473
University of Illinois, Urbana Champaign - CS - 473
CS 473ug: AlgorithmsMahesh Viswanathan vmahesh@cs.uiuc.edu 3232 Siebel CenterUniversity of Illinois, Urbana-ChampaignSpring 2008ViswanathanCS473ugPreliminaries NP Cook-Levin TheoremPart I NP CompletenessViswanathanCS473ugPreliminari
University of Illinois, Urbana Champaign - CS - 473
CS 473ug: AlgorithmsMahesh Viswanathan vmahesh@cs.uiuc.edu 3232 Siebel CenterUniversity of Illinois, Urbana-ChampaignSpring 2008ViswanathanCS473ugMaximum Flow Algorithms Correctness and Analysis Polynomial Time AlgorithmsPart I Network Fl
University of Illinois, Urbana Champaign - CS - 473
Basic Graph Theory Breadth First search Depth First Search Directed GraphsCS 473ug: AlgorithmsMahesh Viswanathan vmahesh@cs.uiuc.edu 3232 Siebel CenterUniversity of Illinois, Urbana-ChampaignSpring 2008ViswanathanCS473ugBasic Graph Theory
University of Illinois, Urbana Champaign - CS - 473
Head-Banging Session 8CS 473: AlgorithmsProblem 1. Tonian Paths: The problem of Finding a Tonian path in a graph is that of nding a simple path that goes through more than half the vertices of a graph. (Where a simple path is one with no repeated v
University of Illinois, Urbana Champaign - CS - 473
Head-Banging Session 9CS 473: AlgorithmsProblem 1. Consider a set A = {a1 , ., an } and a collection B1 , B2 , ., Bm of subsets of A (i.e. Bi A for each i). We say that a set H A is a hitting set for the collection B1 , B2 , .Bm if H contains at
University of Illinois, Urbana Champaign - CS - 473
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University of Illinois, Urbana Champaign - CS - 473
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University of Illinois, Urbana Champaign - CS - 473
CS 473ug: AlgorithmsMahesh Viswanathan vmahesh@cs.uiuc.edu 3232 Siebel CenterUniversity of Illinois, Urbana-ChampaignSpring 2008ViswanathanCS473ugThe Problem Algorithmic Solution Running Time AnalysisPart I Closest PairViswanathanCS47
University of Illinois, Urbana Champaign - CS - 473
Head-Banging Session 11CS 473: AlgorithmsProblem 1. 3-Coloring: Model the 3-Coloring problem with an Integer Linear Program. The 3-Coloring problem is: Given a graph, is it possible to color each node either red, green, or blue, so that no two node
University of Illinois, Urbana Champaign - CS - 473
Head-Banging Session 7CS 473: AlgorithmsProblem 1. Which of the following statements are true and which are false? Justify your answer. 1. If all directed edges in a network have distinct capacities, then there is a unique max ow. 2. If we replace
University of Illinois, Urbana Champaign - CS - 273
CS 273: Intro to Theory of Computation, Spring 2008 Problem Set 4 SolutionsThis homework contains four problems. As usual, please submit each problem on a separate sheet of paper. Turn in your homework at Elaine Wilson's office (3229 Siebel). 1. NFA
University of Illinois, Urbana Champaign - CS - 273
CS 273: Intro to Theory of Computation, Spring 2008 Problem Set 12 Solutions1. Decidable problems. Prove that L is a decidable language: L= D, k D accepts no string of length k, and D is a NFA .Solution:L is decidable because we can design a dec
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Homework 2 CS414, Multimedia Systems (Instructor: Klara Nahrstedt)Posted: April 18, 2008 Due: April 25, 2008 at 11:59pm CSTImportant InstructionsThis homework assignment should be done individually. Penalties for cheating as described in the grad
University of Illinois, Urbana Champaign - CS - 273
CS 273: Intro to Theory of Computation, Spring 2008 Problem Set 7 Due Monday, March 3rd, 4pm.Version: 1.01 This homework contains four problems. Please submit each on a separate sheet of paper. This will help us grade your homeworks more quickly. Tu
University of Illinois, Urbana Champaign - CS - 273
CS 273: Intro to Theory of Computation, Spring 2008 Problem Set 5 SolutionsSolution: 1: The original NFA. 2: Normalizing it.=3: Remove state A. 4: Redrawn without old edges.==5: Removing B.=16: Redrawn. =7: Removing C and redrawn.
University of Illinois, Urbana Champaign - CS - 273
CS 273: Intro to Theory of Computation, Spring 2008 Problem Set 1 (due Tuesday, January 22nd, 4pm)1. Set theory (10 points) Let A = {1, 2, 3} , B = {, {1}, {2} and C = {1, 2, {1, 2} . Compute A B, A B, B C, A C, A B, A C, C - A, C - B, A B C
University of Illinois, Urbana Champaign - CS - 273
CS 273: Intro to Theory of Computation, Spring 2008 Problem Set 8 Solutions1. Extract language from PDA. Give the language of the following PDA. b, , , b, , c, a ,\$ b, a ,\$ a, aa , \$b, Solution:L = an bm |m| 2|n| an bm ck |k| = 2n,
University of Illinois, Urbana Champaign - CS - 273
CS 273: Intro to Theory of Computation, Spring 2008 Problem Set 14 Solutions1. Dovetailing (a) Briey sketch an algorithm for enumerating all Turing machine encodings. Remember that each encoding is just a string, with some specic internal syntax (e.
University of Illinois, Urbana Champaign - CS - 273
CS 273: Intro to Theory of Computation, Spring 2008 Problem Set 5 (due Monday, February 18th, 4pm)This homework contains only one problem, intended to help you study for the rst midterm. As usual, please turn in your homework at Elaine Wilsons oce (
University of Illinois, Urbana Champaign - CS - 579
PCPLecture 26 And Hardness of ApproximationThursday, April 24, 20081Promise ProblemsDecision problems, but with dont cares Specied by a Yes set and a No set, disjoint A TM is said to decide a promise problem if it correctly answers Yes or No
University of Illinois, Urbana Champaign - CS - 273
CS 273 Lecture 26: Posts Correspondence Problem and Tilings24 April 2008This lecture covers Posts Correspondence Problem (section 5.2 in Sipser). Undecidability of this problem implies the undecidability of CFG ambiguity. We will also see how to si
University of Illinois, Urbana Champaign - CS - 598
CS 598CSC: Approximation Algorithms Instructor: Chandra ChekuriLecture date: January 23, 2009 Scribe: Sungjin ImIn the previous lecture, we had a quick overview of several basic aspects of approximation algorithms. We also addressed approximation
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CS 373: Theory of Computation Sariel Har-Peled and Madhusudan ParthasarathyProblem Set 0Due: Thursday Jan 29 at 12:30 in class (i.e., SC 1105) This homework contains four problems (and one extra credit problem). Please follow the homework forma
University of Illinois, Urbana Champaign - CS - 373
CS 273 Lecture 26: Posts Correspondence Problem and Tilings24 April 2008This lecture covers Posts Correspondence Problem (section 5.2 in Sipser). Undecidability of this problem implies the undecidability of CFG ambiguity. We will also see how to si
University of Illinois, Urbana Champaign - CS - 373
CS 273, Lecture 22 Reductions10 April 2008This lecture covers basic undecidability reductions. This is the rst half of Sipser section 5.1 (through most of p. 192).1What is a reduction?Last lecture we proved that ATM is undecidable. Now that w
University of Illinois, Urbana Champaign - CS - 373
CS 273 Lecture : Review of topics coveredThis review of the class notes was written by Madhusudan Parthasarathy.1IntroductionThe theory of computation is perhaps the fundamental theory of computer science. It sets out to dene, mathematically,
University of Illinois, Urbana Champaign - CS - 373
CS 373: Theory of Computation Sariel Har-Peled and Madhusudan ParthasarathyDiscussion 2: Examples of DFAs27 January 2009Purpose: This discussion demonstrates a few constructions of DFAs. How-ever, its main purpose is to show how to move fr
University of Illinois, Urbana Champaign - CS - 373
CS 273, Lecture 24 Linear Bounded Automata17 April 2008This lecture covers Linear Bounded Automata, an interesting compromise in power between Turing machines and the simpler automata (DFAs, NFAs, PDAs). We will use LBAs to show two CFG grammar pro
University of Illinois, Urbana Champaign - CS - 579
Complexity Homework 1Released: January 27, 2008 Due: February 10, 2008 For problems that involve nondeterministic complexity classes, the solutions maybe simpler when phrased in terms of certicates (instead of non-determinism). Problem 1: (a) Let L1
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Homework Problem Set 1CS 477 Spring 2009Assigned: February 3, 2009 Due: February 10, 2009Instructions: You are welcome to collaborate while working out the ideas for a solution. However, you must write down the solutions independently, and must
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Some Sample Triangle MeshesWhat Polygon Meshes AreExplicit mesh description a list of polygonal faces F = ( f1, f2, , fn ) each polygon fi is a list of points this is sometimes called a polygon soup model vertices will be duplicated several ti
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