Running Time. Analysis of Algorithms. Experimental Studies. Limitations of Experiments
|
|
- Jasper Hutchinson
- 5 years ago
- Views:
Transcription
1 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 ruig time of a algorithm typically grows with the iput size. Average case time is ofte difficult to determie. We focus o the worst case ruig time.! Easier to aalyze! Crucial to applicatios such as games, fiace ad robotics Ruig Time best case average case worst case Iput Size Aalysis of Algorithms 2 Experimetal Studies Limitatios of Experimets Write a program implemetig the 8000 algorithm 7000 Ru the program with 6000 iputs of varyig size ad 5000 compositio 4000 Use a method like 3000 System.curretTimeMillis() to get a accurate measure 2000 of the actual ruig time 00 Plot the results 0 Time (ms) Iput Size Aalysis of Algorithms 3 It is ecessary to implemet the algorithm, which may be difficult Results may ot be idicative of the ruig time o other iputs ot icluded i the experimet. I order to compare two algorithms, the same hardware ad software eviromets must be used Aalysis of Algorithms 4
2 Theoretical Aalysis Uses a high-level descriptio of the algorithm istead of a implemetatio. Characterizes ruig time as a fuctio of the iput size,. Takes ito accout all possible iputs. Allows us to evaluate the speed of a algorithm idepedet of the hardware/software eviromet. Pseudocode High-level descriptio of a algorithm More structured tha Eglish prose Less detailed tha a program Preferred otatio for describig algorithms Hides program desig issues See pages ad i Corme et al. Example: fid max elemet of a array Algorithm arraymax(a, ) Iput array A of itegers Output maximum elemet of A curretmax! A[0] for i! 1 to " 1 do if A[i] > curretmax the curretmax! A[i] retur curretmax Aalysis of Algorithms 5 Aalysis of Algorithms 6 Pseudocode Details The Radom Access Machie (RAM) Model Cotrol flow! if the [else ]! while do! repeat util! for do! Idetatio replaces braces Method declaratio Algorithm method (arg [, arg ]) Iput Output Method call var.method (arg [, arg ]) Retur value retur expressio Expressios! Assigmet (like = i Java) = Equality testig (like == i Java) 2 Superscripts ad other mathematical formattig allowed A CPU A potetially ubouded bak of memory cells, each of which ca hold a arbitrary umber or characters 0 12 Memory cells are umbered ad accessig ay cell i memory takes uit time. Aalysis of Algorithms 7 Aalysis of Algorithms 8
3 Seve Importat Fuctios Seve fuctios that ofte appear i algorithm aalysis:! Costat # 1! Logarithmic # log! Liear #! N-Log-N # log! Quadratic # 2! Cubic # 3! Expoetial # 2 I a log-log chart, the slope of the lie correspods to the growth rate of the fuctio T ( ) 1E+30 1E+28 1E+26 1E+24 1E+22 1E+20 1E+18 1E+16 1E+14 1E+12 1E+ 1E+8 1E+6 1E+4 1E+2 1E+0 Cubic Quadratic Liear 1E+0 1E+2 1E+4 1E+6 1E+8 1E+ Aalysis of Algorithms 9 Primitive Operatios Basic computatios performed by a algorithm Idetifiable i pseudocode Largely idepedet from the programmig laguage Exact defiitio ot importat (we will see why later) Assumed to take a costat amout of time i the RAM model Examples:! Evaluatig a expressio! Assigig a value to a variable! Idexig ito a array! Callig a method! Returig from a method Aalysis of Algorithms Coutig Primitive Operatios By ispectig the pseudocode, we ca determie the maximum umber of primitive operatios executed by a algorithm, as a fuctio of the iput size. See 2.2 i Corme et al. Algorithm arraymax(a, ) # operatios curretmax! A[0] 2 for i! 1 to " 1 do 2 if A[i] > curretmax the 2( " 1) curretmax! A[i] 2( " 1) { icremet couter i } 2( " 1) retur curretmax 1 Total 8 " 2 Estimatig Ruig Time Algorithm arraymax executes 8 " 2 primitive operatios i the worst case. Defie: a = Time take by the fastest primitive operatio b = Time take by the slowest primitive operatio Let T() be worst-case time of arraymax. The a (8 " 2) $ T() $ b(8 " 2) Hece, the ruig time T() is bouded by two liear fuctios Aalysis of Algorithms 11 Aalysis of Algorithms 12
4 Growth Rate of Ruig Time Chagig the hardware/ software eviromet!.. affects T() by a costat factor, but!.. does ot alter the growth rate of T(). The liear growth rate of the ruig time T() is a itrisic property of algorithm arraymax. Costat Factors The growth rate is ot affected by! costat factors or! lower-order terms Examples T ( )! is a liear fuctio! is a quadratic fuctio 1E+26 1E+24 1E+22 1E+20 1E+18 1E+16 1E+14 1E+12 1E+ 1E+8 1E+6 1E+4 1E+2 1E+0 Quadratic Quadratic Liear Liear 1E+0 1E+2 1E+4 1E+6 1E+8 1E+ Aalysis of Algorithms 13 Aalysis of Algorithms 14 Big-Oh Notatio Give fuctios f() ad g(), we say that f() is O(g()) if there are positive costats c ad 0 such that f() $ cg() for % 0 Example: 2 + is O()! 2 + $ c! (c " 2) %! % /(c " 2)! Pick c = 3 ad 0 =,000 1, ,000 Aalysis of Algorithms 15 Big-Oh Notatio Give fuctios f() ad g(), we say that f() is O(g()) if there are positive costats c ad 0 such that f() $ cg() for % 0 Example: 2 + is O()! 2 + $ c! (c " 2) %! % /(c " 2)! Pick c = 3 ad 0 = See 3.1 i Corme et al.,000 1, ,000 Aalysis of Algorithms 16
5 Big-Oh Example Example: the fuctio 2 is ot O()! 2 $ c! $ c! The above iequality caot be satisfied sice c must be a costat 1,000,000 0,000,000 1, ^ ,000 Aalysis of Algorithms 17 More Big-Oh Examples is O() eed c > 0 ad 0 % 1 such that 7-2 $ c for % 0 this is true for c = 7 ad 0 = 1! is O( 3 ) eed c > 0 ad 0 % 1 such that $ c 3 for % 0 this is true for c = 4 ad 0 = 21! 3 log log + 5 is O(log ) eed c > 0 ad 0 % 1 such that 3 log + 5 $ c log for % 0 this is true for c = 8 ad 0 = 2 Aalysis of Algorithms 18 Big-Oh ad Growth Rate The big-oh otatio gives a upper boud o the growth rate of a fuctio The statemet f() is O(g()) meas that the growth rate of f() is o more tha the growth rate of g() We ca use the big-oh otatio to rak fuctios accordig to their growth rate g() grows more f() grows more Same growth f() is O(g()) Yes No Yes g() is O(f()) No Yes Yes Big-Oh Rules If is f() a polyomial of degree d, the f() is O( d ), i.e.,! Drop lower-order terms! Drop costat factors Use the smallest possible class of fuctios! Say 2 is O() istead of 2 is O( 2 ) Use the simplest expressio of the class! Say is O() istead of is O(3) Aalysis of Algorithms 19 Aalysis of Algorithms 20
6 Asymptotic Algorithm Aalysis The asymptotic aalysis of a algorithm determies the ruig time i big-oh otatio. To perform the asymptotic aalysis! We fid the worst-case umber of primitive operatios executed as a fuctio of the iput size.! We express this fuctio with big-oh otatio. Example:! We determie that algorithm arraymax executes at most 8 " 2 primitive operatios.! We say that algorithm arraymax rus i O() time. Sice costat factors ad lower-order terms are evetually dropped ayhow, we ca disregard them whe coutig primitive operatios. Computig Prefix Averages We further illustrate asymptotic aalysis with two algorithms for prefix averages The i-th prefix average of a array X is average of the first (i + 1) elemets of X: A[i] = (X[0] + X[1] + + X[i])/(i+1) Computig the array A of prefix averages of aother array X has applicatios to fiacial aalysis X A Aalysis of Algorithms 21 Aalysis of Algorithms 22 Prefix Averages (Quadratic) The followig algorithm computes prefix averages i quadratic time by applyig the defiitio Algorithm prefixaverages1(x, ) Iput array X of itegers Output array A of prefix averages of X #operatios A! ew array of itegers for i! 0 to " 1 do s! X[0] for j! 1 to i do ( " 1) s! s + X[j] ( " 1) A[i]! s / (i + 1) retur A 1 Arithmetic Progressio The ruig time of prefixaverages1 is O( ) The sum of the first itegers is ( + 1) / 2! There is a simple visual proof of this fact 2 Thus, algorithm prefixaverages1 rus i 1 O( 2 ) time Aalysis of Algorithms 23 Aalysis of Algorithms 24
7 Prefix Averages (Liear) The followig algorithm computes prefix averages i liear time by keepig a ruig sum Algorithm prefixaverages2(x, ) Iput array X of itegers Output array A of prefix averages of X #operatios A! ew array of itegers s! 0 1 for i! 0 to " 1 do s! s + X[i] A[i]! s / (i + 1) retur A 1 Algorithm prefixaverages2 rus i O() time Aalysis of Algorithms 25 Math you eed to Review Summatios Logarithms ad Expoets Proof techiques (Basic probability) properties of logarithms: log b (xy) = log b x + log b y log b (x/y) = log b x - log b y log b xa = alog b x log b a = log x a/log x b properties of expoetials: a (b+c) = a b a c a bc = (a b ) c a b /a c = a (b-c) b = a log a b b c = a c*log a b Aalysis of Algorithms 26 Relatives of Big-Oh big-omega! f() is &(g()) if there is a costat c > 0 ad a iteger costat 0 % 1 such that f() % c g() for % 0! Lower bouds big-theta! f() is '(g()) if there are costats c > 0 ad c > 0 ad a iteger costat 0 % 1 such that c g() $ f() $ c g() for % 0! Tight bouds Ituitio for Asymptotic Notatio Big-Oh! f() is O(g()) if f() is asymptotically less tha or equal to g() big-omega! f() is &(g()) if f() is asymptotically greater tha or equal to g() big-theta! f() is '(g()) if f() is asymptotically equal to g() Aalysis of Algorithms 27 Aalysis of Algorithms 28
8 Example Uses of the Relatives of Big-Oh! 5 2 is &( 2 ) f() is &(g()) if there is a costat c > 0 ad a iteger costat 0 % 1 such that f() % c g() for % 0 let c = 5 ad 0 = 1! 5 2 is &() f() is &(g()) if there is a costat c > 0 ad a iteger costat 0 % 1 such that f() % c g() for % 0 let c = 1 ad 0 = 1! 5 2 is '( 2 ) f() is '(g()) if it is &( 2 ) ad O( 2 ). We have already see the former, for the latter recall that f() is O(g()) if there is a costat c > 0 ad a iteger costat 0 % 1 such that f() < c g() for % 0 Let c = 5 ad 0 = 1 Aalysis of Algorithms 29
Analysis of Algorithms
Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Ruig Time Most algorithms trasform iput objects ito output objects. The
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 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 informationOutline and Reading. Analysis of Algorithms. Running Time. Experimental Studies. Limitations of Experiments. Theoretical Analysis
Outlie ad Readig Aalysis of Algorithms Iput Algorithm Output Ruig time ( 3.) Pseudo-code ( 3.2) Coutig primitive operatios ( 3.3-3.) Asymptotic otatio ( 3.6) Asymptotic aalysis ( 3.7) Case study Aalysis
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 informationAnalysis of Algorithms
Aalysis of Algorithms Ruig Time of a algorithm Ruig Time Upper Bouds Lower Bouds Examples Mathematical facts Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite
More informationAnalysis of Algorithms
Presetatio for use with the textbook, Algorithm Desig ad Applicatios, by M. T. Goodrich ad R. Tamassia, Wiley, 2015 Aalysis of Algorithms Iput 2015 Goodrich ad Tamassia Algorithm Aalysis of Algorithms
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 informationElementary maths for GMT. Algorithm analysis Part I
Elementary maths for GMT Algorithm analysis Part I Algorithms An algorithm is a step-by-step procedure for solving a problem in a finite amount of time Most algorithms transform input objects into output
More informationLECTURE 9 Data Structures: A systematic way of organizing and accessing data. --No single data structure works well for ALL purposes.
LECTURE 9 Data Structures: A systematic way of organizing and accessing data. --No single data structure works well for ALL purposes. Input Algorithm Output An algorithm is a step-by-step procedure for
More informationAnalysis of Algorithms
Analysis of Algorithms Data Structures and Algorithms Acknowledgement: These slides are adapted from slides provided with Data Structures and Algorithms in C++ Goodrich, Tamassia and Mount (Wiley, 2004)
More informationData Structures Lecture 8
Fall 2017 Fang Yu Software Security Lab. Dept. Management Information Systems, National Chengchi University Data Structures Lecture 8 Recap What should you have learned? Basic java programming skills Object-oriented
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 informationHow do we evaluate algorithms?
F2 Readig referece: chapter 2 + slides Algorithm complexity Big O ad big Ω To calculate ruig time Aalysis of recursive Algorithms Next time: Litterature: slides mostly The first Algorithm desig methods:
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 information9/10/2018 Algorithms & Data Structures Analysis of Algorithms. Siyuan Jiang, Sept
9/10/2018 Algorithms & Data Structures Analysis of Algorithms Siyuan Jiang, Sept. 2018 1 Email me if the office door is closed Siyuan Jiang, Sept. 2018 2 Grades have been emailed github.com/cosc311/assignment01-userid
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 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 informationOutline and Reading. Analysis of Algorithms 1
Outline and Reading Algorithms Running time ( 3.1) Pseudo-code ( 3.2) Counting primitive operations ( 3.4) Asymptotic notation ( 3.4.1) Asymptotic analysis ( 3.4.2) Case study ( 3.4.3) Analysis of Algorithms
More informationA graphical view of big-o notation. c*g(n) f(n) f(n) = O(g(n))
ca see that time required to search/sort grows with size of We How do space/time eeds of program grow with iput size? iput. time: cout umber of operatios as fuctio of iput Executio size operatio Assigmet:
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 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 informationCOMP Parallel Computing. PRAM (1): The PRAM model and complexity measures
COMP 633 - Parallel Computig Lecture 2 August 24, 2017 : The PRAM model ad complexity measures 1 First class summary This course is about parallel computig to achieve high-er performace o idividual problems
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. Counting Sort Analysis of Algorithms
Algorithm Coutig Sort Aalysis of Algorithms Assumptios: records Coutig sort Each record cotais keys ad data All keys are i the rage of 1 to k Space The usorted list is stored i A, the sorted list will
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 information2. ALGORITHM ANALYSIS
2. ALGORITHM ANALYSIS computatioal tractability survey of commo ruig times 2. ALGORITHM ANALYSIS computatioal tractability survey of commo ruig times Lecture slides by Kevi Waye Copyright 2005 Pearso-Addiso
More informationAlgorithms and Data Structures
Algorithm Analysis Page 1 - Algorithm Analysis Dr. Fall 2008 Algorithm Analysis Page 2 Outline Textbook Overview Analysis of Algorithm Pseudo-Code and Primitive Operations Growth Rate and Big-Oh Notation
More informationCSE 417: Algorithms and Computational Complexity
Time CSE 47: Algorithms ad Computatioal Readig assigmet Read Chapter of The ALGORITHM Desig Maual Aalysis & Sortig Autum 00 Paul Beame aalysis Problem size Worst-case complexity: max # steps algorithm
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 informationA Generalized Set Theoretic Approach for Time and Space Complexity Analysis of Algorithms and Functions
Proceedigs of the 10th WSEAS Iteratioal Coferece o APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 316 A Geeralized Set Theoretic Approach for Time ad Space Complexity Aalysis of Algorithms
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 informationData Structures and Algorithms
Berner Fachhochschule - Technik und Informatik Data Structures and Algorithms Topic 1: Algorithm Analysis Philipp Locher FS 2018 Outline Course and Textbook Overview Analysis of Algorithm Pseudo-Code and
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 informationANALYSIS OF ALGORITHMS
ANALYSIS OF ALGORITHMS Running Time Pseudo-Code Asymptotic Notation Asymptotic Analysis Mathematical facts T(n) n = 4 Input Algorithm Output 1 Average Case vs. Worst Case Running Time of an Algorithm An
More informationPython Programming: An Introduction to Computer Science
Pytho Programmig: A Itroductio to Computer Sciece Chapter 1 Computers ad Programs 1 Objectives To uderstad the respective roles of hardware ad software i a computig system. To lear what computer scietists
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 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 informationCIS 121 Data Structures and Algorithms with Java Spring Stacks, Queues, and Heaps Monday, February 18 / Tuesday, February 19
CIS Data Structures ad Algorithms with Java Sprig 09 Stacks, Queues, ad Heaps Moday, February 8 / Tuesday, February 9 Stacks ad Queues Recall the stack ad queue ADTs (abstract data types from lecture.
More informationHash Tables. 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, 2015 Hash Tables xkcd. http://xkcd.com/221/. Radom Number. Used with permissio uder Creative
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 informationChapter 4 The Datapath
The Ageda Chapter 4 The Datapath Based o slides McGraw-Hill Additioal material 24/25/26 Lewis/Marti Additioal material 28 Roth Additioal material 2 Taylor Additioal material 2 Farmer Tae the elemets that
More informationCS473-Algorithms I. Lecture 2. Asymptotic Notation. CS 473 Lecture 2 1
CS473-Algorithms I Lecture Asymptotic Notatio CS 473 Lecture 1 O-otatio (upper bouds) f() = O(g()) if positive costats c, 0 such that e.g., = O( 3 ) 0 f() cg(), 0 c 3 c c = 1 & 0 = or c = & 0 = 1 Asymptotic
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 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 informationChapter 5. Functions for All Subtasks. Copyright 2015 Pearson Education, Ltd.. All rights reserved.
Chapter 5 Fuctios for All Subtasks Copyright 2015 Pearso Educatio, Ltd.. All rights reserved. Overview 5.1 void Fuctios 5.2 Call-By-Referece Parameters 5.3 Usig Procedural Abstractio 5.4 Testig ad Debuggig
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 informationAlgorithm Efficiency
Algorithm Effiiey Exeutig ime Compariso of algorithms to determie whih oe is better approah implemet algorithms & reord exeutio time Problems with this approah there are may tasks ruig ourretly o a omputer
More informationAlgorithms Chapter 3 Growth of Functions
Algorithms Chapter 3 Growth of Fuctios Istructor: Chig Chi Li 林清池助理教授 chigchi.li@gmail.com Departmet of Computer Sciece ad Egieerig Natioal Taiwa Ocea Uiversity Outlie Asymptotic otatio Stadard otatios
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 informationChapter 4. Procedural Abstraction and Functions That Return a Value. Copyright 2015 Pearson Education, Ltd.. All rights reserved.
Chapter 4 Procedural Abstractio ad Fuctios That Retur a Value Copyright 2015 Pearso Educatio, Ltd.. All rights reserved. Overview 4.1 Top-Dow Desig 4.2 Predefied Fuctios 4.3 Programmer-Defied Fuctios 4.4
More informationCS 11 C track: lecture 1
CS 11 C track: lecture 1 Prelimiaries Need a CMS cluster accout http://acctreq.cms.caltech.edu/cgi-bi/request.cgi Need to kow UNIX IMSS tutorial liked from track home page Track home page: http://courses.cms.caltech.edu/courses/cs11/material
More informationChapter 11. Friends, Overloaded Operators, and Arrays in Classes. Copyright 2014 Pearson Addison-Wesley. All rights reserved.
Chapter 11 Frieds, Overloaded Operators, ad Arrays i Classes Copyright 2014 Pearso Addiso-Wesley. All rights reserved. Overview 11.1 Fried Fuctios 11.2 Overloadig Operators 11.3 Arrays ad Classes 11.4
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 informationCounting the Number of Minimum Roman Dominating Functions of a Graph
Coutig the Number of Miimum Roma Domiatig Fuctios of a Graph SHI ZHENG ad KOH KHEE MENG, Natioal Uiversity of Sigapore We provide two algorithms coutig the umber of miimum Roma domiatig fuctios of a graph
More informationCOP4020 Programming Languages. Subroutines and Parameter Passing Prof. Robert van Engelen
COP4020 Programmig Laguages Subrouties ad Parameter Passig Prof. Robert va Egele Overview Parameter passig modes Subroutie closures as parameters Special-purpose parameters Fuctio returs COP4020 Fall 2016
More informationAlgorithm Analysis. October 12, CMPE 250 Algorithm Analysis October 12, / 66
Algorithm Analysis October 12, 2016 CMPE 250 Algorithm Analysis October 12, 2016 1 / 66 Problem Solving: Main Steps 1 Problem definition 2 Algorithm design / Algorithm specification 3 Algorithm analysis
More informationCS200: Hash Tables. Prichard Ch CS200 - Hash Tables 1
CS200: Hash Tables Prichard Ch. 13.2 CS200 - Hash Tables 1 Table Implemetatios: average cases Search Add Remove Sorted array-based Usorted array-based Balaced Search Trees O(log ) O() O() O() O(1) O()
More informationBACHMANN-LANDAU NOTATIONS. Lecturer: Dr. Jomar F. Rabajante IMSP, UPLB MATH 174: Numerical Analysis I 1 st Sem AY
BACHMANN-LANDAU NOTATIONS Lecturer: Dr. Jomar F. Rabajate IMSP, UPLB MATH 174: Numerical Aalysis I 1 st Sem AY 018-019 RANKING OF FUNCTIONS Name Big-Oh Eamples Costat O(1 10 Logarithmic O(log log, log(
More informationAbstract Data Types (ADTs) Stacks. The Stack ADT ( 4.2) Stack Interface in Java
Abstract Data Types (ADTs) tacks A abstract data type (ADT) is a abstractio of a data structure A ADT specifies: Data stored Operatios o the data Error coditios associated with operatios Example: ADT modelig
More informationPolynomial Functions and Models. Learning Objectives. Polynomials. P (x) = a n x n + a n 1 x n a 1 x + a 0, a n 0
Polyomial Fuctios ad Models 1 Learig Objectives 1. Idetify polyomial fuctios ad their degree 2. Graph polyomial fuctios usig trasformatios 3. Idetify the real zeros of a polyomial fuctio ad their multiplicity
More informationChapter 2. C++ Basics. Copyright 2015 Pearson Education, Ltd.. All rights reserved.
Chapter 2 C++ Basics Copyright 2015 Pearso Educatio, Ltd.. All rights reserved. Overview 2.1 Variables ad Assigmets 2.2 Iput ad Output 2.3 Data Types ad Expressios 2.4 Simple Flow of Cotrol 2.5 Program
More informationCOP4020 Programming Languages. Compilers and Interpreters Prof. Robert van Engelen
COP4020 mig Laguages Compilers ad Iterpreters Prof. Robert va Egele Overview Commo compiler ad iterpreter cofiguratios Virtual machies Itegrated developmet eviromets Compiler phases Lexical aalysis Sytax
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 informationcondition w i B i S maximum u i
ecture 10 Dyamic Programmig 10.1 Kapsack Problem November 1, 2004 ecturer: Kamal Jai Notes: Tobias Holgers We are give a set of items U = {a 1, a 2,..., a }. Each item has a weight w i Z + ad a utility
More informationUniversity of Waterloo Department of Electrical and Computer Engineering ECE 250 Algorithms and Data Structures
Uiversity of Waterloo Departmet of Electrical ad Computer Egieerig ECE 250 Algorithms ad Data Structures Midterm Examiatio ( pages) Istructor: Douglas Harder February 7, 2004 7:30-9:00 Name (last, first)
More informationUNIT 4C Iteration: Scalability & Big O. Efficiency
UNIT 4C Iteratio: Scalability & Big O 1 Efficiecy A computer program should be totally correct, but it should also execute as quickly as possible (time-efficiecy) use memory wisely (storage-efficiecy)
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 informationFast Fourier Transform (FFT) Algorithms
Fast Fourier Trasform FFT Algorithms Relatio to the z-trasform elsewhere, ozero, z x z X x [ ] 2 ~ elsewhere,, ~ e j x X x x π j e z z X X π 2 ~ The DFS X represets evely spaced samples of the z- trasform
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 informationChapter 9. Pointers and Dynamic Arrays. Copyright 2015 Pearson Education, Ltd.. All rights reserved.
Chapter 9 Poiters ad Dyamic Arrays Copyright 2015 Pearso Educatio, Ltd.. All rights reserved. Overview 9.1 Poiters 9.2 Dyamic Arrays Copyright 2015 Pearso Educatio, Ltd.. All rights reserved. Slide 9-3
More informationToday s objectives. CSE401: Introduction to Compiler Construction. What is a compiler? Administrative Details. Why study compilers?
CSE401: Itroductio to Compiler Costructio Larry Ruzzo Sprig 2004 Today s objectives Admiistrative details Defie compilers ad why we study them Defie the high-level structure of compilers Associate specific
More informationAppendix D. Controller Implementation
COMPUTER ORGANIZATION AND DESIGN The Hardware/Software Iterface 5 th Editio Appedix D Cotroller Implemetatio Cotroller Implemetatios Combiatioal logic (sigle-cycle); Fiite state machie (multi-cycle, pipelied);
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 informationChapter 10. Defining Classes. Copyright 2015 Pearson Education, Ltd.. All rights reserved.
Chapter 10 Defiig Classes Copyright 2015 Pearso Educatio, Ltd.. All rights reserved. Overview 10.1 Structures 10.2 Classes 10.3 Abstract Data Types 10.4 Itroductio to Iheritace Copyright 2015 Pearso Educatio,
More informationDesigning a learning system
CS 75 Machie Learig Lecture Desigig a learig system Milos Hauskrecht milos@cs.pitt.edu 539 Seott Square, x-5 people.cs.pitt.edu/~milos/courses/cs75/ Admiistrivia No homework assigmet this week Please try
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 informationPractical Session 2 Constants Don t Matter!!!
Practical Sessio 2 Costats Do t Matter!!! Algorithm Aalysis f() = O(g()) Big O f() = Ω(g()) Big Omega There exist c > 0 ad 0 > 0 such that: 0 f() cg() for each 0 There exist c > 0 ad 0 > 0 such that: 0
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 informationThompson s Group F (p + 1) is not Minimally Almost Convex
Thompso s Group F (p + ) is ot Miimally Almost Covex Claire Wladis Thompso s Group F (p + ). A Descriptio of F (p + ) Thompso s group F (p + ) ca be defied as the group of piecewiseliear orietatio-preservig
More informationAccuracy Improvement in Camera Calibration
Accuracy Improvemet i Camera Calibratio FaJie L Qi Zag ad Reihard Klette CITR, Computer Sciece Departmet The Uiversity of Aucklad Tamaki Campus, Aucklad, New Zealad fli006, qza001@ec.aucklad.ac.z r.klette@aucklad.ac.z
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 informationAbstract. Chapter 4 Computation. Overview 8/13/18. Bjarne Stroustrup Note:
Chapter 4 Computatio Bjare Stroustrup www.stroustrup.com/programmig Abstract Today, I ll preset the basics of computatio. I particular, we ll discuss expressios, how to iterate over a series of values
More informationRecursion. Computer Science S-111 Harvard University David G. Sullivan, Ph.D. Review: Method Frames
Uit 4, Part 3 Recursio Computer Sciece S-111 Harvard Uiversity David G. Sulliva, Ph.D. Review: Method Frames Whe you make a method call, the Java rutime sets aside a block of memory kow as the frame of
More informationA log n lower bound to compute any function in parallel Reduction and broadcast in O(log n) time Parallel prefix (scan) in O(log n) time
CS 267 Tricks with Trees Outlie A log lower boud to compute ay fuctio i parallel Reductio ad broadcast i O(log ) time Parallel prefix (sca) i O(log ) time Addig two -bit itegers i O(log ) time Multiplyig
More informationData diverse software fault tolerance techniques
Data diverse software fault tolerace techiques Complemets desig diversity by compesatig for desig diversity s s limitatios Ivolves obtaiig a related set of poits i the program data space, executig the
More informationClasses and Objects. Again: Distance between points within the first quadrant. José Valente de Oliveira 4-1
Classes ad Objects jvo@ualg.pt José Valete de Oliveira 4-1 Agai: Distace betwee poits withi the first quadrat Sample iput Sample output 1 1 3 4 2 jvo@ualg.pt José Valete de Oliveira 4-2 1 The simplest
More informationGC05: Algorithmic Complexity & Computability
GC05: Algorithmic Complexity & Computability This part of the course deals with assessig the time-demad of algorithmic procedures with the aim, where possible, of fidig efficiet solutios to problems. We
More informationCS : Programming for Non-Majors, Summer 2007 Programming Project #3: Two Little Calculations Due by 12:00pm (noon) Wednesday June
CS 1313 010: Programmig for No-Majors, Summer 2007 Programmig Project #3: Two Little Calculatios Due by 12:00pm (oo) Wedesday Jue 27 2007 This third assigmet will give you experiece writig programs that
More informationMajor CSL Write your name and entry no on every sheet of the answer script. Time 2 Hrs Max Marks 70
NOTE:. Attempt all seve questios. Major CSL 02 2. Write your ame ad etry o o every sheet of the aswer script. Time 2 Hrs Max Marks 70 Q No Q Q 2 Q 3 Q 4 Q 5 Q 6 Q 7 Total MM 6 2 4 0 8 4 6 70 Q. Write a
More information1 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 informationNTH, GEOMETRIC, AND TELESCOPING TEST
NTH, GEOMETRIC, AND TELESCOPING TEST Sectio 9. Calculus BC AP/Dual, Revised 08 viet.dag@humbleisd.et /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test SUMMARY OF TESTS FOR SERIES Lookig at the first few
More informationOrder statistics. Order Statistics. Randomized divide-andconquer. Example. CS Spring 2006
406 CS 5633 -- Sprig 006 Order Statistics Carola We Slides courtesy of Charles Leiserso with small chages by Carola We CS 5633 Aalysis of Algorithms 406 Order statistics Select the ith smallest of elemets
More informationChapter 3 Classification of FFT Processor Algorithms
Chapter Classificatio of FFT Processor Algorithms The computatioal complexity of the Discrete Fourier trasform (DFT) is very high. It requires () 2 complex multiplicatios ad () complex additios [5]. As
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 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 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 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 informationMR-2010I %MktBSize Macro 989. %MktBSize Macro
MR-2010I %MktBSize Macro 989 %MktBSize Macro The %MktBSize autocall macro suggests sizes for balaced icomplete block desigs (BIBDs). The sizes that it reports are sizes that meet ecessary but ot sufficiet
More informationCOP4020 Programming Languages. Functional Programming Prof. Robert van Engelen
COP4020 Programmig Laguages Fuctioal Programmig Prof. Robert va Egele Overview What is fuctioal programmig? Historical origis of fuctioal programmig Fuctioal programmig today Cocepts of fuctioal programmig
More information