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Course: CS 6520, Fall 2008
School: Georgia Tech
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Word Count: 629

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6520: CS Computational Complexity Problem Set 1 Due March 4, 2008 Problem 1 Give a Karp reduction from CLIQUE to SAT. Problem 2 Let Quadratic be the problem of deciding whether a given system of quadratic multivariate polynomial equations with integer coecients has a solution modulo 2. Prove that Quadratic is NP-Complete. Problem 3 An NP minimization problem is dened by an objective function Obj : N for...

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6520: CS Computational Complexity Problem Set 1 Due March 4, 2008 Problem 1 Give a Karp reduction from CLIQUE to SAT. Problem 2 Let Quadratic be the problem of deciding whether a given system of quadratic multivariate polynomial equations with integer coecients has a solution modulo 2. Prove that Quadratic is NP-Complete. Problem 3 An NP minimization problem is dened by an objective function Obj : N for which there is a constant c and an algorithm that for every x, y {0, 1} , computes Obj(x, y) in time O(|x|c ). For such an objective function Obj, the corresponding NP minimization problem is: Given x , nd a y such that Obj(x, y) is minimized. Prove that P = NP if and only if every NP minimization problem has a polynomial-time algorithm.1 Problem 4 Prove that for each i N, i -SAT is P -Complete. i Problem 5 Consider the Factoring problem: Given a natural number N , express N as a product of its prime factors. To date no polynomial-time algorithm for Factoring is known, and the conjectured hardness of Factoring has been the basis of several public-key cryptosystems. 1. Prove that if P = NP coNP, then Factoring can be solved by a polynomial-time algorithm. You may nd the following facts useful: An NP maximization problem can be dened similarly, for which an analogous result holds. 1 1 The Fundamental Theorem of Arithmetics: Every integer greater than one can be decomposed into a product of prime factors, and moreover such a decomposition is unique up to the order of the prime factors. There is a polynomial-time algorithm for deciding whether a given integer is a prime. 2. Prove that unless NP = coNP, Factoring is not NP-Hard under Cook reduction. Conclude that unless NP = PH, Factoring is not NP-Hard under Cook reduction. Problem 6 A language L is self-reducible if there is a oracle polynomial-time machine M such that (a) M L decides L, and (b) on every input x, M only makes queries of length strictly smaller than |x|. For instance, as shown in class, SAT and TQBF are self-reducible. Prove that every self-reducible language is in PSPACE. Problem 7 1. A directed graph G = (V, E) is strongly connected if for every pair of vertices u, v V , there is a path from u to v in G. Prove that the problem of deciding whether a given directed graph is strongly connected is NL-Complete. 2. Prove that 2-SAT is NL-Complete. 3. Let Bipartite be the language consisting of bipartite graphs. Prove that Bipartite NL. 4. Let USTCONN be the following problem: Given an undirected graph G = (V, E) and two vertices s, t V , decide whether there is a path from s to t in G. Prove that USTCONN and Bipartite are computationally equivalent under log-space reductions. 2 Problem 8 1. Let s(n) be a space-constructible function. Prove that a language L NSPACE(s(n)) if and only if there is a deterministic Turing machine M with a read-once2 certicate tape such that for every x ...

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Georgia Tech - CS - 2008
Supporting Time-Sensitive Applications on a Commodity OSAshvin Goel, Luca Abeni, Charles Krasic, Jim Snow, Jonathan Walpole Department of Computer Science and Engineering Oregon Graduate Institute, Portland{ashvin,luca,jsnow,krasic,walpole}@cse.ogi
Georgia Tech - CS - 4220
Supporting Time-Sensitive Applications on a Commodity OSAshvin Goel, Luca Abeni, Charles Krasic, Jim Snow, Jonathan Walpole Department of Computer Science and Engineering Oregon Graduate Institute, Portland{ashvin,luca,jsnow,krasic,walpole}@cse.ogi
Georgia Tech - CS - 8803
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Georgia Tech - CS - 4237
Security Handshake Pitfalls (II)CS 4237Establishing Session KeysAuthentication handshakes to securely establish session keys Usingshared secret Using public keys One-way public key (only Alice needs to have keys) Lamports hash1Session
Georgia Tech - CS - 4237
IP SecurityCS 4237OutlineObjectives IPSec architecture &amp; concepts IPSec authentication header IPSec encapsulating security payload1IPSEC ObjectivesBand-aid for IPv4 Spoofinga problem Not designed with security or authentication in
Georgia Tech - CS - 2008
CS 3240: Languages and ComputationProblem Set 3 SolutionsProblem 1Show that the following languages are not regular. L1 = {www : w {a, b} }. Solution: We prove that L1 is nonregular by the pumping lemma. Let p be any given positive integer, and
Georgia Tech - CS - 3240
CS 3240: Languages and ComputationProblem Set 3 SolutionsProblem 1Show that the following languages are not regular. L1 = {www : w {a, b} }. Solution: We prove that L1 is nonregular by the pumping lemma. Let p be any given positive integer, and
Georgia Tech - CS - 2008
A New Design of Bloom Filter for Packet Inspection SpeedupYang Chen, Abhishek Kumar, and Jun (Jim) Xu College of Computing, Georgia Institute of Technology {yangchen, akumar, jx}@cc.gatech.eduAbstract- Bloom filter is a space-efficient randomized d
Georgia Tech - CS - 6250
A New Design of Bloom Filter for Packet Inspection SpeedupYang Chen, Abhishek Kumar, and Jun (Jim) Xu College of Computing, Georgia Institute of Technology {yangchen, akumar, jx}@cc.gatech.eduAbstract- Bloom filter is a space-efficient randomized d
Georgia Tech - CS - 2008
Georgia Tech - CS - 6250
Georgia Tech - CS - 8803
u x xh p e q q # ! q x q x u q p p jh q g u e p g x u e # Igewgwe ris&quot;\$e~ig\$ trsiw`&quot;wg g! x e ! # ! x h h # q g e ! q x q g x s # e h q u q e p g x u e # ! u e e j ! g q e u x q u ! u q u e 4t\$ww\$!tiwti\$iwziyge7wti&quot;wgpgri
Georgia Tech - CS - 8803
jj #pj #u!p d y # gjpu p g d! y g euj! swipw&quot;g m hCw\$jwi&quot;hwlCwfl\$&quot;hws m hCf}nwwlCwwyj m w(iewipwCe m iq m tyf&quot;CwXvsh&quot;f! # u ! p d jy u e ! g r e u e p g ! g e # g y# j i # e u u g m y e i e ! r ! d e j t j u e y u u g e i e u j
Georgia Tech - CS - 2008
Name:CS 3510: Design and Analysis of Algorithms Summer 2008 Quiz IIInstructions: Please write your name in the space provided at the upper left corner of this page. This quiz has 5 questions on the following 5 pages. For each question, write your
Georgia Tech - CS - 3510
Name:CS 3510: Design and Analysis of Algorithms Summer 2008 Quiz IIInstructions: Please write your name in the space provided at the upper left corner of this page. This quiz has 5 questions on the following 5 pages. For each question, write your
Georgia Tech - CS - 7210
BZFlags Features, Design and Issues Topics covered in this presentationInstallation Game Design Architecture Some relevant files InstallationRequirements OpenGL 1.0+ libSDL 1.2+ libCURLBuilding from the source http:/my.bzflag.org/w
Georgia Tech - CS - 2008
Genetic Algorithm in Kernighan and Richie C/* * * * Genetics algorithm example taken from &quot;The Amateur Scientist&quot; column of * * Scientific American, July 1992, pages 114-116 by Rick L. Riolo. * * * * This version of genes.c is written in the K&amp;R ver
Georgia Tech - CS - 6390
Genetic Algorithm in Kernighan and Richie C/* * * * Genetics algorithm example taken from &quot;The Amateur Scientist&quot; column of * * Scientific American, July 1992, pages 114-116 by Rick L. Riolo. * * * * This version of genes.c is written in the K&amp;R ver
Georgia Tech - CS - 2008
SOFTWARE REUSE Using existing software artifacts during the construction of a new software system - Krueger Code (libraries, idioms); Designs (patterns, architectural styles) Problems Abstracting, parameterization, compiling from specification t
Georgia Tech - CS - 6330
SOFTWARE REUSE Using existing software artifacts during the construction of a new software system - Krueger Code (libraries, idioms); Designs (patterns, architectural styles) Problems Abstracting, parameterization, compiling from specification t
Georgia Tech - CS - 2008
SORT EXERCISE Sorting is a well-understood concept that is surprisingly hard to specify precisely For this exercise, try to formally specify a SORT routine that takes a vector of integer values as input and returns a sorted vector of integer values
Georgia Tech - CS - 6330
SORT EXERCISE Sorting is a well-understood concept that is surprisingly hard to specify precisely For this exercise, try to formally specify a SORT routine that takes a vector of integer values as input and returns a sorted vector of integer values
Georgia Tech - CS - 2008
GOAL The overarching goal is to satisfy your customer This is one definition of quality This means having confidence that what you build is what the customer wanted There are various approaches for raising confidence; none that provide any guara
Georgia Tech - CS - 6330
GOAL The overarching goal is to satisfy your customer This is one definition of quality This means having confidence that what you build is what the customer wanted There are various approaches for raising confidence; none that provide any guara
Georgia Tech - CS - 2008
RefactoringMartin Fowler. Refactoring / Improving the Design of Existing Code. Addison Wesley, 1999.RefactoringRefactoring is the process of changing a software system in such a way that it does not alter the external behavior of the code yet imp
Georgia Tech - CS - 6330
RefactoringMartin Fowler. Refactoring / Improving the Design of Existing Code. Addison Wesley, 1999.RefactoringRefactoring is the process of changing a software system in such a way that it does not alter the external behavior of the code yet imp
Georgia Tech - CS - 2008
MUTATION TESTING White-box, error-based testing technique Built-in adequacy criteria Tool support The goal of mutation testing is to build an adequate sets of tests Finding faults is a side-effectERROR-BASED TESTING Devise tests that mimic th
Georgia Tech - CS - 6330
MUTATION TESTING White-box, error-based testing technique Built-in adequacy criteria Tool support The goal of mutation testing is to build an adequate sets of tests Finding faults is a side-effectERROR-BASED TESTING Devise tests that mimic th
Georgia Tech - CS - 6330
RefactoringMartin Fowler. Refactoring / Improving the Design of Existing Code. Addison Wesley, 1999.RefactoringRefactoring is the process of changing a software system in such a way that it does not alter the external behavior of the code yet imp
Georgia Tech - CS - 6330
What are Application Generators? They generate programs from high-level specifications Can also generate documentation and test data They have proven successful when they are confined to a limited domain (application area) This area used to be c
Georgia Tech - CS - 2008
Domain Engineering Domain Engineering is the activity of collecting, organizing, and storing past experience in building systems or parts of systems in a particular domain in the form of reusable assets (i.e., reusable work products), as well as pro