Taking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling


 Buck Blair
 1 years ago
 Views:
Transcription
1 Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria {maheswar, tambe, bowrig, jppearce, DCOP Formulatios for DiMES: Proofs of Propositios TSA Time Slots as ariables): This method reflects a atural first step whe cosiderig schedulig issues. Let us defie a DCOP where a variable x t) represets the th resource s tth time slot. Thus, we have N T variables. Each variable ca a tae o a value of the idex of a evet for which it is a required resource, or the value 0 to idicate that o evet will be assiged for that particular time slot: x t) {0} { {1,..., K} : R A }. It is atural to distribute the variables i a maer such that {x 1),..., x T)} belog to a aget represetig the schedule of the th resource. Propositio 1 The DCOP formulatio with time slots as variables, where the costrait betwee variables x 1 t 1 ) ad x 2 t 2 ) whe x 1 t 1 ) = ad x 2 t 2 ) = 2 taes o the utility f 1, t 1, ; 2, t 2, 2 ) = ) ) ) I {1 2 } 1 I{t1 t 2 } 1 I{P1 P 2 A {1,...,K}} fiter, 2 ) + I {1 = 2 } 1 I{t1 =t 2 } fitra 1 ; t 1, ; t 2, 2 ) where the iteraget costrait utilities are f iter, 2 ) = MI {1 2 }I {A A 2 } ad itraaget costraits fully coected amog the time slots of a sigle aget) for t 1 < t 2 w.l.o.g. are f itra ; t 1, ; t 2, 2 ) = M 0, t 2 t 1 < L, 2 M 0, t 2 t 1 L, 2 = g; t 1, ; t 2, 2 ) otherwise where g; t 1, ; t 2, 2 ) = 1 [ 0 t 1 ) ) I {1 0} + 2 i 0 T 1 t 2) ) ] I {2 0} with M > NT max where N is the umber of participats, T is the umber of time slots ad max = max,, yields the optimal solutio to the Distributed MultiEvet Schedulig DiMES) problem. ) Proof. The fuctio f ca be aalyzed as follows. The first term i the factor I {1 2 } 1 I{t1 t 2 } idetifies a iteraget costrait ad the secod term reflects that iteraget costraits exist oly across idetical time slots. The factor 1 I {t1 t 2 }) implies that a iteraget li exists betwee x1 t) ad x 2 t), oly if there exists a evet that requires both P 1 ad P 2 as attedees. The factor I {1 = 2 } idetifies a itraaget costrait ad subsequetly ) 1 I {t1 =t 2 } expresses that there is o costrait from a variable to itself. The fuctio f iter ) idicates that a pealty of M is assessed if the agets assig differet evets for the same time 2 ) ad the evets force a participat to be at two evets at the same time A A 2 ). 1 1 For otatioal cosistecy i the situatio where a agets decides ot to assig a evet to a particular time slot, we defie A 0 :=. 1
2 The fuctio f itra ) assesses a pealty to esure that at least L slots must be assiged cotiguously whe evet E is scheduled 0, t 2 t 1 < L, 2 ). Also, a pealty uder the secod coditio 0, t 2 t 1 L, 2 = ) is assessed to esure that o more tha L slots are assiged ad also to esure that the same evet is ot scheduled twice. If oe of these coditios are met, the utility o the costrait is the differece i the values for attedig the evets ad the value of eepig the time slots uassiged divided by T 1 the umber of outgoig itraaget lis). The sum of all costrait utilities excludig the pealties) is N T 1 =1 s=1 t=s+1 T g; s, x i s); t, x t)) N T 1 T =1 s=1 t=s+1 2 max N T 1 = =1 N T 1 T =1 s=1 t=s+1 1 [ x i s) T 1 ] + x it) TT 1) 2 max 2 T 1 = NT max = M Thus, the total utility if a pealty is icurred is opositive. Cosequetly, it would be better to assig a value of 0 to all variables ad obtai a global utility of zero tha cosider a solutio with a pealty. This implies that ay optimal solutios to the DCOP will ot icur ay pealties. The absece of iteraget pealties imply, for ay, {t : x 1 t) = E } = {t : x 2 t) = E }, 1, 2 such that P 1, P 2 A. This expresses that if a evet is scheduled all participats will have assiged the idetical set of time slots to that evet. The absece of itraaget pealties implies that if P A, {t : x t) = E } = {t0,..., t 0 + L 1} where t0 = mi{t : x t) = E }. This expresses that a evet will be scheduled i cotiguous slots totalig the exact legth of the evet. If we defie S E ) := {t : x t) = E } where P A, the because two evets caot be assiged to the same time slot, we have S E ) S E 2 ) =, 2 {1,..., K}, 2, A A 2. 1) Summig over the lis, we have the followig global utility: N T 1 T =1 s=1 t=s+1 1 [ x s) 0 s) ) I {x s) 0} + x t) 0 t) ) ] I {x t) 0} T 1 Because the itraaget variables are fully coected, each variable has T 1 outgoig lis. Rewritig the previous expressio as a sum over time slots, we have the global utility as: N T x t) 0 t) ) N K I {x t) 0} = I {i A } 0 t) ) K = R t) ). 2) =1 t=1 =1 =1 t S E ) =1 A t S E ) The first equality i 2) is obtaied by partitioig the time domai for each participat ito scheduled evets ad the secod equality by switchig the order of the first two summatios. The solutio to the DCOP will determie a schedule that maximizes the fial expressio i 2) which, whe coupled with the coditios i 1) implied by the absece of pealties, is idetical to the multievet schedulig problem MESP). EA Evets as ariables): We ote that the graph structure of TSA grows as the time rage cosidered icreases or the size of the time quatizatio iterval decreases, leadig to a deser graph. A alterate approach is to cosider the evets as the decisio variables. Let us defie a DCOP where the variable x represets the startig time for evet E. Each of the K variables ca tae o a value from the time slot rage that is sufficietly early to allow for the required legth of the evet or 0 which idicates that a evet is ot scheduled: x {0, 1,..., T L + 1}, = 1,..., K. If a variable x taes o a value t 0, the it is assumed that for all required resources of that evet A ), the time slots {t,..., t + L 1} must be assiged to the evet E. It would be logical to assig each variable/evet to the aget of oe of the required resources for the evet. Propositio 2 The DCOP formulatio with evets as variables where a costrait betwee variables x ad x 2 whe x = t 1 ad x 2 = t 2 exists if the two evets have a commo required attedee A A 2 for 2 ), ad 2
3 cosequetly taes o the utility f, t 1 ; 2, t 2 ) = M t 1 0, t 2 0, t 1 t 2 t 1 + L M t 1 0, t 2 0, t 2 t 1 t 2 + L 2 g, t 1 ; 2, t 2 ) otherwise. where g, t 1 ; 2, t 2 ) = β I {t1 0} L 0 t 1 + l 1) ) + β 2 I {t2 0} A l=1 L 2 A 2 l=1 2 0 t 2 + l 1) ) for β i = 1/ K=1 I {A A i } 1 ) multiplicative iverse of the umber of outgoig lis) with M > NT max where N is the umber of participats, T is the umber of time slots ad max = max,, yields the optimal solutio to the Distributed MultiEvet Schedulig DiMES) problem. Proof. The first two coditios of the fuctio f state that a pealty of M is assessed if the variable assigmets cause a schedulig coflict. If o pealties are imposed, the utility gai for schedulig the evet the differece betwee the rewards for all the attedees a the values for eepig the utilized times uassiged) is distributed uiformly over the outgoig lis from the variable. Let us assume that a pealty is icurred because of a schedulig coflict betwee the variables x = t 1 0 ad x 2 = t 2 0. By settig x = 0 decidig ot to schedule evet E ), the global utility will chage by at least β 2 L 2 2 A 2 l=1 0 t 2 + l 1) ) K=1 I {A A i } 2 L K=1 I {A A i } 1 A l=1 0 t 1 + l 1) ) M where the term i the first set of bracets represets the value of the li whe x = 0 ad the term i the secod set of bracets represets the value of the lie whe x = t 1 0. We ca see that the utility chage is greater tha L M > M NT max > 0. A l=1 Thus, the optimal solutio to the DCOP caot have ay pealties o the lis as we ca always mae a utility gai by ot schedulig a evet that leads to a pealty. Let us defie S E ) = {t,..., t + L 1} if t 0 ad S E ) = if t = 0. Because t + L 1 T which S E ) T. Furthermore, we ow that the optimal solutio will have o coflicts, i.e. S E ) S E 2 ) =, 2 {1,..., K}, 2, A A 2. 3) Sice we have o pealties, the uiformly distributed gais from schedulig ca be aggregated. The global utility will be the sum of the rewards for the scheduled evets mius the values of the utilized times: K L 0 t + l 1) ) = K 0 t) ). =1 A l=1 =1 A t S E ) The DCOP solutio maximizes the previous expressio. This alog with coditio 3), is idetical to the MESP problem. PEA Private Evets as ariables): We ote that i EA, if a aget is to mae a decisio for a evet as a variable, it must be edowed with both the authority to mae assigmets for multiple resources as well as have valuatio iformatio for all required resources. There are settigs where resources, though part of a team, are uwillig or uable to cede this authority or iformatio. To address this, we cosider a modificatio of EA that protects these iterests. Let us defie a set of variables X := {x : A } where x {0, 1,..., T L +1} deotes the startig time for 3
4 evet E i the schedule of R which is a required resource for the evet. If x = 0, the R is choosig ot to schedule E. We the costruct a DCOP with the variable set X := =1 K X. Let us ow defie a set X := {xm X : m = } X which is the collectio of variables pertaiig to the th resource. Clearly, X > 0, otherwise the resource is ot required i ay evet. If X = 1, let X := X {x} 0 where xi 0 0 is a dummy variable. Otherwise, X := X. The DCOP partitios the variables i X to a aget represetig the th resource s iterests. Let all the variables withi X itraaget lis) be fully coected. The additio of the dummy variable to sets X with cardiality oe is to esure that itraaget lis exist for all agets. Iteraget lis exist betwee the variables for all participats of a give evet, i.e., all the variables i X are fully coected. Propositio 3 The DCOP formulatio with private evets as variables, where the costrait betwee the variables x 1 ad x 2 2 whe x 1 = t 1 ad x 2 2 = t 2 taes o the utility f 1,, t 1 ; 2, 2, t 2 ) = MI {1 2 }I {1 = 2 }I {t1 t 2 } + I {1 = 2 }I {1 2 } f iter 1 ;, t 1 ; 2, t 2 ). 4) where f iter ;, t 1 ; 2, t 2 ) = M t 1 0, t 2 0, t 1 t 2 t 1 + L M t 1 0, t 2 0, t 2 t 1 t 2 + L 2 g;, t 1 ; 2, t 2 ) otherwise ad g;, t 1 ; 2, t 2 ) = 1 Z t 1 ) + Z 2 t 2 ) ) where Z i t i ) = X 1 L i l=1 i 0 t i + l 1) ) I {ti 0} with M > NT max where N is the umber of participats, T is the umber of time slots ad max = max,, yields the optimal solutio to the Distributed MultiEvet Schedulig DiMES) problem. Proof. The first term i 4) characterizes that a pealty of M is assessed o a iteraget li 1 2 ) for a commo evet = 2 ) for which the same startig time is ot selected t 1 t 2 ) by the coected resources. The latter term i 4) addresses itraaget costraits 1 = 2 ) betwee differet evets 2 ) where the li utility f iter ) esures that a pealty is icurred o a itraaget costrait if a schedulig coflict is created. Otherwise, the utility gai for a resource assigig a viable time for a evet is uiformly distributed amog the outgoig itraaget lis as deoted i g ). Let us assume that a pealty is icurred o a iteraget costrait. This implies that the required resources for a particular evet could ot agree o a commo time to start. Sice the total utility gai excludig pealties) for holdig a evet E caot exceed L max NT max < M, A t=1 there exists a solutio for the DCOP where the evet is ot scheduled which is at least as good as that with the evet scheduled. Let us ow assume that a pealty is icurred o a itraaget li. This implies that a aget has chose startig times for two evets that causes the same time slot to be assiged to two evets. By similar logic, the pealty M is sufficietly large such that by choosig ot to schedule oe of the evets ad allowig all other agets to choose ot to schedule that evet thereby avoidig icurrig a iteraget pealty), we obtai a higher quality solutio. The above aalysis implies that the optimal DCOP solutio is void of assigmets that would activate a pealty. Thus, x = x m, m, A. Give E ad some A, let us defie S E ) = if x = 0, ad S E ) = {x,..., x + L 1} if x 0. The, we have S E ) S E 2 ) =, 2 {1,..., K}, 2, A A 2. 5) 4
5 Otherwise, a pealty would have bee assessed. The global utility is the the sum of all itraaget lis devoid of pealties g )), which ca be represeted as N K =1 =1 = I { A } K L =1 A l=1 1 K X 1 Z x) =1 I { A } 0 x + l 1) ) I {x 0} = N K 1 = I { A }Zx ) K =1 =1 =1 A t S E ) 0 t) ). The solutio to the DCOP maximizes the previous expressio, which whe coupled with the o coflict coditio i 5) is idetical to the DiMES problem. 5
SECTION 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 information{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers
. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationLesson 12. Sequences and Series
Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or
More informationLearning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.
Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the
More informationrepresented by 4! different arrangements of boxes, divide by 4! to get ways
Problem Set #6 solutios A juggler colors idetical jugglig balls red, white, ad blue (a I how may ways ca this be doe if each color is used at least oce? Let us preemptively color oe ball i each color,
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More information1 The Binomial Theorem: Another Approach
The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets
More informationMATH 361 Homework 9. Royden Royden Royden
MATH 61 Homework 9 Royde..9 First, we show that for ay subset E of the real umbers, E c + y = E + y) c traslatig the complemet is equivalet to the complemet of the traslated set). Without loss of geerality,
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationif A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S,
Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σalgebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatio
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationMultiplexers and Demultiplexers
I this lesso, you will lear about: Multiplexers ad Demultiplexers 1. Multiplexers 2. Combiatioal circuit implemetatio with multiplexers 3. Demultiplexers 4. Some examples Multiplexer A Multiplexer (see
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationRiemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, oegative fuctio o the closed iterval [a, b] Fid
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationwhen n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.
Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have
More informationA Gentle Introduction to Algorithms: Part II
A Getle Itroductio to Algorithms: Part II Cotets of Part I:. Merge: (to merge two sorted lists ito a sigle sorted list.) 2. Bubble Sort 3. Merge Sort: 4. The BigO, BigΘ, BigΩ otatios: asymptotic bouds
More informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationEngineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
More informationCOMPARISON OF THE EFFICIENCY OF SCONTROL CHART AND EWMAS 2 CONTROL CHART FOR THE CHANGES IN A PROCESS
COMPARISON OF THE EFFICIENCY OF SCONTROL CHART AND EWMAS CONTROL CHART FOR THE CHANGES IN A PROCESS Supraee Lisawadi Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techoology, Thammasat
More informationMeasures of Central Tendency
Measures of Cetral Tedecy A studet s grade will be determied by exam grades ( each exam couts twice ad there are three exams, HW average (couts oce, fial exam ( couts three times. Fid the average if the
More informationEstimating Probability Distributions by Observing Betting Practices
5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,
More informationLecture 7: Borel Sets and Lebesgue Measure
EE50: Probability Foudatios for Electrical Egieers JulyNovember 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,
More informationTHE ABRACADABRA PROBLEM
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected
More informationGeometric Sequences and Series. Geometric Sequences. Definition of Geometric Sequence. such that. a2 4
3330_0903qxd /5/05 :3 AM Page 663 Sectio 93 93 Geometric Sequeces ad Series 663 Geometric Sequeces ad Series What you should lear Recogize, write, ad fid the th terms of geometric sequeces Fid th partial
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 200617 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät LudwigMaximiliasUiversität Müche Olie at http://epub.ub.uimueche.de/940/
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationAdvanced Probability Theory
Advaced Probability Theory Math5411 HKUST Kai Che (Istructor) Chapter 1. Law of Large Numbers 1.1. σalgebra, measure, probability space ad radom variables. This sectio lays the ecessary rigorous foudatio
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationFourier Series and the Wave Equation Part 2
Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries
More informationB1. Fourier Analysis of Discrete Time Signals
B. Fourier Aalysis of Discrete Time Sigals Objectives Itroduce discrete time periodic sigals Defie the Discrete Fourier Series (DFS) expasio of periodic sigals Defie the Discrete Fourier Trasform (DFT)
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationReview for College Algebra Final Exam
Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 14. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i
More informationARITHMETIC AND GEOMETRIC PROGRESSIONS
Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives
More informationhp calculators HP 12C Statistics  average and standard deviation Average and standard deviation concepts HP12C average and standard deviation
HP 1C Statistics  average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationA CHARACTERIZATION OF MINIMAL ZEROSEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 A CHARACTERIZATION OF MINIMAL ZEROSEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS Scott T. Chapma 1 Triity Uiversity, Departmet
More information1. Introduction. Scheduling Theory
. Itroductio. Itroductio As a idepedet brach of Operatioal Research, Schedulig Theory appeared i the begiig of the 50s. I additio to computer systems ad maufacturig, schedulig theory ca be applied to may
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationBASIC STATISTICS. Discrete. Mass Probability Function: P(X=x i ) Only one finite set of values is considered {x 1, x 2,...} Prob. t = 1.
BASIC STATISTICS 1.) Basic Cocepts: Statistics: is a sciece that aalyzes iformatio variables (for istace, populatio age, height of a basketball team, the temperatures of summer moths, etc.) ad attempts
More informationConfidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More informationDesigning Incentives for Online Question and Answer Forums
Desigig Icetives for Olie Questio ad Aswer Forums Shaili Jai School of Egieerig ad Applied Scieces Harvard Uiversity Cambridge, MA 0238 USA shailij@eecs.harvard.edu Yilig Che School of Egieerig ad Applied
More informationSection IV.5: Recurrence Relations from Algorithms
Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by
More informationRecursion and Recurrences
Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,
More informationπ d i (b i z) (n 1)π )... sin(θ + )
SOME TRIGONOMETRIC IDENTITIES RELATED TO EXACT COVERS Joh Beebee Uiversity of Alaska, Achorage Jauary 18, 1990 Sherma K Stei proves that if si π = k si π b where i the b i are itegers, the are positive
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationReview of Fourier Series and Its Applications in Mechanical Engineering Analysis
ME 3 Applied Egieerig Aalysis Chapter 6 Review of Fourier Series ad Its Applicatios i Mechaical Egieerig Aalysis TaiRa Hsu, Professor Departmet of Mechaical ad Aerospace Egieerig Sa Jose State Uiversity
More informationIrreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationDistributions of Order Statistics
Chapter 2 Distributios of Order Statistics We give some importat formulae for distributios of order statistics. For example, where F k: (x)=p{x k, x} = I F(x) (k, k + 1), I x (a,b)= 1 x t a 1 (1 t) b 1
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a otforprofit membership associatio whose missio is to coect studets to college success
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationLesson 15 ANOVA (analysis of variance)
Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationSequences II. Chapter 3. 3.1 Convergent Sequences
Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,
More informationMeasurable Functions
Measurable Fuctios Dug Le 1 1 Defiitio It is ecessary to determie the class of fuctios that will be cosidered for the Lebesgue itegratio. We wat to guaratee that the sets which arise whe workig with these
More information3 Basic Definitions of Probability Theory
3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationA Recursive Formula for Moments of a Binomial Distribution
A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by
More informationChapter Gaussian Elimination
Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio
More information8.1 Arithmetic Sequences
MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first
More informationAlternatives To Pearson s and Spearman s Correlation Coefficients
Alteratives To Pearso s ad Spearma s Correlatio Coefficiets Floreti Smaradache Chair of Math & Scieces Departmet Uiversity of New Mexico Gallup, NM 8730, USA Abstract. This article presets several alteratives
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationKey Ideas Section 81: Overview hypothesis testing Hypothesis Hypothesis Test Section 82: Basics of Hypothesis Testing Null Hypothesis
Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, Pvalue Type I Error, Type II Error, Sigificace Level, Power Sectio 81: Overview Cofidece Itervals (Chapter 7) are
More informationx(x 1)(x 2)... (x k + 1) = [x] k n+m 1
1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationSearching Algorithm Efficiencies
Efficiecy of Liear Search Searchig Algorithm Efficiecies Havig implemeted the liear search algorithm, how would you measure its efficiecy? A useful measure (or metric) should be geeral, applicable to ay
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationTAYLOR SERIES, POWER SERIES
TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the
More informationDynamic House Allocation
Dyamic House Allocatio Sujit Gujar 1 ad James Zou 2 ad David C. Parkes 3 Abstract. We study a dyamic variat o the house allocatio problem. Each aget ows a distict object (a house) ad is able to trade its
More informationINVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More informationUniversal coding for classes of sources
Coexios module: m46228 Uiversal codig for classes of sources Dever Greee This work is produced by The Coexios Project ad licesed uder the Creative Commos Attributio Licese We have discussed several parametric
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationReview: Classification Outline
Data Miig CS 341, Sprig 2007 Decisio Trees Neural etworks Review: Lecture 6: Classificatio issues, regressio, bayesia classificatio Pretice Hall 2 Data Miig Core Techiques Classificatio Clusterig Associatio
More informationHypothesis Tests Applied to Means
The Samplig Distributio of the Mea Hypothesis Tests Applied to Meas Recall that the samplig distributio of the mea is the distributio of sample meas that would be obtaied from a particular populatio (with
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationDivide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016
CS 6, Lecture 3 Divide ad Coquer, Solvig Recurreces, Iteger Multiplicatio Scribe: Juliaa Cook (05, V Williams Date: April 6, 06 Itroductio Today we will cotiue to talk about divide ad coquer, ad go ito
More informationCooleyTukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Cosider a legth sequece x[ with a poit DFT X[ where Represet the idices ad as +, +, Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Usig these
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More information1.3 Binomial Coefficients
18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to
More informationConfidence Intervals for One Mean with Tolerance Probability
Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with
More information