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3/24/08 NP-complete sequencing pro...
NP: problems whose solution can be eficiently (poly-time) veried. Ex: Compositeness: Given n-bit number x, in x composite? Hint: a pair y,z > 1 such that yz = x. NP-Complete: a problem X which is in NP and every other problem Y in NP has a poly-time reduction FROM Y TO X. Ex. Ind. Set, 3 SAT. Alt def: a problem X which is in NP and there exists NPC problem Y which has a poly-time reduction from Y to X. 2 easy reductions from ind. set. (1) CLIQUE: Given undirected graph G = (V, E) and a number k > 0. Output yes if G has a subgraph H with k vertices and every 2 vertices of H are joined by an edge. (H is a k-clique.) Take (G, k) an instance of IND. SET. Construct G which has same vertex set and (u,v) E(G ) <=> (u,v) E(G). (1) Reduction runs in poly-time. YES! O(n2) (2) If G has k-ind set, G has a k-clique. (3) If G has no k-ind set, G has no k-clique. (contrapos.) (2) VERTEX COVER: A vertex cover of an undirected graph G=(V,E) is a set of S vertices such that every edge has 1 endpt in S. Given (G,k) does G have a vertex cover of k vertices? rewrite: A vertex cover is a set S such that every edge has 1 endpt in S. S is a vertex cover <=> S is an indep. set. (G, k) is a yes inst. of VTX COVER <=> (G, n-k) is yes inst. of IND SET. where n = # of vertices of G. Hamiltonian Cycle: Given directed G=(V,E) Does there exist a sequence v1, v2, ..., vn
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containing every vertex once and only once such that (vi, vi+1)E i and (vn, v1)E. YES
NO
Reduce from 3SAT. variable gadget? Any object that has exactly 2 settings In the absence of clause gadgets the settings of dierent vars. dont conict. clause gadget? Bidirectional path:
(These correspond to vars.) LR corresp. to true RL corresp. to false Given 3SAT instance with x1, ..., xn and clauses c1, ..., cm Suppose c1 = (x1 x2 xn) Either P1 or P2 or Pn .
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Claim; This graph has exactly 2n Ham. cycles, corresponding to choosing or on each path.
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Cornell - CS - 482

283/26/08 - NP-complete partitioning pro.Reduction from 3SAT to HAM. CYCLE Theorem. HAM CYCLE is NP-Complete Proof. (1) Its in NP. Show me the cycle, and I can verify it in O(n) time. (2) The reduction runs in poly(n) time. Let b = length of each

Cornell - CS - 482

293/28/08 - NP-complete coloring proble.NP-complete problems 3SAT, k-SAT (k > 3) IND. SET CLIQUE VERTEX COVER HAMILTONIAN PATH/CYCLE TRAV. SALES. PROB. Reduce FROM . TO (e.g. set cover) if you could use set cover to solve trav salesman in poly tim

Cornell - CS - 482

112/8/08 - Divide and conquer algorithms.Lecture: Divide and conquer algorithms in computational geometry: nding the closest pair of points Reading: Chapter 5.4 cs482-02-08-08-Audio.mp4 Closest pair of points Given n numbers x1, x2, . , xn Find i,

Cornell - CS - 478

CS 478 Machine Learning: Homework 2Suggested Solutions1Separable or Not?(a) See the following tree:(b) If we draw a large enough sample, there would be at least two points on each of the four positions. Since there is no noise in the label,

Cornell - CS - 478

CS 478 Machine Learning: Homework 1Suggested Solutions1kNN Decision Boundaries(a)(b) (c) It would be classied as circle. Adding one point is enough, say a cross at exactly (1,-1). (d) It would be classied as circle. The ve closest points ar

Cornell - CS - 482

132/13/08 - Computing RNA secondary str.RNA: a single-stranded molecule made up of {A, C, G, U} Secondary structure: certain base pairs on same molecule match up Constraints: (i) A pairs only with U C pairs only with G U pairs only with A G pairs

Cornell - CS - 482

Introduction to Algorithms CS 482, Spring 2008Solution Set 9(1) Let T be the number of trucks used by the algorithm, and suppose the trucks are labeled 1, 2, . . . , T in the order that the algorithm loads them up. Observe that for i = 1, 2, . .

Cornell - CS - 482

303/31/08 - NP-complete numerical probl.NP-Complete Problems 3SAT Vtx Cover Indep. Set (even in deg 3) Clique Hamiltonian Path Set cover/packing Traveling salesman 3D matching Subset Sum. 3-D Matching Given sets X, Y, Z, with n elts each. Given 3

Cornell - LING - 404

41/23/08 - Case Theory; distribution of PRO4 levels of structure D structure move S-structure move LF PFFor next week: read rst 2 chapters Londau D-Structure and S-structure connected by move-alpha Passive: _ was arrested John Johni was arrest

Cornell - LING - 404

51/28/08 - Binding TheoryThe binding theory divides into three types of things anaphors, pronouns, R-expressions The binding theory has three conditions. A. An anaphor must be bound in its governing category. C-Command condition: *Himselfi pinche

Cornell - LING - 404

61/30/08 - PROReformulated the Case Filter as Visibility Condition New problem: It is still the case that it is not possible for PRO to occur in a nonCase position. Just restating the Case Filter doesnt entirely solve the problems, because we stil

Cornell - LING - 404

82/11/08 - proling404-02-11-08-Audio.mp4 PRO [-a, +p] John spoke. He spoke. *spoke. Italian: (Gianni) ha parlato. Empty pronominal element. Johni said [that hei spoke.] Giannii ha detto che ei parlato pro <Gianni,pro> cannot form a chain. wh-trace

Cornell - LING - 404

92/13/08 - more proWhat can we do about languages where there is no morphology to tell us about pro? e.g. Chinese ability to drop subject in presence of rich morphology. BUT in Chinese etc. no morphology at all - what licenses dropping subject? Yo

Cornell - LING - 404

102/18/08 - A-MovementWe have been mainly talking so far about A-movement movement of subjects and objects into the subject position. A movement exemplied by wh-movement I wonder [who John talked to _ ] embedded clause - the who winds up to the le

Cornell - LING - 404

112/20/08 - SubjacencyHomework back on Monday - also new hw due Ross Island Constraint Want to derive binding constraints from a more general principle Chomsky: Subjacency attempts to unify a subset of the constraints allows us to dispense with th

Cornell - LING - 404

123/3/08 - Relativized MinimalityTrace eect you cant extract out of a subject position when there is an overt complementizer Chomsky introduces new barrier Minimality barrier X can sometimes be barriersWho did you say [t [C that t read the book

Cornell - CS - 482

Solution Set for CS 482, Prelim 2 April 8, 2008 Questions in red, solutions in black. PROBLEM 1 (20 points)PART A (15 points) Find a maximum flow and minimum s-t cut in the flow network G shown here. The source and sink are s and t, respectively. Th

Cornell - CS - 482

Introduction to Algorithms CS 482, Spring 2008Solution Set 2(1) The algorithm restores the websites in decreasing order of ci /ti , where ci is the rate of lost dollars per hour for site i, and ti is the number of hours to nish the job. Analysis

Cornell - CS - 482

Introduction to Algorithms CS 482, Spring 2008Solution Set 3(1) We describe an algorithm Test(S) whose input is a set S of bank cards and whose output is: a bank card x S such that more than half the elements of S are equivalent to x, if any su

Cornell - CS - 482

Introduction to Algorithms CS 482, Spring 2008Solution Set 3(1) (a) Heres a counterexample in which n = 3. Week 1 Week 2 Week 3 1 1 1 0 3 10ihiThe optimal solution picks the low-stress job in week 1 and the high-stress job in week 3. The gr

Cornell - CS - 482

384/21/08 - Randomized algorithmsRandomized Algorithms Review Ch. 13.12, read 13.1 - 6. A probability space consists of a sample set (nite in CS 482) and a probability 0 Pr(x) 1 for every x , s.t. x(x) = 1. Examples. Flipping coin n times = {

Cornell - CS - 482

CS 482 FINAL EXAM SOLUTION SET(1) (10 points) Each of the following statements is false. Give a counterexample to each of them. (1a) (5 points) If G is any graph with non-negative edge costs, and e is any edge such that every minimum-cost spanning

Cornell - CS - 482

Introduction to Algorithms CS 482, Spring 2008Solution Set 7(1) First, we prove that Party Invitation is in NP. There is a polynomial-time verier that takes an instance I of Party Invitation consisting of numbers n, k, lists Pi (1 i k), and va

Cornell - CS - 482

Introduction to Algorithms CS 482, Spring 2008Solution Set 5(1) (a) If e is contained in a min-cut (A, B), let C be the capacity of this cut. Every max-ow f has value v(f ) = C, and such a ow must saturate every edge of the min-cut, including edg

Cornell - CS - 482

Introduction to Algorithms CS 482, Spring 2008Solution Set 6(1) (a) To compute whether the blood on hand meets the projected demand, one can construct a ow network with 10 vertices. A super-source s. For each blood type x, a pair of vertices ux

Cornell - CS - 482

Introduction to Algorithms CS 482, Spring 2008Solution Set 8(1) The algorithm resembles the Greedy-Balance algorithm given in Section 11.1. For each machine Mi , we maintain a variable Ti which keeps tracks of the total amount of time required fo

Cornell - CS - 4410

CS4410 - Fall 2008 Assignment 1 SolutionQ1. Give three examples of an explicit hardware mechanism that is motivated by specific OS services. Answer: Atomic operations for synchronization. Kernel/user mode, base/limit registers, protected instruction

Cornell - CS - 4410

CS4410 - Fall 2008 Homework 2 Solution Due September 23, 11:59PMQ1. Explain what goes wrong in the following version of Dekkers Algorithm:CSEnter(int i) { inside[i] = true; while(inside[j]) { inside[i] = false; while(turn = j) continue; inside[i] =

Cornell - CS - 482

172/22/08 - Introduction to network owNetwork Flow Def: A ow network is a directed graph G = (V, E) with edge capacities ce (e E), ce 0 (usually integers) vertices s (source), t (sink)Notations: A vertex v s, t is an internal node. In(v) = {e

Cornell - CS - 482

162/20/08 - Review of dynamic programm.Chapter 6, Problems 4, 5, 11 Prelim tomorrow 7:30-9, Upson B17 Chapter 1, 4, 5, 6 Design techniques Greedy, Divide and Conquer, Dynamic Programming Key algorithms Gale-Shapley, Kruskal, Prim, Bellman-Ford Key

Cornell - CS - 482

602/22 - Chapter 7.1 - The Maximum-Flo.Graphs to model transportation networks - networks whose edges carry some sort of trafic and whose nodes act as switches passing trafic between dierent edges a highway system - edges are highways, nodes are i

Cornell - CS - 482

152/18/08 - Bellman-Ford shortest path al.Shortest Paths with Negative Edge Lengths (Bellman-Ford) Example: Dijkstra can go wrong.Can you add a constant to every edge length and then run Dijkstra?15-1You cant have negative cycles ExamplesC

Cornell - CS - 482

142/15/08 - Edit distanceHomework 2 in Upson 360 by 2pm today Prelim next week, Thursday evening - 2 algorithm questions, questions about lectures Dynamic programming review questions Techniques: (1) Greedy (2) Div. + Conq (3) Dyn. Prog. Ch. 4-6 I

Cornell - CS - 482

31/21/08 - Stable Matching, part 1Lecture: Stable matching, part 1. Reading: Chapter 1.1; Chapters 2,3 (1) Formulating algorithmic problems as mathematically wellpoised questions (2) Distinguishing easy from hard (3) General techniques for designi

Cornell - CS - 482

Introduction to Algorithms CS 482, Spring 2008Solution Set 9(1) There are many correct solutions; here is one. The graph has two vertices s, t and n triples of vertices (ui , vi , wi )n , for a total of 3n + 2 vertices. There are 5n edges, specie

Cornell - CS - 482

122/11/08 - Introduction to dynamic progr.Lecture: Introduction to dynamic programming; weighted interval scheduling Reading: Chapter 6.1 Chapter 6: Dynamic Programming First example: Weighted Interval Scheduling Goal: nd a set of non-overlapping

Cornell - CS - 482

92/4/08 - Divide and conquer algorithms:.Lecture: Divide and conquer algorithms: Multiplying polynomials and integers (Karatsuba's algorithm) Reading: Chapter 5.1, 5.2, 5.5 Divide & Conquer Mergesort Binary search Quicksort Algebra + Geometry Poly

Cornell - CS - 482

82/1/08 - Implementing Kruskals algorit.Lecture: Implementing Kruskal's algorithm using union-nd. Reading: Chapter 4.6 Kruskals Algorithm Sort edges by increasing cost O(mlogn) T< For each edge (u, v) in this order if u, v are not in same componen

Cornell - CS - 482

71/30/08 - Minimum spanning tree; Krus.Lecture: Minimum spanning tree; Kruskal's algorithm. Reading: Chapter 4.5 cs482-01-30-08-Audio.mp4 Minimum spanning tree Given G = (V,E) undirected, connected Costs ce > 0 for every e E. Goal: output connect

Cornell - CS - 482

61/28/08 - Five Representative ProblemsLecture: Five representative problems Reading: Chapter 1.2 Earliest Start Time with Pre-emption (1) Preprocess the input to remove jobs whose interval contains another jobs interval (2) Sort the remaining job

Cornell - CS - 482

51/25/08 - Greedy Scheduling AlgorithmsLecture: Greedy scheduling algorithms Reading: Chapter 4.1, 4.2 Interval Scheduling (jobs) Input: A set of pairs (si, ti)Ti=1 0 si ti T, si, ti N Output: A feasible schedule: a subset of the pairs, s.t. t

USC - REL - 121G

*Women at age 7, remain in women quarter. Food preparation, salad bread, but not meat. Learn skills of spinning and weaving(clothes). *Weaving: peplos(dress), piece of cloth, 5 to 6 feet. 2 steps: 1)Talasiourgein: working with allotment.cleaning, com

USC - MKT - buad307

Budget:I think the ideal allocation strategy for advertising, promotion, and marketing research is 30% from gross profit. Huggies is under Kimberly-Clark corp. (note: gaz, I couldnt find the specific financial statement for huggies, so the statement

USC - MKT - buad307

Focus group

USC - MKT - buad307

1.What is Integrated Marketing Communications? Give examples to support your answer. A concept of marketing communications planning that recognizes the added value of a comprehensive plan that evaluates the strategic roles of a variety of communicati

USC - REL - 121g

1)Literary ethnography: Historical writings. Provide descriptive study of human societies. 2)Independent peasant: 3)Chora: 4)Kyrios: 5)Gorgias: 6)Mageiros: 7)Dorkon: 8)Lycainion: 9)Syntrophos: 10)Aristocratic household: 11)Eros: 12)Kottalos: 13)Prima

Texas College - HIST - 106

Theodore Roosevelt (Teddy) - TR became president in 1901. - He led an endlessly fascinating life, advocated the strenuous life, and largely created the modern presidency: he had many carriers, politician, cattle rancher, mountain climber, soldier, go

Texas College - HIST - 106

National Origins Act: It was an anti immigration legislation. Hostility against immigration had been growing since the late 19th century. Many people felt that US was bring swapped by undesirable immigrants. The opstition to immigration had been gro

Texas College - HIST - 106

- The Open Door Notes: The US interest grew in the eastern Asia and China. Chinas government was very weak, and it was unable to control its borders. As a result, empyreal powers moved in like Germany, France, Russia, Japan, England. The empyreal pow

Texas College - HIST - 106

Woodrow Wilson: He believed in a strong activist government. He also believed that the government should do things and the president should do things. He believed that he should be Americas president and to be Americas destiny. He didnt like to compr

Texas College - HIST - 106

Threats to the system: Rise of Germany Nationalism in the Balkans Security By 1914, the continent had bound itself into two competing blocks: a. Entente They were France, Russia and Britain. There were smaller forces involved. b. Central Powers They

Texas College - HIST - 106

Origins: Muckrakers: Who and what they did: They helped to make progressivism a national movement. Journalists and novelists who investigated and revealed the seamy details of corporate wrong doing, corruption in politics and the harsh realities of s

Texas College - HIST - 106

The treaty of Versailles: As part of the treaty several new nations were created in Europe and the middle east. Ugoslavia, Chekoslovakia, Poland, finland. The League of Nations: Wilson essentially scarified the rest of the 14 points for this point. T

Texas College - HIST - 106

Major industries that had fuels the American economy were weakening. Thus didnt have the power to stimulate the economy as before. New industries such as chemicals, appliances, food processing werent yet strong enough to prevent depression or to ease

Texas College - HIST - 106

Loans and Peace: American opinion: Many Americans of German and Irish ancestors favored Germany in the war. The most officials in the government favored France due to common language, economic ties and culture. They viewed Britain and France as those

Texas College - HIST - 106

Results of Prohibition: Positive: - Alcohol consumption declined as did - arrest for public drunkness and - death caused by alcohol. Negative: -millions of Americans broke the law - Strenghthened organized crime - Undermined public morality by encour

Texas College - HIST - 106

Progressivism: By the 1890, people are fed up and they realize that something should be done for all the problems in America. Progressivism was the solution. However there was not a unified movement which agreed upon a set of goals. It was diverse mo

Virginia Tech - ENGE - 1024

Math Review Worksheet ENGE 1024 Fall 2006Instructions This is to be completed in 30 minutes using a scientific calculator and no reference material. Conversions 12 inches = 1 foot 3 feet = 1 yard 1 meter = 3.281 feet 1 mile = 5,280 feet Questions 1

Virginia Tech - STAT - 2004

Virginia Tech - STAT - 2004

Stat 2004 Practice Exam 1. In 1798 the English scientist Henry Cavendish measured the density of the earth with great care. It is common practice to repeat careful measurements several times and use the mean as the final result. Cavendish repeated hi

Virginia Tech - CHEM - 1035

Chemistry 1035 Fall-06 Test 2 Name_ Form A1. Which of the following salts is Insoluble in water? a) Sodium sulfate b) Potassium acetate c) Lithium carbonate d) All of these are soluble in water 2. Given the following pairs of solutions, which one w