Dynamic Programming. Sequence Of Decisions
|
|
- Rebecca Blankenship
- 5 years ago
- Views:
Transcription
1 Dyamic Programmig Sequece of decisios. Problem state. Priciple of optimality. Dyamic Programmig Recurrece Equatios. Solutio of recurrece equatios. Sequece Of Decisios As i the greedy method, the solutio to a problem is viewed as the result of a sequece of decisios. Ulike the greedy method, decisios are ot made i a greedy ad bidig maer.
2 0/1 Kapsack Problem Let x i = 1 whe item i is selected ad let x i = 0 whe item i is ot selected. maximize p i x i i = 1 subject to w i x i <= c i = 1 ad x i = 0 or 1 for all i All profits ad weights are positive. Sequece Of Decisios Decide the x i values i the order x 1, x 2, x 3,, x. Decide the x i values i the order x, x -1, x -2,, x 1. Decide the x i values i the order x 1, x, x 2, x -1, Or ay other order.
3 Problem State The state of the 0/1 kapsack problem is give by ƒ the weights ad profits of the available items ƒ the capacity of the kapsack Whe a decisio o oe of the x i values is made, the problem state chages. ƒ item i is o loger available ƒ the remaiig kapsack capacity may be less Problem State Suppose that decisios are made i the order x 1, x 2, x 3,, x. The iitial state of the problem is described by the pair (1, c). ƒ Items 1 through are available (the weights, profits ad are implicit). ƒ The available kapsack capacity is c. Followig the first decisio the state becomes oe of the followig: ƒ (2, c) whe the decisio is to set x 1 = 0. ƒ (2, c-w 1 ) whe the decisio is to set x 1 = 1.
4 Problem State Suppose that decisios are made i the order x, x -1, x -2,, x 1. The iitial state of the problem is described by the pair (, c). ƒ Items 1 through are available (the weights, profits ad first item idex are implicit). ƒ The available kapsack capacity is c. Followig the first decisio the state becomes oe of the followig: ƒ (-1, c) whe the decisio is to set x = 0. ƒ (-1, c-w ) whe the decisio is to set x = 1. Priciple Of Optimality A optimal solutio satisfies the followig property: ƒ No matter what the first decisio, the remaiig decisios are optimal with respect to the state that results from this decisio. Dyamic programmig may be used oly whe the priciple of optimality holds.
5 0/1 Kapsack Problem Suppose that decisios are made i the order x 1, x 2, x 3,, x. Let x 1 = a 1, x 2 = a 2, x 3 = a 3,, x = a be a optimal solutio. If a 1 = 0, the followig the first decisio the state is (2, c). a 2, a 3,, a must be a optimal solutio to the kapsack istace give by the state (2,c). x 1 = a 1 = 0 maximize p i x i i = 2 subject to w i x i <= c i = 2 ad x i = 0 or 1 for all i If ot, this istace has a better solutio b 2, b 3,, b. p i b i > p i a i i = 2 i = 2
6 x 1 = a 1 = 0 x 1 = a 1, x 2 = b 2, x 3 = b 3,, x = b is a better solutio to the origial istace tha is x 1 = a 1, x 2 = a 2, x 3 = a 3,, x = a. So x 1 = a 1, x 2 = a 2, x 3 = a 3,, x = a caot be a optimal solutio a cotradictio with the assumptio that it is optimal. x 1 = a 1 = 1 Next, cosider the case a 1 = 1. Followig the first decisio the state is (2, c-w 1 ). a 2, a 3,, a must be a optimal solutio to the kapsack istace give by the state (2,c -w 1 ).
7 x 1 = a 1 = 1 maximize p i x i i = 2 subject to w i x i <= c- w 1 i = 2 ad x i = 0 or 1 for all i If ot, this istace has a better solutio b 2, b 3,, b. p i b i > p i a i i = 2 i = 2 x 1 = a 1 = 1 x 1 = a 1, x 2 = b 2, x 3 = b 3,, x = b is a better solutio to the origial istace tha is x 1 = a 1, x 2 = a 2, x 3 = a 3,, x = a. So x 1 = a 1, x 2 = a 2, x 3 = a 3,, x = a caot be a optimal solutio a cotradictio with the assumptio that it is optimal.
8 0/1 Kapsack Problem Therefore, o matter what the first decisio, the remaiig decisios are optimal with respect to the state that results from this decisio. The priciple of optimality holds ad dyamic programmig may be applied. Dyamic Programmig Recurrece Let f(i,y) be the profit value of the optimal solutio to the kapsack istace defied by the state (i,y). ƒ Items i through are available. ƒ Available capacity is y. For the time beig assume that we wish to determie oly the value of the best solutio. ƒ Later we will worry about determiig the x i s that yield this maximum value. Uder this assumptio, our task is to determie f(1,c).
9 Dyamic Programmig Recurrece f(,y) is the value of the optimal solutio to the kapsack istace defied by the state (,y). ƒ Oly item is available. ƒ Available capacity is y. If w <= y, f(,y) = p. If w > y, f(,y) = 0. Dyamic Programmig Recurrece Suppose that i <. f(i,y) is the value of the optimal solutio to the kapsack istace defied by the state (i,y). ƒ Items i through are available. ƒ Available capacity is y. Suppose that i the optimal solutio for the state (i,y), the first decisio is to set x i = 0. From the priciple of optimality (we have show that this priciple holds for the kapsack problem), it follows that f(i,y) = f(i+1,y).
10 Dyamic Programmig Recurrece The oly other possibility for the first decisio is x i = 1. The case x i = 1 ca arise oly whe y >= w i. From the priciple of optimality, it follows that f(i,y) = f(i+1,y-w i ) + p i. Combiig the two cases, we get ƒ f(i,y) = f(i+1,y) wheever y < w i. ƒ f(i,y) = max{f(i+1,y), f(i+1,y-w i ) + p i }, y >= w i. Recursive Code f(i,y) */ private static it f(it i, it y) { if (i == ) retur (y < w[])? 0 : p[]; if (y < w[i]) retur f(i + 1, y); retur Math.max(f(i + 1, y), f(i + 1, y - w[i]) + p[i]); }
11 Recursio Tree f(1,c) f(2,c) f(2,c-w 1 ) f(3,c) f(3,c-w 2 ) f(3,c-w 1 ) f(3,c-w 1 w 2 ) f(4,c) f(4,c-w 3 ) f(4,c-w 2 ) f(4,c-w 1 w 3 ) f(5,c) f(5,c-w 1 w 3 w 4 ) Time Complexity Let t() be the time required whe items are available. t(0) = t(1) = a, where a is a costat. Whe t > 1, t() <= 2t(-1) + b, where b is a costat. t() = O(2 ). Solvig dyamic programmig recurreces recursively ca be hazardous to ru time.
12 Reducig Ru Time f(1,c) f(2,c) f(2,c-w 1 ) f(3,c) f(3,c-w 2 ) f(3,c-w 1 ) f(3,c-w 1 w 2 ) f(4,c) f(4,c-w 3 ) f(4,c-w 2 ) f(5,c) f(4,c-w 1 w 3 ) f(5,c-w 1 w 3 w 4 ) Time Complexity Level i of the recursio tree has up to 2 i-1 odes. At each such ode a f(i,y) is computed. Several odes may compute the same f(i,y). We ca save time by ot recomputig already computed f(i,y)s. Save computed f(i,y)s i a dictioary. ƒ Key is (i, y) value. ƒ f(i, y) is computed recursively oly whe (i,y) is ot i the dictioary. ƒ Otherwise, the dictioary value is used.
13 Iteger Weights Assume that each weight is a iteger. The kapsack capacity c may also be assumed to be a iteger. Oly f(i,y)s with 1 <= i <= ad 0 <= y <= c are of iterest. Eve though level i of the recursio tree has up to 2 i-1 odes, at most c+1 represet differet f(i,y)s. Iteger Weights Dictioary Use a array farray[][] as the dictioary. farray[1:][0:c] farray[i][y] = -1 iff f(i,y) ot yet computed. This iitializatio is doe before the recursive method is ivoked. The iitializatio takes O(c) time.
14 No Recomputatio Code private static it f(it i, it y) { if (farray[i][y] >= 0) retur farray[i][y]; if (i == ) {farray[i][y] = (y < w[])? 0 : p[]; retur farray[i][y];} if (y < w[i]) farray[i][y] = f(i + 1, y); else farray[i][y] = Math.max(f(i + 1, y), f(i + 1, y - w[i]) + p[i]); retur farray[i][y]; } Time Complexity t() = O(c). Aalysis doe i text. Good whe c is small relative to 2. = 3, c = w = [100102, , 6327] p = [102, 505, 5] 2 = 8 c =
Dynamic Programming. Sequence Of Decisions. 0/1 Knapsack Problem. Sequence Of Decisions
Dyamic Programmig Sequece Of Decisios Sequece of decisios. Problem state. Priciple of optimality. Dyamic Programmig Recurrece Equatios. Solutio of recurrece equatios. As i the greedy method, the solutio
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 informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationRecursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
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 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 informationand each factor on the right is clearly greater than 1. which is a contradiction, so n must be prime.
MATH 324 Summer 200 Elemetary Number Theory Solutios to Assigmet 2 Due: Wedesday July 2, 200 Questio [p 74 #6] Show that o iteger of the form 3 + is a prime, other tha 2 = 3 + Solutio: If 3 + is a prime,
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 informationA 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,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationMarkov Decision Processes
Markov Decisio Processes Defiitios; Statioary policies; Value improvemet algorithm, Policy improvemet algorithm, ad liear programmig for discouted cost ad average cost criteria. Markov Decisio Processes
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 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 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 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 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 informationDifferent kinds of Mathematical Induction
Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true}
More informationSolutions for the Exam 9 January 2012
Mastermath ad LNMB Course: Discrete Optimizatio Solutios for the Exam 9 Jauary 2012 Utrecht Uiversity, Educatorium, 15:15 18:15 The examiatio lasts 3 hours. Gradig will be doe before Jauary 23, 2012. Studets
More informationOptimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
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 informationA New Solution Method for the Finite-Horizon Discrete-Time EOQ Problem
This is the Pre-Published Versio. A New Solutio Method for the Fiite-Horizo Discrete-Time EOQ Problem Chug-Lu Li Departmet of Logistics The Hog Kog Polytechic Uiversity Hug Hom, Kowloo, Hog Kog Phoe: +852-2766-7410
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 informationsubject to A 1 x + A 2 y b x j 0, j = 1,,n 1 y j = 0 or 1, j = 1,,n 2
Additioal Brach ad Boud Algorithms 0-1 Mixed-Iteger Liear Programmig The brach ad boud algorithm described i the previous sectios ca be used to solve virtually all optimizatio problems cotaiig iteger variables,
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 informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationAnalytic Continuation
Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for
More informationSection 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations
Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?
More informationRead carefully the instructions on the answer book and make sure that the particulars required are entered on each answer book.
THE UNIVERSITY OF WARWICK FIRST YEAR EXAMINATION: Jauary 2009 Aalysis I Time Allowed:.5 hours Read carefully the istructios o the aswer book ad make sure that the particulars required are etered o each
More informationA MATHEMATICA PACKAGE FOR COMPUTING ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF P-FINITE RECURRENCE EQUATIONS. 1. The Problem
A MATHEMATICA PACKAGE FOR COMPUTING ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF P-FINITE RECURRENCE EQUATIONS MANUEL KAUERS Abstract. We describe a simple package for computig a fudametal system of certai formal
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 informationChapter 6. Advanced Counting Techniques
Chapter 6 Advaced Coutig Techiques 6.: Recurrece Relatios Defiitio: A recurrece relatio for the sequece {a } is a equatio expressig a i terms of oe or more of the previous terms of the sequece: a,a2,a3,,a
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 informationFundamental Algorithms
Fudametal Algorithms Chapter 2b: Recurreces Michael Bader Witer 2014/15 Chapter 2b: Recurreces, Witer 2014/15 1 Recurreces Defiitio A recurrece is a (i-equality that defies (or characterizes a fuctio i
More informationLinear Programming and the Simplex Method
Liear Programmig ad the Simplex ethod Abstract This article is a itroductio to Liear Programmig ad usig Simplex method for solvig LP problems i primal form. What is Liear Programmig? Liear Programmig is
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 informationLangford s Problem. Moti Ben-Ari. Department of Science Teaching. Weizmann Institute of Science.
Lagford s Problem Moti Be-Ari Departmet of Sciece Teachig Weizma Istitute of Sciece http://www.weizma.ac.il/sci-tea/beari/ c 017 by Moti Be-Ari. This work is licesed uder the Creative Commos Attributio-ShareAlike
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 information2.4 - Sequences and Series
2.4 - Sequeces ad Series Sequeces A sequece is a ordered list of elemets. Defiitio 1 A sequece is a fuctio from a subset of the set of itegers (usually either the set 80, 1, 2, 3,... < or the set 81, 2,
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 informationUnit 6: Sequences and Series
AMHS Hoors Algebra 2 - Uit 6 Uit 6: Sequeces ad Series 26 Sequeces Defiitio: A sequece is a ordered list of umbers ad is formally defied as a fuctio whose domai is the set of positive itegers. It is commo
More informationCSE 202 Homework 1 Matthias Springer, A Yes, there does always exist a perfect matching without a strong instability.
CSE 0 Homework 1 Matthias Spriger, A9950078 1 Problem 1 Notatio a b meas that a is matched to b. a < b c meas that b likes c more tha a. Equality idicates a tie. Strog istability Yes, there does always
More informationFind a formula for the exponential function whose graph is given , 1 2,16 1, 6
Math 4 Activity (Due by EOC Apr. ) Graph the followig epoetial fuctios by modifyig the graph of f. Fid the rage of each fuctio.. g. g. g 4. g. g 6. g Fid a formula for the epoetial fuctio whose graph is
More informationHoggatt and King [lo] defined a complete sequence of natural numbers
REPRESENTATIONS OF N AS A SUM OF DISTINCT ELEMENTS FROM SPECIAL SEQUENCES DAVID A. KLARNER, Uiversity of Alberta, Edmoto, Caada 1. INTRODUCTION Let a, I deote a sequece of atural umbers which satisfies
More informationMath 2112 Solutions Assignment 5
Math 2112 Solutios Assigmet 5 5.1.1 Idicate which of the followig relatioships are true ad which are false: a. Z Q b. R Q c. Q Z d. Z Z Z e. Q R Q f. Q Z Q g. Z R Z h. Z Q Z a. True. Every positive iteger
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 informationMATH 324 Summer 2006 Elementary Number Theory Solutions to Assignment 2 Due: Thursday July 27, 2006
MATH 34 Summer 006 Elemetary Number Theory Solutios to Assigmet Due: Thursday July 7, 006 Departmet of Mathematical ad Statistical Scieces Uiversity of Alberta Questio [p 74 #6] Show that o iteger of the
More informationMath 140A Elementary Analysis Homework Questions 1
Math 14A Elemetary Aalysis Homewor Questios 1 1 Itroductio 1.1 The Set N of Natural Numbers 1 Prove that 1 2 2 2 2 1 ( 1(2 1 for all atural umbers. 2 Prove that 3 11 (8 5 4 2 for all N. 4 (a Guess a formula
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 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 informationIterative Techniques for Solving Ax b -(3.8). Assume that the system has a unique solution. Let x be the solution. Then x A 1 b.
Iterative Techiques for Solvig Ax b -(8) Cosider solvig liear systems of them form: Ax b where A a ij, x x i, b b i Assume that the system has a uique solutio Let x be the solutio The x A b Jacobi ad Gauss-Seidel
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 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 informationInteger Programming (IP)
Iteger Programmig (IP) The geeral liear mathematical programmig problem where Mied IP Problem - MIP ma c T + h Z T y A + G y + y b R p + vector of positive iteger variables y vector of positive real variables
More informationOptimization Methods MIT 2.098/6.255/ Final exam
Optimizatio Methods MIT 2.098/6.255/15.093 Fial exam Date Give: December 19th, 2006 P1. [30 pts] Classify the followig statemets as true or false. All aswers must be well-justified, either through a short
More informationRADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify
Example 1 Simplify 1.2A Radical Operatios a) 4 2 b) 16 1 2 c) 16 d) 2 e) 8 1 f) 8 What is the relatioship betwee a, b, c? What is the relatioship betwee d, e, f? If x = a, the x = = th root theorems: RADICAL
More informationMath 475, Problem Set #12: Answers
Math 475, Problem Set #12: Aswers A. Chapter 8, problem 12, parts (b) ad (d). (b) S # (, 2) = 2 2, sice, from amog the 2 ways of puttig elemets ito 2 distiguishable boxes, exactly 2 of them result i oe
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 informationMassachusetts Institute of Technology
6.0/6.3: Probabilistic Systems Aalysis (Fall 00) Problem Set 8: Solutios. (a) We cosider a Markov chai with states 0,,, 3,, 5, where state i idicates that there are i shoes available at the frot door i
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 information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationIP Reference guide for integer programming formulations.
IP Referece guide for iteger programmig formulatios. by James B. Orli for 15.053 ad 15.058 This documet is iteded as a compact (or relatively compact) guide to the formulatio of iteger programs. For more
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationLINEAR RECURSION RELATIONS - LESSON FOUR SECOND-ORDER LINEAR RECURSION RELATIONS
LINEAR RECURSION RELATIONS - LESSON FOUR SECOND-ORDER LINEAR RECURSION RELATIONS BROTHER ALFRED BROUSSEAU St. Mary's College, Califoria Give a secod-order liear recursio relatio (.1) T. 1 = a T + b T 1,
More information5. Solving recurrences
5. Solvig recurreces Time Complexity Alysis of Merge Sort T( ) 0 if 1 2T ( / 2) otherwise sortig oth hlves mergig Q. How to prove tht the ru-time of merge sort is O( )? A. 2 Time Complexity Alysis of Merge
More informationa 2 +b 2 +c 2 ab+bc+ca.
All Problems o the Prize Exams Sprig 205 The source for each problem is listed below whe available; but eve whe the source is give, the formulatio of the problem may have bee chaged. Solutios for the problems
More information[ 47 ] then T ( m ) is true for all n a. 2. The greatest integer function : [ ] is defined by selling [ x]
[ 47 ] Number System 1. Itroductio Pricile : Let { T ( ) : N} be a set of statemets, oe for each atural umber. If (i), T ( a ) is true for some a N ad (ii) T ( k ) is true imlies T ( k 1) is true for all
More information) n. ALG 1.3 Deterministic Selection and Sorting: Problem P size n. Examples: 1st lecture's mult M(n) = 3 M ( È
Algorithms Professor Joh Reif ALG 1.3 Determiistic Selectio ad Sortig: (a) Selectio Algorithms ad Lower Bouds (b) Sortig Algorithms ad Lower Bouds Problem P size fi divide ito subproblems size 1,..., k
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationPrinciple Of Superposition
ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give
More informationV. Adamchik 1. Recursions. Victor Adamchik Fall of x n1. x n 2. Here are a few first values of the above sequence (coded in Mathematica)
V. Adamchik Recursios Victor Adamchik Fall of 2005 Pla. Covergece of sequeces 2. Fractals 3. Coutig biary trees Covergece of Sequeces I the previous lecture we cosidered a cotiued fractio for 2 : 2 This
More informationIt is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.
Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable
More information1. Introduction. Notice that we have taken the terms of the harmonic series, and subtracted them from
06/5/00 EAANGING TEM O A HAMONIC-LIKE EIE Mathematics ad Computer Educatio, 35(00), pp 36-39 Thomas J Osler Mathematics Departmet owa Uiversity Glassboro, NJ 0808 osler@rowaedu Itroductio The commutative
More informationHomework 2. Show that if h is a bounded sesquilinear form on the Hilbert spaces X and Y, then h has the representation
omework 2 1 Let X ad Y be ilbert spaces over C The a sesquiliear form h o X Y is a mappig h : X Y C such that for all x 1, x 2, x X, y 1, y 2, y Y ad all scalars α, β C we have (a) h(x 1 + x 2, y) h(x
More informationMA131 - Analysis 1. Workbook 3 Sequences II
MA3 - Aalysis Workbook 3 Sequeces II Autum 2004 Cotets 2.8 Coverget Sequeces........................ 2.9 Algebra of Limits......................... 2 2.0 Further Useful Results........................
More informationRecursive Algorithm for Generating Partitions of an Integer. 1 Preliminary
Recursive Algorithm for Geeratig Partitios of a Iteger Sug-Hyuk Cha Computer Sciece Departmet, Pace Uiversity 1 Pace Plaza, New York, NY 10038 USA scha@pace.edu Abstract. This article first reviews the
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 informationMath 210A Homework 1
Math 0A Homework Edward Burkard Exercise. a) State the defiitio of a aalytic fuctio. b) What are the relatioships betwee aalytic fuctios ad the Cauchy-Riema equatios? Solutio. a) A fuctio f : G C is called
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
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 informationLinearly Independent Sets, Bases. Review. Remarks. A set of vectors,,, in a vector space is said to be linearly independent if the vector equation
Liearly Idepedet Sets Bases p p c c p Review { v v vp} A set of vectors i a vector space is said to be liearly idepedet if the vector equatio cv + c v + + c has oly the trivial solutio = = { v v vp} The
More informationMathematics review for CSCI 303 Spring Department of Computer Science College of William & Mary Robert Michael Lewis
Mathematics review for CSCI 303 Sprig 019 Departmet of Computer Sciece College of William & Mary Robert Michael Lewis Copyright 018 019 Robert Michael Lewis Versio geerated: 13 : 00 Jauary 17, 019 Cotets
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 informationProperties and Tests of Zeros of Polynomial Functions
Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by
More informationStochastic Matrices in a Finite Field
Stochastic Matrices i a Fiite Field Abstract: I this project we will explore the properties of stochastic matrices i both the real ad the fiite fields. We first explore what properties 2 2 stochastic matrices
More informationCOS 341 Discrete Mathematics. Exponential Generating Functions and Recurrence Relations
COS 341 Discrete Mathematics Epoetial Geeratig Fuctios ad Recurrece Relatios 1 Tetbook? 1 studets said they eeded the tetbook, but oly oe studet bought the tet from Triagle. If you do t have the book,
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationSigma notation. 2.1 Introduction
Sigma otatio. Itroductio We use sigma otatio to idicate the summatio process whe we have several (or ifiitely may) terms to add up. You may have see sigma otatio i earlier courses. It is used to idicate
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 informationProbability theory and mathematical statistics:
N.I. Lobachevsky State Uiversity of Nizhi Novgorod Probability theory ad mathematical statistics: Law of Total Probability. Associate Professor A.V. Zorie Law of Total Probability. 1 / 14 Theorem Let H
More information( ) ( ), (S3) ( ). (S4)
Ultrasesitivity i phosphorylatio-dephosphorylatio cycles with little substrate: Supportig Iformatio Bruo M.C. Martis, eter S. Swai 1. Derivatio of the equatios associated with the mai model From the differetial
More informationsubcaptionfont+=small,labelformat=parens,labelsep=space,skip=6pt,list=0,hypcap=0 subcaption ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, 2/16/2016
subcaptiofot+=small,labelformat=pares,labelsep=space,skip=6pt,list=0,hypcap=0 subcaptio ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, /6/06. Self-cojugate Partitios Recall that, give a partitio λ, we may
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 informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More information3.1 & 3.2 SEQUENCES. Definition 3.1: A sequence is a function whose domain is the positive integers (=Z ++ )
3. & 3. SEQUENCES Defiitio 3.: A sequece is a fuctio whose domai is the positive itegers (=Z ++ ) Examples:. f() = for Z ++ or, 4, 6, 8, 0,. a = +/ or, ½, / 3, ¼, 3. b = /² or, ¼, / 9, 4. c = ( ) + or
More informationROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND
Pacific-Asia Joural of Mathematics, Volume 5, No., Jauary-Jue 20 ROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND SHAKEEL JAVAID, Z. H. BAKHSHI & M. M. KHALID ABSTRACT: I this paper, the roll cuttig problem
More informationYou may work in pairs or purely individually for this assignment.
CS 04 Problem Solvig i Computer Sciece OOC Assigmet 6: Recurreces You may work i pairs or purely idividually for this assigmet. Prepare your aswers to the followig questios i a plai ASCII text file or
More informationDATA STRUCTURES I, II, III, AND IV
Data structures DATA STRUCTURES I, II, III, AND IV I. Amortized Aalysis II. Biary ad Biomial Heaps III. Fiboacci Heaps IV. Uio Fid Static problems. Give a iput, produce a output. Ex. Sortig, FFT, edit
More information