CS161 Design and Analysis of Algorithms. Administrative
|
|
- Emily Harrington
- 6 years ago
- Views:
Transcription
1 CS161 Desig ad Aalysis of Algorithms Da Boeh 1 Admiistrative Lecture 1, April 3, 1 Web page Hadouts» Aoucemets» Late breakig ews Gradig ad course requiremets» Midterm/fial/hw» Project» Late HW policy» Importace of readable HW» Collaboratio Probability - Ch. 6., pp READ NOW! 1
2 Why Study Algorithms? (why cs161?) Bag of tricks» Sortig» Data structures: queues/stacks/heaps/trees» Search Methodology - how to desig algorithms» Divide & coquer» Recursive algorithms» Radomized algorithms» Dyamic programmig Useful abstractios.» Schedulig classes graphs.» Job assigmet balls ad boxes. Higher-level way of approachig problems 3 How to compare algorithms? Code ad ru - experimet» Iputs?» Parameters?» Bad implemetatios? Average case» what is average iput?? Worst case» Asymptotics» rough idea o performace» aalytical depedece betwee parameters 4
3 Example from Ch. Isertio sort: for j = to key = A(j) i=j-1 while i > ad A(i) > key A(i+1) = A(i) A(i) = key i-- ed ed Example: About Pseudo-Code Not really a program, just a outlie Eough details to establish the ruig time ad correctess. No error-hadlig mechaisms. Eve pseudo-code is too complicated! Note that for a trivial algorithms it obscures what is really goig o... The i-place part is a optimizatio. We could start by a simpler descriptio:» Go over the umbers oe-by-oe, startig from the first, copy to ew array.» Each time copy to the correct place i the ew array.» I order to create empty space, shift the umbers that are larger tha the curretly cosidered umber oe cell to the right. 6 3
4 Aalysis Correctess ad termiatio. Ruig time:» Depeds o iput size» iput properties Wat a upper boud o:» Worst case: max T(), ay iput.» Expected: E[T()], iput take from a distributio. which?? example: sortig arrivig TCP/IP packets they are mostly sorted already.» Best case: Ca be used to argue that the algorithm is really bad. (ay algorithm ca be rewritte to have a excellet best case performace) 7 Back to isertio sort Isertio sort: for j = to key = A(j) i=j-1 while i > ad A(i) > key A(i+1) = A(i) A(i) = key i-- ed ed Simplified algorithm:» Go over the umbers oe-by-oe, startig from the first, copy to ew array.» Each time copy to the correct place i the ew array.» I order to create empty space, shift the umbers that are larger tha the curretly cosidered umber oe cell to the right Â( t j -1) j= times t j each 8 4
5 Aalysis Best ruig time: Outer loop always executed, Ier loop - ot executed if iput already sorted. Assume each operatio takes 1 time uit - approximatio. + ( - 1) + ( - 1) + t + ( t - 1) + ( - 1) t j worst case ª j  fi t = j j= Would like to formalize this statemet!  ( + 1) This domiates!  j j j= j= Do we really eed to pay close attetio to all the idices i the summatios? Maybe some or them are ot really importat?? 9 Formalizatio ( +1) How to formalize that was the mai issue?? The aswer is asymptotic aalysis:» Igore machie-depedet costats.» Look at growth of T() as Ituitio: drop low-order terms eg: =Θ( 4) Idea: as, Θ( ) becomes better (faster) tha Θ( 4) 1 5
6 Back to isertio sort aalysis Ier loop was Q(j) T () Θ ( t ) ( ) ( j =Θ tj =Θ ) Is this formal? NO! Example, usig the same logic: Θ (1) +Θ (1) =Θ(1) seems to imply that Θ (1) =Θ(1) Icorrect! i= 1 We eed formalizatio!?? Aother example: log ~ 1/1 11 Asymptotics big-oh otatio: f ( ) = O( g( )) cost c, s.t. : f ( ) cg( ) Example: = O( 6 ) but ot vice versa!! = is ot equality but membership i a set. Set otatio is cumbersome: O( g( )) = { f ( ) cost c, s.t. : f ( ) cg( )} What do we mea by f ( ) = O( ) + h( ) = O( ), f ( ) = h( ) + We are too lazy to specify h() exactly! 1 6
7 Asymptotics Small-oh otatio: f ( ) = o( g( )) cost c, s.t. : f ( ) < cg( ) Differeces from big-oh Prove that = o( ): Give c, lets take = / c for, c c c ( c ) = > QED g() f() Factor f() g() f=o(g) i both cases! 13 Omega otatio Big-Omega: f ( ) = Ω( g( )) cost c, s.t. : cg( ) f ( ) Small-omega: f ( ) = ω( g( )) cost c, s.t. : cg( ) < f ( ) O: o: < W: : > 14 7
8 Trasitivity etc. Most rules apply: Example: trasitivity a b, b c a c f = Og ( ), g= Oh () f = Oh () Proof: f = Og ( ) cost c1, 1 s.t. 1 : f() cg 1 () g= Oh () cost c, s.t. : g () ch () Take 3= max( 1, ), c3= cc 1 The: 3: f() cg 1 () cch 1 () = ch 3 () f() = Og ( ( )) QED Not all rules apply! f, g s.t. f O( g) ad g O( f ) example: f =, g = 1 + si 15 Theta otatio Theta: f ( ) = Θ( g( )) cost c, c, s.t. : c g( ) f ( ) c g( ) 1 1 Ofte cofused with Big-Oh otatio! Example: / = Θ( ) Proof: take = 8, the for : / / 4+ 8 / 4 = / 4 O the other had, we have: / < / Thus: / 4 / / i.e. c = 1/ 4, c = 1/. 1 Claim: Low order terms do ot matter. Needs a proof! (HW?) 16 8
9 Simple Theorem Claim f = O g ad g = O f f = Θ g ( ) ( ( )) ( ) ( ( )) ( ) ( ( )) Proof: 1, c1 s.t. 1: f ( ) c1 g( ), c s.t. : g( ) c f ( ) max( 1, ): 1 c g ( ) f ( ) c 1 g ( ) QED 17 Summary Remember the defiitios. Formally prove from defiitios. Use ituitio from the properties of,, etc. 18 9
Advanced Course of Algorithm Design and Analysis
Differet complexity measures Advaced Course of Algorithm Desig ad Aalysis Asymptotic complexity Big-Oh otatio Properties of O otatio Aalysis of simple algorithms A algorithm may may have differet executio
More informationAlgorithm Analysis. Chapter 3
Data Structures Dr Ahmed Rafat Abas Computer Sciece Dept, Faculty of Computer ad Iformatio, Zagazig Uiversity arabas@zu.edu.eg http://www.arsaliem.faculty.zu.edu.eg/ Algorithm Aalysis Chapter 3 3. Itroductio
More informationDesign and Analysis of ALGORITHM (Topic 2)
DR. Gatot F. Hertoo, MSc. Desig ad Aalysis of ALGORITHM (Topic 2) Algorithms + Data Structures = Programs Lessos Leared 1 Our Machie Model: Assumptios Geeric Radom Access Machie (RAM) Executes operatios
More informationCS 270 Algorithms. Oliver Kullmann. Growth of Functions. Divide-and- Conquer Min-Max- Problem. Tutorial. Reading from CLRS for week 2
Geeral remarks Week 2 1 Divide ad First we cosider a importat tool for the aalysis of algorithms: Big-Oh. The we itroduce a importat algorithmic paradigm:. We coclude by presetig ad aalysig two examples.
More informationAnalysis of Algorithms. Introduction. Contents
Itroductio The focus of this module is mathematical aspects of algorithms. Our mai focus is aalysis of algorithms, which meas evaluatig efficiecy of algorithms by aalytical ad mathematical methods. We
More informationAlgorithm Analysis. Algorithms that are equally correct can vary in their utilization of computational resources
Algorithm Aalysis Algorithms that are equally correct ca vary i their utilizatio of computatioal resources time ad memory a slow program it is likely ot to be used a program that demads too much memory
More informationCh3. Asymptotic Notation
Ch. Asymptotic Notatio copyright 006 Preview of Chapters Chapter How to aalyze the space ad time complexities of program Chapter Review asymptotic otatios such as O, Ω, Θ, o for simplifyig the aalysis
More informationData Structures and Algorithm. Xiaoqing Zheng
Data Structures ad Algorithm Xiaoqig Zheg zhegxq@fudaeduc What are algorithms? A sequece of computatioal steps that trasform the iput ito the output Sortig problem: Iput: A sequece of umbers
More informationCSI 5163 (95.573) ALGORITHM ANALYSIS AND DESIGN
CSI 5163 (95.573) ALGORITHM ANALYSIS AND DESIGN CSI 5163 (95.5703) ALGORITHM ANALYSIS AND DESIGN (3 cr.) (T) Topics of curret iterest i the desig ad aalysis of computer algorithms for graphtheoretical
More informationCS583 Lecture 02. Jana Kosecka. some materials here are based on E. Demaine, D. Luebke slides
CS583 Lecture 02 Jaa Kosecka some materials here are based o E. Demaie, D. Luebke slides Previously Sample algorithms Exact ruig time, pseudo-code Approximate ruig time Worst case aalysis Best case aalysis
More informationSorting Algorithms. Algorithms Kyuseok Shim SoEECS, SNU.
Sortig Algorithms Algorithms Kyuseo Shim SoEECS, SNU. Desigig Algorithms Icremetal approaches Divide-ad-Coquer approaches Dyamic programmig approaches Greedy approaches Radomized approaches You are ot
More informationExamples: data compression, path-finding, game-playing, scheduling, bin packing
Algorithms - Basic Cocepts Algorithms so what is a algorithm, ayway? The dictioary defiitio: A algorithm is a well-defied computatioal procedure that takes iput ad produces output. This class will deal
More informationCOMP26120: Introducing Complexity Analysis (2018/19) Lucas Cordeiro
COMP60: Itroduig Complexity Aalysis (08/9) Luas Cordeiro luas.ordeiro@mahester.a.uk Itroduig Complexity Aalysis Textbook: Algorithm Desig ad Appliatios, Goodrih, Mihael T. ad Roberto Tamassia (hapter )
More informationClassification of problem & problem solving strategies. classification of time complexities (linear, logarithmic etc)
Classificatio of problem & problem solvig strategies classificatio of time complexities (liear, arithmic etc) Problem subdivisio Divide ad Coquer strategy. Asymptotic otatios, lower boud ad upper boud:
More informationLecture 3: Asymptotic Analysis + Recurrences
Lecture 3: Asymptotic Aalysis + Recurreces Data Structures ad Algorithms CSE 373 SU 18 BEN JONES 1 Warmup Write a model ad fid Big-O for (it i = 0; i < ; i++) { for (it j = 0; j < i; j++) { System.out.pritl(
More informationCS 332: Algorithms. Linear-Time Sorting. Order statistics. Slide credit: David Luebke (Virginia)
1 CS 332: Algorithms Liear-Time Sortig. Order statistics. Slide credit: David Luebke (Virgiia) Quicksort: Partitio I Words Partitio(A, p, r): Select a elemet to act as the pivot (which?) Grow two regios,
More informationITEC 360 Data Structures and Analysis of Algorithms Spring for n 1
ITEC 360 Data Structures ad Aalysis of Algorithms Sprig 006 1. Prove that f () = 60 + 5 + 1 is Θ ( ). 60 + 5 + 1 60 + 5 + = 66 for 1 Take C 1 = 66 f () = 60 + 5 + 1 is O( ) Sice 60 + 5 + 1 60 for 1 If
More informationDesign and Analysis of Algorithms
Desig ad Aalysis of Algorithms Probabilistic aalysis ad Radomized algorithms Referece: CLRS Chapter 5 Topics: Hirig problem Idicatio radom variables Radomized algorithms Huo Hogwei 1 The hirig problem
More informationChapter 22 Developing Efficient Algorithms
Chapter Developig Efficiet Algorithms 1 Executig Time Suppose two algorithms perform the same task such as search (liear search vs. biary search). Which oe is better? Oe possible approach to aswer this
More informationModel of Computation and Runtime Analysis
Model of Computatio ad Rutime Aalysis Model of Computatio Model of Computatio Specifies Set of operatios Cost of operatios (ot ecessarily time) Examples Turig Machie Radom Access Machie (RAM) PRAM Map
More informationMathematical Foundation. CSE 6331 Algorithms Steve Lai
Mathematical Foudatio CSE 6331 Algorithms Steve Lai Complexity of Algorithms Aalysis of algorithm: to predict the ruig time required by a algorithm. Elemetary operatios: arithmetic & boolea operatios:
More informationModel of Computation and Runtime Analysis
Model of Computatio ad Rutime Aalysis Model of Computatio Model of Computatio Specifies Set of operatios Cost of operatios (ot ecessarily time) Examples Turig Machie Radom Access Machie (RAM) PRAM Map
More informationCS:3330 (Prof. Pemmaraju ): Assignment #1 Solutions. (b) For n = 3, we will have 3 men and 3 women with preferences as follows: m 1 : w 3 > w 1 > w 2
Shiyao Wag CS:3330 (Prof. Pemmaraju ): Assigmet #1 Solutios Problem 1 (a) Cosider iput with me m 1, m,..., m ad wome w 1, w,..., w with the followig prefereces: All me have the same prefereces for wome:
More information4.3 Growth Rates of Solutions to Recurrences
4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.
More informationOPTIMAL ALGORITHMS -- SUPPLEMENTAL NOTES
OPTIMAL ALGORITHMS -- SUPPLEMENTAL NOTES Peter M. Maurer Why Hashig is θ(). As i biary search, hashig assumes that keys are stored i a array which is idexed by a iteger. However, hashig attempts to bypass
More informationCS161 Handout 05 Summer 2013 July 10, 2013 Mathematical Terms and Identities
CS161 Hadout 05 Summer 2013 July 10, 2013 Mathematical Terms ad Idetities Thaks to Ady Nguye ad Julie Tibshirai for their advice o this hadout. This hadout covers mathematical otatio ad idetities that
More informationCSE 4095/5095 Topics in Big Data Analytics Spring 2017; Homework 1 Solutions
CSE 09/09 Topics i ig Data Aalytics Sprig 2017; Homework 1 Solutios Note: Solutios to problems,, ad 6 are due to Marius Nicolae. 1. Cosider the followig algorithm: for i := 1 to α log e do Pick a radom
More informationCS / MCS 401 Homework 3 grader solutions
CS / MCS 401 Homework 3 grader solutios assigmet due July 6, 016 writte by Jāis Lazovskis maximum poits: 33 Some questios from CLRS. Questios marked with a asterisk were ot graded. 1 Use the defiitio of
More informationLecture 9: Hierarchy Theorems
IAS/PCMI Summer Sessio 2000 Clay Mathematics Udergraduate Program Basic Course o Computatioal Complexity Lecture 9: Hierarchy Theorems David Mix Barrigto ad Alexis Maciel July 27, 2000 Most of this lecture
More informationThis Lecture. Divide and Conquer. Merge Sort: Algorithm. Merge Sort Algorithm. MergeSort (Example) - 1. MergeSort (Example) - 2
This Lecture Divide-ad-coquer techique for algorithm desig. Example the merge sort. Writig ad solvig recurreces Divide ad Coquer Divide-ad-coquer method for algorithm desig: Divide: If the iput size is
More informationCOMP285 Midterm Exam Department of Mathematics
COMP85 Midterm Exam Departmet of Mathematics Fall 010/011 - November 8, 010 Name: Studet Number: Please fiish withi 90 miutes. All poits above 100 are cosidered as bous poit. You ca reach maximal 1 poits.
More informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
More informationThis is an introductory course in Analysis of Variance and Design of Experiments.
1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hard-copy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class
More informationCS 332: Algorithms. Quicksort
CS 33: Aorithms Quicsort David Luebe //03 Homewor Assiged today, due ext Wedesday Will be o web page shortly after class Go over ow David Luebe //03 Review: Quicsort Sorts i place Sorts O( ) i the average
More informationSums, products and sequences
Sums, products ad sequeces How to write log sums, e.g., 1+2+ (-1)+ cocisely? i=1 Sum otatio ( sum from 1 to ): i 3 = 1 + 2 + + If =3, i=1 i = 1+2+3=6. The ame ii does ot matter. Could use aother letter
More informationA recurrence equation is just a recursive function definition. It defines a function at one input in terms of its value on smaller inputs.
CS23 Algorithms Hadout #6 Prof Ly Turbak September 8, 200 Wellesley College RECURRENCES This hadout summarizes highlights of CLRS Chapter 4 ad Appedix A (CLR Chapters 3 & 4) Two-Step Strategy for Aalyzig
More informationThe Growth of Functions. Theoretical Supplement
The Growth of Fuctios Theoretical Supplemet The Triagle Iequality The triagle iequality is a algebraic tool that is ofte useful i maipulatig absolute values of fuctios. The triagle iequality says that
More informationTest One (Answer Key)
CS395/Ma395 (Sprig 2005) Test Oe Name: Page 1 Test Oe (Aswer Key) CS395/Ma395: Aalysis of Algorithms This is a closed book, closed otes, 70 miute examiatio. It is worth 100 poits. There are twelve (12)
More informationCSE Introduction to Parallel Processing. Chapter 3. Parallel Algorithm Complexity
Dr. Izadi CSE-40533 Itroductio to Parallel Processig Chapter 3 Parallel Algorithm Complexity Review algorithm complexity ad various complexity classes Itroduce the otios of time ad time-cost optimality
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 informationA Probabilistic Analysis of Quicksort
A Probabilistic Aalysis of Quicsort You are assumed to be familiar with Quicsort. I each iteratio this sortig algorithm chooses a pivot ad the, by performig comparisios with the pivot, splits the remaider
More informationCIS 121 Data Structures and Algorithms with Java Spring Code Snippets and Recurrences Monday, February 4/Tuesday, February 5
CIS 11 Data Structures ad Algorithms with Java Sprig 019 Code Sippets ad Recurreces Moday, February 4/Tuesday, February 5 Learig Goals Practice provig asymptotic bouds with code sippets Practice solvig
More informationCS166 Handout 02 Spring 2018 April 3, 2018 Mathematical Terms and Identities
CS166 Hadout 02 Sprig 2018 April 3, 2018 Mathematical Terms ad Idetities Thaks to Ady Nguye ad Julie Tibshirai for their advice o this hadout. This hadout covers mathematical otatio ad idetities that may
More informationAnalysis of Algorithms -Quicksort-
Aalysis of Algorithms -- Adreas Ermedahl MRTC (Mälardales Real-Time Research Ceter) adreas.ermedahl@mdh.se Autum 2004 Proposed by C.A.R. Hoare i 962 Worst- case ruig time: Θ( 2 ) Expected ruig time: Θ(
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationChapter 2. Asymptotic Notation
Asyptotic Notatio 3 Chapter Asyptotic Notatio Goal : To siplify the aalysis of ruig tie by gettig rid of details which ay be affected by specific ipleetatio ad hardware. [1] The Big Oh (O-Notatio) : It
More informationDesign and Analysis of Algorithms
Desig ad Aalysis of Algorithms CSE 53 Lecture 9 Media ad Order Statistics Juzhou Huag, Ph.D. Departmet of Computer Sciece ad Egieerig Dept. CSE, UT Arligto CSE53 Desig ad Aalysis of Algorithms Medias ad
More informationSpectral Partitioning in the Planted Partition Model
Spectral Graph Theory Lecture 21 Spectral Partitioig i the Plated Partitio Model Daiel A. Spielma November 11, 2009 21.1 Itroductio I this lecture, we will perform a crude aalysis of the performace of
More informationMachine Learning Brett Bernstein
Machie Learig Brett Berstei Week 2 Lecture: Cocept Check Exercises Starred problems are optioal. Excess Risk Decompositio 1. Let X = Y = {1, 2,..., 10}, A = {1,..., 10, 11} ad suppose the data distributio
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More informationSYDE 112, LECTURE 2: Riemann Sums
SYDE, LECTURE : Riema Sums Riema Sums Cosider the problem of determiig the area below the curve f(x) boud betwee two poits a ad b. For simple geometrical fuctios, we ca easily determie this based o ituitio.
More informationMerge and Quick Sort
Merge ad Quick Sort Merge Sort Merge Sort Tree Implemetatio Quick Sort Pivot Item Radomized Quick Sort Adapted from: Goodrich ad Tamassia, Data Structures ad Algorithms i Java, Joh Wiley & So (1998). Ruig
More informationHand Out: Analysis of Algorithms. September 8, Bud Mishra. In general, there can be several algorithms to solve a problem; and one is faced
Had Out Aalysis of Algorithms September 8, 998 Bud Mishra c Mishra, February 9, 986 Itroductio I geeral, there ca be several algorithms to solve a problem; ad oe is faced with the problem of choosig a
More informationAlgorithms and Data Structures Lecture IV
Algorithms ad Data Structures Lecture IV Simoas Šalteis Aalborg Uiversity simas@cs.auc.dk September 5, 00 1 This Lecture Aalyzig the ruig time of recursive algorithms (such as divide-ad-coquer) Writig
More informationCS 171 Lecture Outline October 09, 2008
CS 171 Lecture Outlie October 09, 2008 The followig theorem comes very hady whe calculatig the expectatio of a radom variable that takes o o-egative iteger values. Theorem: Let Y be a radom variable that
More informationHashing and Amortization
Lecture Hashig ad Amortizatio Supplemetal readig i CLRS: Chapter ; Chapter 7 itro; Sectio 7.. Arrays ad Hashig Arrays are very useful. The items i a array are statically addressed, so that isertig, deletig,
More informationWRITTEN ASSIGNMENT 1 ANSWER KEY
CISC 65 Itrodutio Desig ad Aalysis of Algorithms WRITTEN ASSIGNMENT ANSWER KEY. Problem -) I geeral, this problem requires f() = some time period be solve for a value. This a be doe for all ase expet lg
More informationw (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.
2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For
More informationLecture 2 February 8, 2016
MIT 6.854/8.45: Advaced Algorithms Sprig 206 Prof. Akur Moitra Lecture 2 February 8, 206 Scribe: Calvi Huag, Lih V. Nguye I this lecture, we aalyze the problem of schedulig equal size tasks arrivig olie
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationNotes for Lecture 11
U.C. Berkeley CS78: Computatioal Complexity Hadout N Professor Luca Trevisa 3/4/008 Notes for Lecture Eigevalues, Expasio, ad Radom Walks As usual by ow, let G = (V, E) be a udirected d-regular graph with
More informationMachine Learning Theory Tübingen University, WS 2016/2017 Lecture 12
Machie Learig Theory Tübige Uiversity, WS 06/07 Lecture Tolstikhi Ilya Abstract I this lecture we derive risk bouds for kerel methods. We will start by showig that Soft Margi kerel SVM correspods to miimizig
More informationHomework 5 Solutions
Homework 5 Solutios p329 # 12 No. To estimate the chace you eed the expected value ad stadard error. To do get the expected value you eed the average of the box ad to get the stadard error you eed the
More informationAverage-Case Analysis of QuickSort
Average-Case Aalysis of QuickSort Comp 363 Fall Semester 003 October 3, 003 The purpose of this documet is to itroduce the idea of usig recurrece relatios to do average-case aalysis. The average-case ruig
More informationData Structures Lecture 9
Fall 2017 Fag Yu Software Security Lab. Dept. Maagemet Iformatio Systems, Natioal Chegchi Uiversity Data Structures Lecture 9 Midterm o Dec. 7 (9:10-12:00am, 106) Lec 1-9, TextBook Ch1-8, 11,12 How to
More informationNotes for Lecture 5. 1 Grover Search. 1.1 The Setting. 1.2 Motivation. Lecture 5 (September 26, 2018)
COS 597A: Quatum Cryptography Lecture 5 (September 6, 08) Lecturer: Mark Zhadry Priceto Uiversity Scribe: Fermi Ma Notes for Lecture 5 Today we ll move o from the slightly cotrived applicatios of quatum
More informationAlgorithms. Elementary Sorting. Dong Kyue Kim Hanyang University
Algorithms Elemetary Sortig Dog Kyue Kim Hayag Uiversity dqkim@hayag.a.kr Cotets Sortig problem Elemetary sortig algorithms Isertio sort Merge sort Seletio sort Bubble sort Sortig problem Iput A sequee
More informationCSI 2101 Discrete Structures Winter Homework Assignment #4 (100 points, weight 5%) Due: Thursday, April 5, at 1:00pm (in lecture)
CSI 101 Discrete Structures Witer 01 Prof. Lucia Moura Uiversity of Ottawa Homework Assigmet #4 (100 poits, weight %) Due: Thursday, April, at 1:00pm (i lecture) Program verificatio, Recurrece Relatios
More informationAlgorithms 演算法. Multi-threaded Algorithms
演算法 Multi-threaded Professor Chie-Mo James Li 李建模 Graduate Istitute of Electroics Egieerig Natioal aiwa Uiversity Outlie Multithreaded, CH7 7. Basics 7. Matrix Multiplicatio 7.3 Merge sort Leoardo Fiboacci
More informationDiscrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22
CS 70 Discrete Mathematics for CS Sprig 2007 Luca Trevisa Lecture 22 Aother Importat Distributio The Geometric Distributio Questio: A biased coi with Heads probability p is tossed repeatedly util the first
More informationLecture 11: Pseudorandom functions
COM S 6830 Cryptography Oct 1, 2009 Istructor: Rafael Pass 1 Recap Lecture 11: Pseudoradom fuctios Scribe: Stefao Ermo Defiitio 1 (Ge, Ec, Dec) is a sigle message secure ecryptio scheme if for all uppt
More information1 Review and Overview
DRAFT a fial versio will be posted shortly CS229T/STATS231: Statistical Learig Theory Lecturer: Tegyu Ma Lecture #3 Scribe: Migda Qiao October 1, 2013 1 Review ad Overview I the first half of this course,
More informationCS 253: Algorithms. Syllabus. Chapter 1. Appendix A
CS 5: Algorithms Syllabus Chapter Appedi A Syllabus Istructor : Firet Ercal - Office: CS 4 Phoe: 4-4857 E-mail & URL : ercal@mst.edu http://web.mst.edu/~ercal/ide.html Meetig Times : :50 pm. M W F Office
More information1 Hash tables. 1.1 Implementation
Lecture 8 Hash Tables, Uiversal Hash Fuctios, Balls ad Bis Scribes: Luke Johsto, Moses Charikar, G. Valiat Date: Oct 18, 2017 Adapted From Virgiia Williams lecture otes 1 Hash tables A hash table is a
More informationIntro to Learning Theory
Lecture 1, October 18, 2016 Itro to Learig Theory Ruth Urer 1 Machie Learig ad Learig Theory Comig soo 2 Formal Framework 21 Basic otios I our formal model for machie learig, the istaces to be classified
More informationLecture 2: Uncomputability and the Haling Problem
CSE 200 Computability ad Complexity Wedesday, April 3, 2013 Lecture 2: Ucomputability ad the Halig Problem Istructor: Professor Shachar Lovett Scribe: Dogcai She 1 The Uiversal Turig Machie I the last
More informationDivide and Conquer. 1 Overview. 2 Multiplying Bit Strings. COMPSCI 330: Design and Analysis of Algorithms 1/19/2016 and 1/21/2016
COMPSCI 330: Desig ad Aalysis of Algorithms 1/19/2016 ad 1/21/2016 Lecturer: Debmalya Paigrahi Divide ad Coquer Scribe: Tiaqi Sog 1 Overview I this lecture, a importat algorithm desig techique called divide-ad-coquer
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationSome examples of vector spaces
Roberto s Notes o Liear Algebra Chapter 11: Vector spaces Sectio 2 Some examples of vector spaces What you eed to kow already: The te axioms eeded to idetify a vector space. What you ca lear here: Some
More informationUnderstanding Samples
1 Will Moroe CS 109 Samplig ad Bootstrappig Lecture Notes #17 August 2, 2017 Based o a hadout by Chris Piech I this chapter we are goig to talk about statistics calculated o samples from a populatio. We
More informationMA131 - Analysis 1. Workbook 2 Sequences I
MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................
More informationDepartment of Informatics Prof. Dr. Michael Böhlen Binzmühlestrasse Zurich Phone:
Departmet of Iformatics Prof. Dr. Michael Böhle Bizmühlestrasse 14 8050 Zurich Phoe: +41 44 635 4333 Email: boehle@ifi.uzh.ch Iformatik II Midterm1 Sprig 018 3.03.018 Advice You have 90 miutes to complete
More informationSTAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6)
STAT 350 Hadout 9 Samplig Distributio, Cetral Limit Theorem (6.6) A radom sample is a sequece of radom variables X, X 2,, X that are idepedet ad idetically distributed. o This property is ofte abbreviated
More informationLecture Note 8 Point Estimators and Point Estimation Methods. MIT Spring 2006 Herman Bennett
Lecture Note 8 Poit Estimators ad Poit Estimatio Methods MIT 14.30 Sprig 2006 Herma Beett Give a parameter with ukow value, the goal of poit estimatio is to use a sample to compute a umber that represets
More informationA New Method to Order Functions by Asymptotic Growth Rates Charlie Obimbo Dept. of Computing and Information Science University of Guelph
A New Method to Order Fuctios by Asymptotic Growth Rates Charlie Obimbo Dept. of Computig ad Iformatio Sciece Uiversity of Guelph ABSTRACT A ew method is described to determie the complexity classes of
More informationLecture 5: April 17, 2013
TTIC/CMSC 350 Mathematical Toolkit Sprig 203 Madhur Tulsiai Lecture 5: April 7, 203 Scribe: Somaye Hashemifar Cheroff bouds recap We recall the Cheroff/Hoeffdig bouds we derived i the last lecture idepedet
More informationIntroduction to Algorithms 6.046J/18.401J LECTURE 3 Divide and conquer Binary search Powering a number Fibonacci numbers Matrix multiplication
Itroductio to Algorithms 6.046J/8.40J LECTURE 3 Divide ad coquer Biary search Powerig a umber Fiboacci umbers Matrix multiplicatio Strasse s algorithm VLSI tree layout Prof. Charles E. Leiserso The divide-ad-coquer
More informationSkip Lists. Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015 S 3 S S 1
Presetatio for use with the textbook, Algorithm Desig ad Applicatios, by M. T. Goodrich ad R. Tamassia, Wiley, 2015 Skip Lists S 3 15 15 23 10 15 23 36 Skip Lists 1 What is a Skip List A skip list for
More informationLecture 16: Monotone Formula Lower Bounds via Graph Entropy. 2 Monotone Formula Lower Bounds via Graph Entropy
15-859: Iformatio Theory ad Applicatios i TCS CMU: Sprig 2013 Lecture 16: Mootoe Formula Lower Bouds via Graph Etropy March 26, 2013 Lecturer: Mahdi Cheraghchi Scribe: Shashak Sigh 1 Recap Graph Etropy:
More informationDivide & Conquer. Divide-and-conquer algorithms. Conventional product of polynomials. Conventional product of polynomials.
Divide-ad-coquer algorithms Divide & Coquer Strategy: Divide the problem ito smaller subproblems of the same type of problem Solve the subproblems recursively Combie the aswers to solve the origial problem
More informationMatriculation number: You have 90 minutes to complete the exam of InformatikIIb. The following rules apply:
Departmet of Iformatics Prof. Dr. Michael Böhle Bizmühlestrasse 14 8050 Zurich Phoe: +41 44 635 4333 Email: boehle@ifi.uzh.ch AlgoDat Midterm1 Sprig 016 08.04.016 Name: Matriculatio umber: Advice You have
More informationMechatronics. Time Response & Frequency Response 2 nd -Order Dynamic System 2-Pole, Low-Pass, Active Filter
Time Respose & Frequecy Respose d -Order Dyamic System -Pole, Low-Pass, Active Filter R 4 R 7 C 5 e i R 1 C R 3 - + R 6 - + e out Assigmet: Perform a Complete Dyamic System Ivestigatio of the Two-Pole,
More information18.440, March 9, Stirling s formula
Stirlig s formula 8.44, March 9, 9 The factorial fuctio! is importat i evaluatig biomial, hypergeometric, ad other probabilities. If is ot too large,! ca be computed directly, by calculators or computers.
More informationRead through these prior to coming to the test and follow them when you take your test.
Math 143 Sprig 2012 Test 2 Iformatio 1 Test 2 will be give i class o Thursday April 5. Material Covered The test is cummulative, but will emphasize the recet material (Chapters 6 8, 10 11, ad Sectios 12.1
More informationRandomized Algorithms I, Spring 2018, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 25, 2018)
Radomized Algorithms I, Sprig 08, Departmet of Computer Sciece, Uiversity of Helsiki Homework : Solutios Discussed Jauary 5, 08). Exercise.: Cosider the followig balls-ad-bi game. We start with oe black
More information6.046 Recitation 5: Binary Search Trees Bill Thies, Fall 2004 Outline
6.046 Recitatio 5: Biary Search Trees Bill Thies, Fall 2004 Outlie My cotact iformatio: Bill Thies thies@mit.edu Office hours: Sat 1-3pm, 36-153 Recitatio website: http://cag.lcs.mit.edu/~thies/6.046/
More informationVector Quantization: a Limiting Case of EM
. Itroductio & defiitios Assume that you are give a data set X = { x j }, j { 2,,, }, of d -dimesioal vectors. The vector quatizatio (VQ) problem requires that we fid a set of prototype vectors Z = { z
More informationSimple Random Sampling!
Simple Radom Samplig! Professor Ro Fricker! Naval Postgraduate School! Moterey, Califoria! Readig:! 3/26/13 Scheaffer et al. chapter 4! 1 Goals for this Lecture! Defie simple radom samplig (SRS) ad discuss
More informationPower and Type II Error
Statistical Methods I (EXST 7005) Page 57 Power ad Type II Error Sice we do't actually kow the value of the true mea (or we would't be hypothesizig somethig else), we caot kow i practice the type II error
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More information