CS 683: Advanced Design and Analysis of Algorithms
|
|
- Clementine Walton
- 5 years ago
- Views:
Transcription
1 CS 683: Advaced Desig ad Aalysis of Algorithms Lecture 6, February 1, 2008 Lecturer: Joh Hopcroft Scribes: Shaomei Wu, Etha Feldma February 7, Threshold for k CNF Satisfiability I the previous lecture, we discussed the threshold for 3 CNF problem ad claimed that for ay k CNF, there is a threshold r k = 2 k l 2 for formula satisfiability. Today s lecture gives a more formal proof for this statemet. Theorem 1 Give a k CNF formula with c clauses(c is a fixed umber) ad variables, assume c = r k, the r k = 2 k l 2 is a threshold for the give formula to be satisfiable. Proof: (1) Pick a radom assigmet S for variables. (2) For ay clause (with k literals), the probability that it is false uder assigmet S is 1 2 k, hece P rob(clause is true uder S) = k. (3) Give c clauses, the formula is true with probability P rob(formula is true) = (1 1 2 k )c (*The clauses are cosidered to be geerated idepedetly.) Whe r k = 2 k l 2 ad c = r k, P rob(formula is true) = (1 1 2 k )c = (1 1 2 k )2k l 2. The secod part of this lecture o graph coectivity was re-stated ad corrected i February 3 s lecture, hece we reduced the amout of cotet i that part. Please refer to later otes for more clear proof ad explaatios 1
2 For large k, (1 1 2 k ) 2k e 1, hece P rob(formula is true) = (1 1 2 k )2k l 2 e (l 2) = 1 2. I other words, the probability that the give formula is satisfied by a radom assigmet of variables is 1 2. Let s use a set of idicator radom variables I 1, I 2,..., I 2 to represet the evets of 1st, 2d,..., ad 2 t h assigmets satisfyig the give formula, respectively. Kowig that each assigmet has the same probability of makig the formula true, the expected values for all I i s are same as E(I 1 ) = E(I 2 ) =... = E(I 2 ) = = 1 2. Therefore, the expected umber of assigmets that satisfy the give formula would be E(I) = E(I 1 ) + E(I 2 ) E(I 2 ) = = 1 Figure 1: Probability of k-cnf beig satisfied by a radom assigmet. As proved i previous lectures, the umber of satisfyig assigemets for the give formula is a mootoe property. Icreasig c will itroduce a decrease of the probability that the formula is true uder a radom assigmet, which meas, whe c is greater tha r k, the expected umber of satisfyig assigmets will be less tha 1. As show i Figure 1, r k = 2 k l 2 is the place where the upperboud of threshold for k-cnf appears. 2
3 For k = 3, Chao ad Fraco [1] preseted a heuristic algorithm based o Uit-Clause rule ad proved that: whe r k 2k k, the probability of the heuristic fidig a solutio is approachig 1; while whe r k 2 k l 2, the probability is approachig 0. For example, assumig there are 10 variables appearig i a 3-CNF formula: if the umber of clauses is below 26, the heuristic algorithm ca almost always fid the satisfyig assigmet; if the umber of clauses is above 55, it would be early impossible for the heuristic algorithm to geerate a solutio. However, the situatio betwee r k = 2k k ad r k = 2 k l 2 is still uclear to us, ad it is possible that there is a secod-momet threshold existig i this rage. Oe guess ca be made by observig the chage of solutio space accordig to r k : startig from r k = 0, the solutio space is a fully-coected graph; alog with the icrease of r k, the solutio space will fall apart ad form a set of coected compoets; after passig the threshold of r k = 2 k l 2, the solutio space becomes very sparse, cosists of isolated vertices. Ivestigatig the satisfiability of k-cnf from the solutio space view brigs us back to the coectivity problem of graphs, which will be discussed i the followig sectio. 2 Whe does a graph become coected? Give a radom G(, p), whe does it become coected? states that G(, p) ca be: 1. G(, p) is coected; 2. G(, p) cosists of isolated vertices; There are three 3. G(, p) has o isolated vertex, but is ot coected either (It is ulikely for this case to happe). 2.1 Disappearace of isolated vertices Specific vertex is isolated with probability (1 p) 1. Threshold at p = log +c Note: P rob(o isolated vertices) was wrog for this lecture because the P rob(isolated vertex) is ot idepedet across all vertices. See lecture otes for 01/25/2008. If isolated vertices are the last to disappear before the graph becomes coected, the the probability that the graph is coected is e e c. 2.2 Disappearace of compoets of size 2 Give 2 vertices u, v coected by a edge. Threshold at p = log +c. 3
4 2 {}}{ u v P rob(compoet of size 2) = (1 p) 2 4 ( 1 log + c 2(log +c) = e = e 2log e 2c = e 2c 2 ) 2 } {{ } 2 Number of coected vertices log. Note: P rob(o compoets of size 2) was wrog for this lecture because the P rob(compoet of size 2) is ot idepedet across coected odes. See lecture otes for 02/04/ Graphs with Giat Compoets p = d, d > 1 Algorithm for traversig coected compoets: 1. Select a vertex v 2. Mark all vertices as udiscovered ad uexplored 3. Mark v as discovered 4. While discovered but uexplored vertex, select oe ad 5. add all adjacet vertices to discovered ad mark the selected vertex explored Let d = 2 for this example. S = total # of discovered vertices. t = iteratio #. F = expected # of vertices discovered per iteratio (frotier). Graph: 4
5 F = S t # of discovered vertices o each iteratio time t Iitially, the algorithm discovers 2 vertex per iteratio (expected degree of each vertex = 2, so o each iteratio we expect to fid 2 more vertices), but also t is icremeted so iitially we expect the lie F to have a slope 1. F is a expected value, so there will be some variatio from the give curve (idicated by the lighter lies above.) If F = 0, the frotier is empty so we just foud the last vertex of a coected compoet, so there exists a coected compoet of size t. As the graph shows, we expect F to be ear 0 oly durig the first few iteratios ad the ear (the giat coected compoet). Differetial Equatio: Solvig for S: ds (1 dt = d s ) S = (1 e d t) Refereces [1] M. T. Chao ad J. Fraco. Probabilistic aalysis of a geeralizatio of the uit clause selectio heuristic for the k-satisfiability problem. Iformatio Scieces, 51: ,
1 Graph Sparsfication
CME 305: Discrete Mathematics ad Algorithms 1 Graph Sparsficatio I this sectio we discuss the approximatio of a graph G(V, E) by a sparse graph H(V, F ) o the same vertex set. I particular, we cosider
More informationOnes Assignment Method for Solving Traveling Salesman Problem
Joural of mathematics ad computer sciece 0 (0), 58-65 Oes Assigmet Method for Solvig Travelig Salesma Problem Hadi Basirzadeh Departmet of Mathematics, Shahid Chamra Uiversity, Ahvaz, Ira Article history:
More informationRandom Graphs and Complex Networks T
Radom Graphs ad Complex Networks T-79.7003 Charalampos E. Tsourakakis Aalto Uiversity Lecture 3 7 September 013 Aoucemet Homework 1 is out, due i two weeks from ow. Exercises: Probabilistic iequalities
More informationLecture 2: Spectra of Graphs
Spectral Graph Theory ad Applicatios WS 20/202 Lecture 2: Spectra of Graphs Lecturer: Thomas Sauerwald & He Su Our goal is to use the properties of the adjacecy/laplacia matrix of graphs to first uderstad
More informationThe Closest Line to a Data Set in the Plane. David Gurney Southeastern Louisiana University Hammond, Louisiana
The Closest Lie to a Data Set i the Plae David Gurey Southeaster Louisiaa Uiversity Hammod, Louisiaa ABSTRACT This paper looks at three differet measures of distace betwee a lie ad a data set i the plae:
More information15-859E: Advanced Algorithms CMU, Spring 2015 Lecture #2: Randomized MST and MST Verification January 14, 2015
15-859E: Advaced Algorithms CMU, Sprig 2015 Lecture #2: Radomized MST ad MST Verificatio Jauary 14, 2015 Lecturer: Aupam Gupta Scribe: Yu Zhao 1 Prelimiaries I this lecture we are talkig about two cotets:
More information6.854J / J Advanced Algorithms Fall 2008
MIT OpeCourseWare http://ocw.mit.edu 6.854J / 18.415J Advaced Algorithms Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.415/6.854 Advaced Algorithms
More informationThe isoperimetric problem on the hypercube
The isoperimetric problem o the hypercube Prepared by: Steve Butler November 2, 2005 1 The isoperimetric problem We will cosider the -dimesioal hypercube Q Recall that the hypercube Q is a graph whose
More informationCh 9.3 Geometric Sequences and Series Lessons
Ch 9.3 Geometric Sequeces ad Series Lessos SKILLS OBJECTIVES Recogize a geometric sequece. Fid the geeral, th term of a geometric sequece. Evaluate a fiite geometric series. Evaluate a ifiite geometric
More informationRecursive Estimation
Recursive Estimatio Raffaello D Adrea Sprig 2 Problem Set: Probability Review Last updated: February 28, 2 Notes: Notatio: Uless otherwise oted, x, y, ad z deote radom variables, f x (x) (or the short
More informationArithmetic Sequences
. Arithmetic Sequeces COMMON CORE Learig Stadards HSF-IF.A. HSF-BF.A.1a HSF-BF.A. HSF-LE.A. Essetial Questio How ca you use a arithmetic sequece to describe a patter? A arithmetic sequece is a ordered
More informationComputational Geometry
Computatioal Geometry Chapter 4 Liear programmig Duality Smallest eclosig disk O the Ageda Liear Programmig Slides courtesy of Craig Gotsma 4. 4. Liear Programmig - Example Defie: (amout amout cosumed
More informationProject 2.5 Improved Euler Implementation
Project 2.5 Improved Euler Implemetatio Figure 2.5.10 i the text lists TI-85 ad BASIC programs implemetig the improved Euler method to approximate the solutio of the iitial value problem dy dx = x+ y,
More informationENGI 4421 Probability and Statistics Faculty of Engineering and Applied Science Problem Set 1 Descriptive Statistics
ENGI 44 Probability ad Statistics Faculty of Egieerig ad Applied Sciece Problem Set Descriptive Statistics. If, i the set of values {,, 3, 4, 5, 6, 7 } a error causes the value 5 to be replaced by 50,
More informationCSC 220: Computer Organization Unit 11 Basic Computer Organization and Design
College of Computer ad Iformatio Scieces Departmet of Computer Sciece CSC 220: Computer Orgaizatio Uit 11 Basic Computer Orgaizatio ad Desig 1 For the rest of the semester, we ll focus o computer architecture:
More information3D Model Retrieval Method Based on Sample Prediction
20 Iteratioal Coferece o Computer Commuicatio ad Maagemet Proc.of CSIT vol.5 (20) (20) IACSIT Press, Sigapore 3D Model Retrieval Method Based o Sample Predictio Qigche Zhag, Ya Tag* School of Computer
More informationSymmetric Class 0 subgraphs of complete graphs
DIMACS Techical Report 0-0 November 0 Symmetric Class 0 subgraphs of complete graphs Vi de Silva Departmet of Mathematics Pomoa College Claremot, CA, USA Chaig Verbec, Jr. Becer Friedma Istitute Booth
More informationLecture 7 7 Refraction and Snell s Law Reading Assignment: Read Kipnis Chapter 4 Refraction of Light, Section III, IV
Lecture 7 7 Refractio ad Sell s Law Readig Assigmet: Read Kipis Chapter 4 Refractio of Light, Sectio III, IV 7. History I Eglish-speakig coutries, the law of refractio is kow as Sell s Law, after the Dutch
More informationPLEASURE TEST SERIES (XI) - 04 By O.P. Gupta (For stuffs on Math, click at theopgupta.com)
wwwtheopguptacom wwwimathematiciacom For all the Math-Gya Buy books by OP Gupta A Compilatio By : OP Gupta (WhatsApp @ +9-9650 350 0) For more stuffs o Maths, please visit : wwwtheopguptacom Time Allowed
More informationAdministrative UNSUPERVISED LEARNING. Unsupervised learning. Supervised learning 11/25/13. Final project. No office hours today
Admiistrative Fial project No office hours today UNSUPERVISED LEARNING David Kauchak CS 451 Fall 2013 Supervised learig Usupervised learig label label 1 label 3 model/ predictor label 4 label 5 Supervised
More informationRecursive Procedures. How can you model the relationship between consecutive terms of a sequence?
6. Recursive Procedures I Sectio 6.1, you used fuctio otatio to write a explicit formula to determie the value of ay term i a Sometimes it is easier to calculate oe term i a sequece usig the previous terms.
More informationBig-O Analysis. Asymptotics
Big-O Aalysis 1 Defiitio: Suppose that f() ad g() are oegative fuctios of. The we say that f() is O(g()) provided that there are costats C > 0 ad N > 0 such that for all > N, f() Cg(). Big-O expresses
More informationCivil Engineering Computation
Civil Egieerig Computatio Fidig Roots of No-Liear Equatios March 14, 1945 World War II The R.A.F. first operatioal use of the Grad Slam bomb, Bielefeld, Germay. Cotets 2 Root basics Excel solver Newto-Raphso
More informationPattern Recognition Systems Lab 1 Least Mean Squares
Patter Recogitio Systems Lab 1 Least Mea Squares 1. Objectives This laboratory work itroduces the OpeCV-based framework used throughout the course. I this assigmet a lie is fitted to a set of poits usig
More informationEvaluation scheme for Tracking in AMI
A M I C o m m u i c a t i o A U G M E N T E D M U L T I - P A R T Y I N T E R A C T I O N http://www.amiproject.org/ Evaluatio scheme for Trackig i AMI S. Schreiber a D. Gatica-Perez b AMI WP4 Trackig:
More informationAlpha Individual Solutions MAΘ National Convention 2013
Alpha Idividual Solutios MAΘ Natioal Covetio 0 Aswers:. D. A. C 4. D 5. C 6. B 7. A 8. C 9. D 0. B. B. A. D 4. C 5. A 6. C 7. B 8. A 9. A 0. C. E. B. D 4. C 5. A 6. D 7. B 8. C 9. D 0. B TB. 570 TB. 5
More informationCubic Polynomial Curves with a Shape Parameter
roceedigs of the th WSEAS Iteratioal Coferece o Robotics Cotrol ad Maufacturig Techology Hagzhou Chia April -8 00 (pp5-70) Cubic olyomial Curves with a Shape arameter MO GUOLIANG ZHAO YANAN Iformatio ad
More informationIMP: Superposer Integrated Morphometrics Package Superposition Tool
IMP: Superposer Itegrated Morphometrics Package Superpositio Tool Programmig by: David Lieber ( 03) Caisius College 200 Mai St. Buffalo, NY 4208 Cocept by: H. David Sheets, Dept. of Physics, Caisius College
More informationLecture 5. Counting Sort / Radix Sort
Lecture 5. Coutig Sort / Radix Sort T. H. Corme, C. E. Leiserso ad R. L. Rivest Itroductio to Algorithms, 3rd Editio, MIT Press, 2009 Sugkyukwa Uiversity Hyuseug Choo choo@skku.edu Copyright 2000-2018
More informationPseudocode ( 1.1) Analysis of Algorithms. Primitive Operations. Pseudocode Details. Running Time ( 1.1) Estimating performance
Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Pseudocode ( 1.1) High-level descriptio of a algorithm More structured
More informationImage Segmentation EEE 508
Image Segmetatio Objective: to determie (etract) object boudaries. It is a process of partitioig a image ito distict regios by groupig together eighborig piels based o some predefied similarity criterio.
More informationA General Framework for Accurate Statistical Timing Analysis Considering Correlations
A Geeral Framework for Accurate Statistical Timig Aalysis Cosiderig Correlatios 7.4 Vishal Khadelwal Departmet of ECE Uiversity of Marylad-College Park vishalk@glue.umd.edu Akur Srivastava Departmet of
More informationLecture 1: Introduction and Strassen s Algorithm
5-750: Graduate Algorithms Jauary 7, 08 Lecture : Itroductio ad Strasse s Algorithm Lecturer: Gary Miller Scribe: Robert Parker Itroductio Machie models I this class, we will primarily use the Radom Access
More informationAnalysis Metrics. Intro to Algorithm Analysis. Slides. 12. Alg Analysis. 12. Alg Analysis
Itro to Algorithm Aalysis Aalysis Metrics Slides. Table of Cotets. Aalysis Metrics 3. Exact Aalysis Rules 4. Simple Summatio 5. Summatio Formulas 6. Order of Magitude 7. Big-O otatio 8. Big-O Theorems
More information27 Refraction, Dispersion, Internal Reflection
Chapter 7 Refractio, Dispersio, Iteral Reflectio 7 Refractio, Dispersio, Iteral Reflectio Whe we talked about thi film iterferece, we said that whe light ecouters a smooth iterface betwee two trasparet
More informationIntro to Scientific Computing: Solutions
Itro to Scietific Computig: Solutios Dr. David M. Goulet. How may steps does it take to separate 3 objects ito groups of 4? We start with 5 objects ad apply 3 steps of the algorithm to reduce the pile
More informationTerm Project Report. This component works to detect gesture from the patient as a sign of emergency message and send it to the emergency manager.
CS2310 Fial Project Loghao Li Term Project Report Itroductio I this project, I worked o expadig exercise 4. What I focused o is makig the real gesture recogizig sesor ad desig proper gestures ad recogizig
More informationSolution printed. Do not start the test until instructed to do so! CS 2604 Data Structures Midterm Spring, Instructions:
CS 604 Data Structures Midterm Sprig, 00 VIRG INIA POLYTECHNIC INSTITUTE AND STATE U T PROSI M UNI VERSI TY Istructios: Prit your ame i the space provided below. This examiatio is closed book ad closed
More informationForce Network Analysis using Complementary Energy
orce Network Aalysis usig Complemetary Eergy Adrew BORGART Assistat Professor Delft Uiversity of Techology Delft, The Netherlads A.Borgart@tudelft.l Yaick LIEM Studet Delft Uiversity of Techology Delft,
More informationParabolic Path to a Best Best-Fit Line:
Studet Activity : Fidig the Least Squares Regressio Lie By Explorig the Relatioship betwee Slope ad Residuals Objective: How does oe determie a best best-fit lie for a set of data? Eyeballig it may be
More informationPython Programming: An Introduction to Computer Science
Pytho Programmig: A Itroductio to Computer Sciece Chapter 6 Defiig Fuctios Pytho Programmig, 2/e 1 Objectives To uderstad why programmers divide programs up ito sets of cooperatig fuctios. To be able to
More informationLecture 18. Optimization in n dimensions
Lecture 8 Optimizatio i dimesios Itroductio We ow cosider the problem of miimizig a sigle scalar fuctio of variables, f x, where x=[ x, x,, x ]T. The D case ca be visualized as fidig the lowest poit of
More informationThe Adjacency Matrix and The nth Eigenvalue
Spectral Graph Theory Lecture 3 The Adjacecy Matrix ad The th Eigevalue Daiel A. Spielma September 5, 2012 3.1 About these otes These otes are ot ecessarily a accurate represetatio of what happeed i class.
More informationA Study on the Performance of Cholesky-Factorization using MPI
A Study o the Performace of Cholesky-Factorizatio usig MPI Ha S. Kim Scott B. Bade Departmet of Computer Sciece ad Egieerig Uiversity of Califoria Sa Diego {hskim, bade}@cs.ucsd.edu Abstract Cholesky-factorizatio
More informationCHAPTER IV: GRAPH THEORY. Section 1: Introduction to Graphs
CHAPTER IV: GRAPH THEORY Sectio : Itroductio to Graphs Sice this class is called Number-Theoretic ad Discrete Structures, it would be a crime to oly focus o umber theory regardless how woderful those topics
More informationn n B. How many subsets of C are there of cardinality n. We are selecting elements for such a
4. [10] Usig a combiatorial argumet, prove that for 1: = 0 = Let A ad B be disjoit sets of cardiality each ad C = A B. How may subsets of C are there of cardiality. We are selectig elemets for such a subset
More informationTUTORIAL Create Playlist Helen Doron Course
TUTORIAL Create Playlist Hele Doro Course TUTY Tutorial Create Playlist Hele Doro Course Writte by Serafii Giampiero (INV SRL) Revised by Raffaele Forgioe (INV SRL) Editio EN - 0 Jue 0-0, INV S.r.l. Cotact:
More informationInformed Search. Russell and Norvig Chap. 3
Iformed Search Russell ad Norvig Chap. 3 Not all search directios are equally promisig Outlie Iformed: use problem-specific kowledge Add a sese of directio to search: work toward the goal Heuristic fuctios:
More information3. b. Present a combinatorial argument that for all positive integers n : : 2 n
. b. Preset a combiatorial argumet that for all positive itegers : : Cosider two distict sets A ad B each of size. Sice they are distict, the cardiality of A B is. The umber of ways of choosig a pair of
More informationSome non-existence results on Leech trees
Some o-existece results o Leech trees László A.Székely Hua Wag Yog Zhag Uiversity of South Carolia This paper is dedicated to the memory of Domiique de Cae, who itroduced LAS to Leech trees.. Abstract
More informationNumerical Methods Lecture 6 - Curve Fitting Techniques
Numerical Methods Lecture 6 - Curve Fittig Techiques Topics motivatio iterpolatio liear regressio higher order polyomial form expoetial form Curve fittig - motivatio For root fidig, we used a give fuctio
More informationSection 7.2: Direction Fields and Euler s Methods
Sectio 7.: Directio ields ad Euler s Methods Practice HW from Stewart Tetbook ot to had i p. 5 # -3 9-3 odd or a give differetial equatio we wat to look at was to fid its solutio. I this chapter we will
More informationExact Minimum Lower Bound Algorithm for Traveling Salesman Problem
Exact Miimum Lower Boud Algorithm for Travelig Salesma Problem Mohamed Eleiche GeoTiba Systems mohamed.eleiche@gmail.com Abstract The miimum-travel-cost algorithm is a dyamic programmig algorithm to compute
More informationRedundancy Allocation for Series Parallel Systems with Multiple Constraints and Sensitivity Analysis
IOSR Joural of Egieerig Redudacy Allocatio for Series Parallel Systems with Multiple Costraits ad Sesitivity Aalysis S. V. Suresh Babu, D.Maheswar 2, G. Ragaath 3 Y.Viaya Kumar d G.Sakaraiah e (Mechaical
More informationCombination Labelings Of Graphs
Applied Mathematics E-Notes, (0), - c ISSN 0-0 Available free at mirror sites of http://wwwmaththuedutw/ame/ Combiatio Labeligs Of Graphs Pak Chig Li y Received February 0 Abstract Suppose G = (V; E) is
More informationHeaps. Presentation for use with the textbook Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015
Presetatio for use with the textbook Algorithm Desig ad Applicatios, by M. T. Goodrich ad R. Tamassia, Wiley, 201 Heaps 201 Goodrich ad Tamassia xkcd. http://xkcd.com/83/. Tree. Used with permissio uder
More informationCSC165H1 Worksheet: Tutorial 8 Algorithm analysis (SOLUTIONS)
CSC165H1, Witer 018 Learig Objectives By the ed of this worksheet, you will: Aalyse the ruig time of fuctios cotaiig ested loops. 1. Nested loop variatios. Each of the followig fuctios takes as iput a
More informationAlgorithm Design Techniques. Divide and conquer Problem
Algorithm Desig Techiques Divide ad coquer Problem Divide ad Coquer Algorithms Divide ad Coquer algorithm desig works o the priciple of dividig the give problem ito smaller sub problems which are similar
More informationThroughput-Delay Scaling in Wireless Networks with Constant-Size Packets
Throughput-Delay Scalig i Wireless Networks with Costat-Size Packets Abbas El Gamal, James Mamme, Balaji Prabhakar, Devavrat Shah Departmets of EE ad CS Staford Uiversity, CA 94305 Email: {abbas, jmamme,
More informationGlobal Support Guide. Verizon WIreless. For the BlackBerry 8830 World Edition Smartphone and the Motorola Z6c
Verizo WIreless Global Support Guide For the BlackBerry 8830 World Editio Smartphoe ad the Motorola Z6c For complete iformatio o global services, please refer to verizowireless.com/vzglobal. Whether i
More informationExamples and Applications of Binary Search
Toy Gog ITEE Uiersity of Queeslad I the secod lecture last week we studied the biary search algorithm that soles the problem of determiig if a particular alue appears i a sorted list of iteger or ot. We
More informationCIS 121 Data Structures and Algorithms with Java Spring Stacks and Queues Monday, February 12 / Tuesday, February 13
CIS Data Structures ad Algorithms with Java Sprig 08 Stacks ad Queues Moday, February / Tuesday, February Learig Goals Durig this lab, you will: Review stacks ad queues. Lear amortized ruig time aalysis
More informationECE4050 Data Structures and Algorithms. Lecture 6: Searching
ECE4050 Data Structures ad Algorithms Lecture 6: Searchig 1 Search Give: Distict keys k 1, k 2,, k ad collectio L of records of the form (k 1, I 1 ), (k 2, I 2 ),, (k, I ) where I j is the iformatio associated
More informationThe golden search method: Question 1
1. Golde Sectio Search for the Mode of a Fuctio The golde search method: Questio 1 Suppose the last pair of poits at which we have a fuctio evaluatio is x(), y(). The accordig to the method, If f(x())
More informationCMPT 125 Assignment 2 Solutions
CMPT 25 Assigmet 2 Solutios Questio (20 marks total) a) Let s cosider a iteger array of size 0. (0 marks, each part is 2 marks) it a[0]; I. How would you assig a poiter, called pa, to store the address
More informationHomework 1 Solutions MA 522 Fall 2017
Homework 1 Solutios MA 5 Fall 017 1. Cosider the searchig problem: Iput A sequece of umbers A = [a 1,..., a ] ad a value v. Output A idex i such that v = A[i] or the special value NIL if v does ot appear
More informationGraphs. Minimum Spanning Trees. Slides by Rose Hoberman (CMU)
Graphs Miimum Spaig Trees Slides by Rose Hoberma (CMU) Problem: Layig Telephoe Wire Cetral office 2 Wirig: Naïve Approach Cetral office Expesive! 3 Wirig: Better Approach Cetral office Miimize the total
More informationProtected points in ordered trees
Applied Mathematics Letters 008 56 50 www.elsevier.com/locate/aml Protected poits i ordered trees Gi-Sag Cheo a, Louis W. Shapiro b, a Departmet of Mathematics, Sugkyukwa Uiversity, Suwo 440-746, Republic
More informationCIS 121 Data Structures and Algorithms with Java Fall Big-Oh Notation Tuesday, September 5 (Make-up Friday, September 8)
CIS 11 Data Structures ad Algorithms with Java Fall 017 Big-Oh Notatio Tuesday, September 5 (Make-up Friday, September 8) Learig Goals Review Big-Oh ad lear big/small omega/theta otatios Practice solvig
More informationBezier curves. Figure 2 shows cubic Bezier curves for various control points. In a Bezier curve, only
Edited: Yeh-Liag Hsu (998--; recommeded: Yeh-Liag Hsu (--9; last updated: Yeh-Liag Hsu (9--7. Note: This is the course material for ME55 Geometric modelig ad computer graphics, Yua Ze Uiversity. art of
More informationLecture Notes 6 Introduction to algorithm analysis CSS 501 Data Structures and Object-Oriented Programming
Lecture Notes 6 Itroductio to algorithm aalysis CSS 501 Data Structures ad Object-Orieted Programmig Readig for this lecture: Carrao, Chapter 10 To be covered i this lecture: Itroductio to algorithm aalysis
More informationComputers and Scientific Thinking
Computers ad Scietific Thikig David Reed, Creighto Uiversity Chapter 15 JavaScript Strigs 1 Strigs as Objects so far, your iteractive Web pages have maipulated strigs i simple ways use text box to iput
More informationA New Morphological 3D Shape Decomposition: Grayscale Interframe Interpolation Method
A ew Morphological 3D Shape Decompositio: Grayscale Iterframe Iterpolatio Method D.. Vizireau Politehica Uiversity Bucharest, Romaia ae@comm.pub.ro R. M. Udrea Politehica Uiversity Bucharest, Romaia mihea@comm.pub.ro
More informationName of the Student: Unit I (Logic and Proofs) 1) Truth Table: Conjunction Disjunction Conditional Biconditional
SUBJECT NAME : Discrete Mathematics SUBJECT CODE : MA 2265 MATERIAL NAME : Formula Material MATERIAL CODE : JM08ADM009 (Sca the above QR code for the direct dowload of this material) Name of the Studet:
More informationWhat are we going to learn? CSC Data Structures Analysis of Algorithms. Overview. Algorithm, and Inputs
What are we goig to lear? CSC316-003 Data Structures Aalysis of Algorithms Computer Sciece North Carolia State Uiversity Need to say that some algorithms are better tha others Criteria for evaluatio Structure
More information( n+1 2 ) , position=(7+1)/2 =4,(median is observation #4) Median=10lb
Chapter 3 Descriptive Measures Measures of Ceter (Cetral Tedecy) These measures will tell us where is the ceter of our data or where most typical value of a data set lies Mode the value that occurs most
More informationA PREDICTION MODEL FOR USER S SHARE ANALYSIS IN DUAL- SIM ENVIRONMENT
GSJ: Computer Sciece ad Telecommuicatios 03 No.3(39) ISSN 5-3 A PRDICTION MODL FOR USR S SHAR ANALYSIS IN DUAL- SIM NVIRONMNT Thakur Sajay, Jai Parag Orietal Uiversity, Idore, Idia sajaymca00@yahoo.com
More informationData Structures and Algorithms. Analysis of Algorithms
Data Structures ad Algorithms Aalysis of Algorithms Outlie Ruig time Pseudo-code Big-oh otatio Big-theta otatio Big-omega otatio Asymptotic algorithm aalysis Aalysis of Algorithms Iput Algorithm Output
More informationName Date Hr. ALGEBRA 1-2 SPRING FINAL MULTIPLE CHOICE REVIEW #1
Name Date Hr. ALGEBRA - SPRING FINAL MULTIPLE CHOICE REVIEW #. The high temperatures for Phoeix i October of 009 are listed below. Which measure of ceter will provide the most accurate estimatio of the
More informationImproved Random Graph Isomorphism
Improved Radom Graph Isomorphism Tomek Czajka Gopal Paduraga Abstract Caoical labelig of a graph cosists of assigig a uique label to each vertex such that the labels are ivariat uder isomorphism. Such
More informationLecture 6. Lecturer: Ronitt Rubinfeld Scribes: Chen Ziv, Eliav Buchnik, Ophir Arie, Jonathan Gradstein
068.670 Subliear Time Algorithms November, 0 Lecture 6 Lecturer: Roitt Rubifeld Scribes: Che Ziv, Eliav Buchik, Ophir Arie, Joatha Gradstei Lesso overview. Usig the oracle reductio framework for approximatig
More informationprerequisites: 6.046, 6.041/2, ability to do proofs Randomized algorithms: make random choices during run. Main benefits:
Itro Admiistrivia. Sigup sheet. prerequisites: 6.046, 6.041/2, ability to do proofs homework weekly (first ext week) collaboratio idepedet homeworks gradig requiremet term project books. questio: scribig?
More information9.1. Sequences and Series. Sequences. What you should learn. Why you should learn it. Definition of Sequence
_9.qxd // : AM Page Chapter 9 Sequeces, Series, ad Probability 9. Sequeces ad Series What you should lear Use sequece otatio to write the terms of sequeces. Use factorial otatio. Use summatio otatio to
More informationINTERSECTION CORDIAL LABELING OF GRAPHS
INTERSECTION CORDIAL LABELING OF GRAPHS G Meea, K Nagaraja Departmet of Mathematics, PSR Egieerig College, Sivakasi- 66 4, Virudhuagar(Dist) Tamil Nadu, INDIA meeag9@yahoocoi Departmet of Mathematics,
More informationName Date Hr. ALGEBRA 1-2 SPRING FINAL MULTIPLE CHOICE REVIEW #2
Name Date Hr. ALGEBRA - SPRING FINAL MULTIPLE CHOICE REVIEW # 5. Which measure of ceter is most appropriate for the followig data set? {7, 7, 75, 77,, 9, 9, 90} Mea Media Stadard Deviatio Rage 5. The umber
More information. Written in factored form it is easy to see that the roots are 2, 2, i,
CMPS A Itroductio to Programmig Programmig Assigmet 4 I this assigmet you will write a java program that determies the real roots of a polyomial that lie withi a specified rage. Recall that the roots (or
More informationLower Bounds for Sorting
Liear Sortig Topics Covered: Lower Bouds for Sortig Coutig Sort Radix Sort Bucket Sort Lower Bouds for Sortig Compariso vs. o-compariso sortig Decisio tree model Worst case lower boud Compariso Sortig
More informationXiaozhou (Steve) Li, Atri Rudra, Ram Swaminathan. HP Laboratories HPL Keyword(s): graph coloring; hardness of approximation
Flexible Colorig Xiaozhou (Steve) Li, Atri Rudra, Ram Swamiatha HP Laboratories HPL-2010-177 Keyword(s): graph colorig; hardess of approximatio Abstract: Motivated b y reliability cosideratios i data deduplicatio
More informationCOSC 1P03. Ch 7 Recursion. Introduction to Data Structures 8.1
COSC 1P03 Ch 7 Recursio Itroductio to Data Structures 8.1 COSC 1P03 Recursio Recursio I Mathematics factorial Fiboacci umbers defie ifiite set with fiite defiitio I Computer Sciece sytax rules fiite defiitio,
More informationn Some thoughts on software development n The idea of a calculator n Using a grammar n Expression evaluation n Program organization n Analysis
Overview Chapter 6 Writig a Program Bjare Stroustrup Some thoughts o software developmet The idea of a calculator Usig a grammar Expressio evaluatio Program orgaizatio www.stroustrup.com/programmig 3 Buildig
More informationA Note on Chromatic Transversal Weak Domination in Graphs
Iteratioal Joural of Mathematics Treds ad Techology Volume 17 Number 2 Ja 2015 A Note o Chromatic Trasversal Weak Domiatio i Graphs S Balamuruga 1, P Selvalakshmi 2 ad A Arivalaga 1 Assistat Professor,
More informationComputer Science Foundation Exam. August 12, Computer Science. Section 1A. No Calculators! KEY. Solutions and Grading Criteria.
Computer Sciece Foudatio Exam August, 005 Computer Sciece Sectio A No Calculators! Name: SSN: KEY Solutios ad Gradig Criteria Score: 50 I this sectio of the exam, there are four (4) problems. You must
More informationMath Section 2.2 Polynomial Functions
Math 1330 - Sectio. Polyomial Fuctios Our objectives i workig with polyomial fuctios will be, first, to gather iformatio about the graph of the fuctio ad, secod, to use that iformatio to geerate a reasoably
More informationRunning Time. Analysis of Algorithms. Experimental Studies. Limitations of Experiments
Ruig Time Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects. The
More informationSorting in Linear Time. Data Structures and Algorithms Andrei Bulatov
Sortig i Liear Time Data Structures ad Algorithms Adrei Bulatov Algorithms Sortig i Liear Time 7-2 Compariso Sorts The oly test that all the algorithms we have cosidered so far is compariso The oly iformatio
More information1. SWITCHING FUNDAMENTALS
. SWITCING FUNDMENTLS Switchig is the provisio of a o-demad coectio betwee two ed poits. Two distict switchig techiques are employed i commuicatio etwors-- circuit switchig ad pacet switchig. Circuit switchig
More informationCS 111: Program Design I Lecture 16: Module Review, Encodings, Lists
CS 111: Program Desig I Lecture 16: Module Review, Ecodigs, Lists Robert H. Sloa & Richard Warer Uiversity of Illiois at Chicago October 18, 2016 Last time Dot otatio ad methods Padas: user maual poit
More informationChapter 1. Introduction to Computers and C++ Programming. Copyright 2015 Pearson Education, Ltd.. All rights reserved.
Chapter 1 Itroductio to Computers ad C++ Programmig Copyright 2015 Pearso Educatio, Ltd.. All rights reserved. Overview 1.1 Computer Systems 1.2 Programmig ad Problem Solvig 1.3 Itroductio to C++ 1.4 Testig
More informationBASED ON ITERATIVE ERROR-CORRECTION
A COHPARISO OF CRYPTAALYTIC PRICIPLES BASED O ITERATIVE ERROR-CORRECTIO Miodrag J. MihaljeviC ad Jova Dj. GoliC Istitute of Applied Mathematics ad Electroics. Belgrade School of Electrical Egieerig. Uiversity
More informationRunning Time ( 3.1) Analysis of Algorithms. Experimental Studies. Limitations of Experiments
Ruig Time ( 3.1) Aalysis of Algorithms Iput Algorithm Output A algorithm is a step- by- step procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
More information