1 Approximation Algorithms

Size: px
Start display at page:

Download "1 Approximation Algorithms"

Transcription

1 CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons to these problems. Three standard approaches nclude: Explotng specal problem structure: perhaps we do not need to solve the general case of the problem but rather a tractable specal verson; Heurstcs: procedures that tend to gve reasonable estmates but for whch no proven guarantees exst; Approxmaton algorthms: procedures whch are proven to gve solutons wthn a factor of optmum. Of these approaches, approxmaton algorthms are arguably the most mathematcally satsfyng, and wll be the subject of dscusson for ths secton. An algorthm s a factor α approxmaton α-approxmaton algorthm) for a problem ff for every nstance of the problem t can fnd a soluton wthn a factor α of the optmum soluton. If the problem at hand s a mnmzaton then α > 1 and ths defnton mples that the soluton found by the algorthm s at most α tmes the optmum soluton. If the problem s a maxmzaton, α < 1 and ths defnton guarantees that the approxmate soluton s at least α tmes the optmum. 1.1 Mnmum Vertex Cover Gven a graph GV, E), the mnmum vertex cover problem s to fnd a subset S V wth mnmum cardnalty such that every edge n E has at least one endpont n S. Algorthm 1 2-Approxmaton Algorthm for Mnmum Vertex Cover Fnd a maxmal matchng M n G. Output the endponts of edges n M: S = e M e. Clam 1 The output of algorthm 1 s feasble. Proof: We prove ths by contradcton: suppose there exsts an edge e = u, v) such that u, v / S. Snce e does not share an endpont wth any of the vertces n M, M {e} s a larger matchng, whch contradcts M beng a maxmal matchng. Lemma 1 Algorthm 1 gves a 2-approxmaton for mnmum vertex cover regardless of the choce of M. Proof: The edges of M are ndependent; thus any feasble cover must take at least one vertex from every edge n M. Ths means that M OP T and then we have: M OP T S = 2 M 2OP T Ths technque of lower boundng the optmum s often key n provng approxmaton factors, as we are usually unable to compute the value of OPT.

2 2 CME 305: Dscrete Mathematcs and Algorthms - Lecture Mnmum Weght Vertex Cover Gven a graph GV, E), and a weght functon W : V R +, the mnmum weght vertex cover problem s to fnd a subset S V that covers all the edges and has mnmum weght W S) = v S wv). We may formulate ths problem as an nteger program as follows: assocate varable x v wth node v, and solve: mnmze: wv)x v s.t. v V x u + x v 1, u, v) E x v {0, 1} v V Solvng ths nteger program s equvalent to solvng mn weght vertex cover, an NP-complete problem. We nstead attempt to solve a smpler problem for whch polynomal-tme algorthms exsts. We modfy the second constrant to read: 0 x v 1 v V. Ths s known as a lnear programmng LP) relaxaton of the nteger program. It s well known that lnear programs can be solved n polynomal tme. Let {x v, v V } be the soluton of the LP. We need to convert ths fractonal soluton to an ntegral one; we use the followng roundng polcy: f x v < 0 then we set x v = 0, otherwse we set x v = 1. Note that { x v, v V } s a feasble soluton. Because {x v, v V } s a feasble soluton of the LP, x v + x u 1, and thus x v 1/2 or x u 1/2 whch mples that x v + x u 1. Lemma 2 Let S = {v V : x v = 1}. Then S gves a 2-approxmaton to mn weght vertex cover,.e. wv) 2wS ) where S s the optmum soluton. v S Proof: Snce the feasble regon of the IP s a subset of the feasble regon of the LP, the optmum of the LP s a lower bound for the optmum of the IP. Moreover, note that our roundng procedure ensures that x v 2x v for all v V, thus: OP T IP ws) = v V wv) x v 2 v V wv)x v = 2OP T LP 2OP T IP. 1.3 Job Schedulng Suppose we have n jobs each of whch take tme t to process and m dentcal machnes on whch to schedule ther completon. Jobs cannot be splt between machnes. For a gven schedulng, let A j be the set of jobs assgned to machne j. Let L j = A j t be the load of machne j. The mnmum makespan schedulng problem s to fnd an assgnment of jobs to machnes that mnmzes the makespan, defned as the maxmum load over all machnes.e. max j L j ). We consder the followng greedy algorthm for ths problem whch sorts the jobs so that t 1 t 2... t n, and teratvely allocates the next job to the machne wth the least load. We note that algorthm 2 need not return an optmal soluton. Consderng jobs of szes {3, 3, 2, 2, 2} to be assgned to two machnes we see that t s no better than a 7/6-approxmaton algorthm. We prove n these

3 CME 305: Dscrete Mathematcs and Algorthms - Lecture 10 3 Algorthm 2 Greedy Approxmaton Algorthm for Job Schedulng j, A j, T j 0 for = 1 to n do j argmn k T k A j = A j {} T j = T j + t end for notes that algorthm 2 has an approxmaton factor of no worse than 3/2; we leave as an exercse to the reader to prove that t s actually a 4/3-approxmaton algorthm. Let T denote the optmal makespan. The followng two facts are self-evdent: T max T 1 m t n t. Clam 2 The soluton of the greedy makespan algorthm s at most 1 m n t + max t Proof: Consder machne j wth maxmum load T j. Let be the last job scheduled on machne j. When was scheduled, j had the smallest load, so j must have had load smaller than the average load. Then, T j = T j t j ) + t j 1 m n t + max t. Ths clam shows mmedately that algorthm 2 s a 2-approxmaton algorthm. Slghtly more careful analyss proves α = 3/2. Lemma 3 The approxmaton factor of the greedy makespan algorthm s at most 3/2. Proof: If there are at most m jobs, the schedulng s optmal snce we put each job on ts own machne. If there are more than m jobs, then by the pgeonhole prncple at least one processor n the optmal schedulng must get 2 of the frst m + 1 jobs. Each of these jobs s at least as bg as t m+1. Thus, T 2t m+1. Take the output of algorthm 2 and consder machne j assgned maxmum load T j. Let be the last job assgned to j. We may assume > m or else the algorthm s output s optmal. Snce the jobs are sorted, t t m+1 T /2 and we have T j = T j t ) + t 1 m n t + t T + T /2 = 3 2 T. =1

4 4 CME 305: Dscrete Mathematcs and Algorthms - Lecture Non-Unform Job Schedulng Consder a more general verson of the mnmum makespan schedulng problem n whch job can have dfferent processng tme on dfferent machnes. Let t j be the tme t takes machne j to process job. We want to mnmze the makespan T = max j x jt j where the varable x j {0, 1} ndcates whether job s assgned to machne j. We may wrte ths as an nteger program wth constrants ensurng that each job s assgned to exactly one machne and that the load of each machne does not exceed T, the makespan. mnmze: s.t. T x j = 1 j x j t j T x j {0, 1} j To approxmate a soluton to ths nteger program, one mght frst try to relax varable x j and let x j [0, 1], however, the soluton of the correspondng LP may be too far from the soluton of the IP. Thus soluton of LP does not serve as a tght lower bound for OPT. For nstance, f we have only 1 job, m machnes, and t 1j = m for every machne j, then OPT= m but the soluton of the LP s 1 by assgnng x 1j = 1 m. The maxmum rato of the optmal IP and LP solutons s called the ntegralty gap. We need to defne the relaxed problem n such a way that the ntegralty gap s small. The dea s to ensure that f t j > T we assgn x j = 0. We do ths by defnng a seres of feasblty LPs wth a makespan parameter T as follows. x j = 1 j x j t j T j 0 x j 1, j x j = 0, j s.t. t j > T Usng bnary search we can obtan the smallest value of T, T, for whch the above feasblty lnear program FLP) has a soluton. We outlne a procedure for roundng the soluton of the FLP wth T = T. Let GJ, M, E) be a bpartte graph defned on the set of jobs and machnes where edge {, j} between job J and machne j M has weght x j. Our clam s that ths allocaton graph can be transformed nto a forest n such a way that the load of every machne remans the same or decreases. Suppose G = J, M, E) s not a tree, thus t has cycle c = j 1, 1, j 2, 2,..., j r, r, j 1 ; suppose we update x 1j 1 to x 1j 1 ɛ 1, we proceed around cycle c and update the weght of edges n the followng way: Snce the total weght of edges ncdent to 1 must add up to one, f we decrease x 1j 1 by ɛ 1, we need to ncrease x 1j 2 by ɛ 1, thus we update x 1j 2 to x 1j 2 + ɛ 1. Now to keep the load of machne 2 less than or equal to T, we decrease x 2j 2 by ɛ 2 = t 1 j 2 t 2 ɛ 1, repeatng ths procedure, we modfy the weghts keepng the j 2 constrants satsfed. The only constrant that may become unsatsfed s the load of machne j 1 ; we decrease the load of j 1 by ɛ 1 va edge { 1, j 1 } and at the end, we may ncrease the load by t 1 j 2 t 2 j 3 t 2 j 2 t 3... t r 1 jr j 3 t r jr ɛ. If t 1 j 2 t 2 j 3 t 2 j 2 t 3... t r 1 jr j 3 t r jr > 1 we use the smple observaton that f we start from {j 1, r } and go around the cycle ) t1 j n that drecton, then we would need to ncrease the weght of { 1, j 1 } by 2 t 1 2 j 3 t 2 j 2 t 3... t r 1 jr j 3 t r jr < 1 thus the total load of 1 would become less that T.

5 CME 305: Dscrete Mathematcs and Algorthms - Lecture 10 5 Usng the above scheme, we are able to decrease the weght of { 1, j 1 } by ɛ keepng the soluton feasble. Repeatng ths reducton, we can make the weght of one of the edges zero, thus we can remove cycle c. We obtan our forest by breakng all the cycles. Suppose now that we have a soluton of the FLP wth T = T whose allocaton graph s a forest, call t x. Note that f job J s a leaf of the tree wth parent j M, then x j = 1, thus there s no j leaf wth fractonal weght. However, we can have fractonal edge {, j} between leaf j M and ts parent J. Suppose has k leaves j 1, j 2,..., j k, we choose one of the leaf machnes unformly at random and assgn job to t wth probablty x j. Then we remove from the graph. We repeat ths procedure of assgnng fractonal load of a parent to one of ts chldren and removng the job from the graph untl all the jobs are assgned. Clam 3 The above roundng procedure produces a factor 2 approxmaton. Proof: Let OP T be the makespan of the optmal assgnment and let T be the mnmum value of T found usng bnary search on FLP. Then, T OP T snce the FLP s clearly feasble usng the optmal assgnment. x s a feasble soluton therefore load of each machne s at most T. Durng the roundng procedure, we add the load of at most one job to each machne because a node can only have one parent n the forest G. Any machne j s load before ths addton s J x jt j T, so the new load of machne j s less than 2T. Hence, the fnal makespan s at most 2T. 1.5 Maxmum Satsfablty We return to the settng of boolean formulas and consder a problem related to satsfablty: for a gven formula n CNF, what s the maxmum number of ts clauses that can be satsfed by assgnng truth values to ts varables? More concretely, suppose we have n varables x 1,..., x n and m clauses C 1,..., C m where C = x x. S + S The maxmum satsfablty problem s to fnd the maxmum number of clauses that may be satsfed by an assgnment x. We frst propose a smple randomzed algorthm to approxmate a soluton to ths problem. Set each x ndependently to be 0 or 1 wth probablty 1/2. Then the probablty that any C s satsfed s 1 2 C. If we let Z denote the event that clause C s satsfed by ths random assgnment and Z = m =1 Z be the total number of satsfed clauses, we may compute: m m E[Z] = E[Z ] = 1 2 C ). =1 In the case that all of our clauses are large,.e. C K for each, then ths randomzed algorthm has an approxmaton rato of 1 2 K n expectaton: =1 m1 2 K ) E[Z] OP T m. An expected approxmaton rato s sometmes undesrable because t does not shed much lght on the probablty that the randomzed algorthm returns a good soluton. Concentraton nequaltes can help us estmate such probabltes, but n some cases we may do even better. Ths algorthm may be derandomzed usng condtonal expectaton as follows.

6 6 CME 305: Dscrete Mathematcs and Algorthms - Lecture 10 Algorthm 3 Derandomzed Approxmaton Algorthm for Maxmum Satsfablty for = 1 to n do Compute 1 2 E[Z x = 1, x 1,..., x 1 ] and 1 2 E[Z x = 0, x 1,..., x 1 ]. Set x = 1 f the frst expresson s larger than the second, set x = 0 otherwse. end for return x For motvaton, we consder the frst step of the algorthm. Note that E[Z] = 1 2 E[Z x 1 = 1] E[Z x 1 = 0]. Both terms on the rght hand sde may be computed smply by summng over all clauses the probablty that C s satsfed gven the nformaton on x 1. The equaton above mples that E[Z x 1 = 1] E[Z] or E[Z x 1 = 0] E[Z]. Thus f we choose the greater expectaton n each step of the algorthm, we wll determnstcally buld up an assgnment x such that E[Z x] E[Z] m1 2 K ) where E[Z x] s no longer an expectaton, but merely an evaluaton. The approxmaton rato of algorthm 3 depends on all of the clauses havng K or more varables. We present a dfferent algorthm for dealng wth the possblty that some of the clauses may be small. It s based on the now famlar concept of LP relaxaton. We wrte down an nteger program for maxmum satsfablty. maxmze: s.t. m =1 q q y j + 1 y j ) j S + q, y j {0, 1} j S, j The varables q correspond to the truth value of each clause C, and the varables y j correspond to the values of each boolean varable x. We relax the last condton to be 0 q, y j 1 n order to get a lnear program. Algorthm 4 LP Relaxaton Algorthm for Maxmum Satsfablty Solve the LP gven above. for j = 1 to n do Independently set x j = { 1 : wth probablty y j 0 : wth probablty 1 y j end for In order to analyze algorthm 4 we consder the probablty that a partcular clause s satsfed; WLOG we

7 CME 305: Dscrete Mathematcs and Algorthms - Lecture 10 7 consder C 1 = x 1... x k. We have q1 = mn{y yk, 1} and by the AM-GM nequalty: P r[c 1 ] = 1 k 1 yj ) j=1 1 1 k k 1 yj ) j=1 ) k 1 1 q 1 k q 1 1 q 1 1 1/e) 1 1 k Ths last lne mples, through the lnearty of expectaton, that ths roundng procedure gves a 1 1/e)- approxmaton for maxmum satsfablty regardless of the sze of the smallest clause. Algorthm 4 may also be derandomzed by the method of condtonal expectatons. These two derandomzed algorthms may be combned to gve a factor 3/4 approxmaton algorthm for maxmum satsfablty. We smply run both algorthms on a gven problem nstance and output the better of the two assgnments. Ths procedure tself may be vewed as a derandomzaton of an algorthm that flps a far con to decde whch randomzed sub-algorthm to run. That algorthm has approxmaton factor 3/4: m m 1 E[Z] = E[Z ] 1 2 C ) =1 =1 ) k ) k 1 1 ) )) C 3/4)m. C

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby s Alg. for Maximal Independent Sets using Pairwise Independence Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

More information

The Greedy Method. Introduction. 0/1 Knapsack Problem

The Greedy Method. Introduction. 0/1 Knapsack Problem The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton

More information

Graph Theory and Cayley s Formula

Graph Theory and Cayley s Formula Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

1 Example 1: Axis-aligned rectangles

1 Example 1: Axis-aligned rectangles COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

More information

Communication Networks II Contents

Communication Networks II Contents 8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

More information

INSTITUT FÜR INFORMATIK

INSTITUT FÜR INFORMATIK INSTITUT FÜR INFORMATIK An EPTAS for schedulng jobs on unform processors: usng an MILP relaxaton wth a constant number of ntegral varables Klaus Jansen Bercht Nr. 1002 February 2010 CHRISTIAN-ALBRECHTS-UNIVERSITÄT

More information

Approximation algorithms for allocation problems: Improving the factor of 1 1/e

Approximation algorithms for allocation problems: Improving the factor of 1 1/e Approxmaton algorthms for allocaton problems: Improvng the factor of 1 1/e Urel Fege Mcrosoft Research Redmond, WA 98052 urfege@mcrosoft.com Jan Vondrák Prnceton Unversty Prnceton, NJ 08540 jvondrak@gmal.com

More information

PLANAR GRAPHS. Plane graph (or embedded graph) A graph that is drawn on the plane without edge crossing, is called a Plane graph

PLANAR GRAPHS. Plane graph (or embedded graph) A graph that is drawn on the plane without edge crossing, is called a Plane graph PLANAR GRAPHS Basc defntons Isomorphc graphs Two graphs G(V,E) and G2(V2,E2) are somorphc f there s a one-to-one correspondence F of ther vertces such that the followng holds: - u,v V, uv E, => F(u)F(v)

More information

Solutions to the exam in SF2862, June 2009

Solutions to the exam in SF2862, June 2009 Solutons to the exam n SF86, June 009 Exercse 1. Ths s a determnstc perodc-revew nventory model. Let n = the number of consdered wees,.e. n = 4 n ths exercse, and r = the demand at wee,.e. r 1 = r = r

More information

Properties of American Derivative Securities

Properties of American Derivative Securities Capter 6 Propertes of Amercan Dervatve Securtes 6.1 Te propertes Defnton 6.1 An Amercan dervatve securty s a sequence of non-negatve random varables fg k g n k= suc tat eac G k s F k -measurable. Te owner

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

Generalizing the degree sequence problem

Generalizing the degree sequence problem Mddlebury College March 2009 Arzona State Unversty Dscrete Mathematcs Semnar The degree sequence problem Problem: Gven an nteger sequence d = (d 1,...,d n ) determne f there exsts a graph G wth d as ts

More information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

Feature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College

Feature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure

More information

greatest common divisor

greatest common divisor 4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

More information

Extending Probabilistic Dynamic Epistemic Logic

Extending Probabilistic Dynamic Epistemic Logic Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set

More information

Joint Scheduling of Processing and Shuffle Phases in MapReduce Systems

Joint Scheduling of Processing and Shuffle Phases in MapReduce Systems Jont Schedulng of Processng and Shuffle Phases n MapReduce Systems Fangfe Chen, Mural Kodalam, T. V. Lakshman Department of Computer Scence and Engneerng, The Penn State Unversty Bell Laboratores, Alcatel-Lucent

More information

8 Algorithm for Binary Searching in Trees

8 Algorithm for Binary Searching in Trees 8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the

More information

v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

v a 1 b 1 i, a 2 b 2 i,..., a n b n i. SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

More information

INSTITUT FÜR INFORMATIK

INSTITUT FÜR INFORMATIK INSTITUT FÜR INFORMATIK Schedulng jobs on unform processors revsted Klaus Jansen Chrstna Robene Bercht Nr. 1109 November 2011 ISSN 2192-6247 CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL Insttut für Informat

More information

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60 BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

More information

Lossless Data Compression

Lossless Data Compression Lossless Data Compresson Lecture : Unquely Decodable and Instantaneous Codes Sam Rowes September 5, 005 Let s focus on the lossless data compresson problem for now, and not worry about nosy channel codng

More information

A Computer Technique for Solving LP Problems with Bounded Variables

A Computer Technique for Solving LP Problems with Bounded Variables Dhaka Unv. J. Sc. 60(2): 163-168, 2012 (July) A Computer Technque for Solvng LP Problems wth Bounded Varables S. M. Atqur Rahman Chowdhury * and Sanwar Uddn Ahmad Department of Mathematcs; Unversty of

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

DEFINING %COMPLETE IN MICROSOFT PROJECT CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

Rate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Priority-based scheduling. States of a process

Rate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Priority-based scheduling. States of a process Dsadvantages of cyclc TDDB47 Real Tme Systems Manual scheduler constructon Cannot deal wth any runtme changes What happens f we add a task to the set? Real-Tme Systems Laboratory Department of Computer

More information

Logical Development Of Vogel s Approximation Method (LD-VAM): An Approach To Find Basic Feasible Solution Of Transportation Problem

Logical Development Of Vogel s Approximation Method (LD-VAM): An Approach To Find Basic Feasible Solution Of Transportation Problem INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN 77-866 Logcal Development Of Vogel s Approxmaton Method (LD- An Approach To Fnd Basc Feasble Soluton Of Transportaton

More information

General Auction Mechanism for Search Advertising

General Auction Mechanism for Search Advertising General Aucton Mechansm for Search Advertsng Gagan Aggarwal S. Muthukrshnan Dávd Pál Martn Pál Keywords game theory, onlne auctons, stable matchngs ABSTRACT Internet search advertsng s often sold by an

More information

Formula of Total Probability, Bayes Rule, and Applications

Formula of Total Probability, Bayes Rule, and Applications 1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the

More information

J. Parallel Distrib. Comput.

J. Parallel Distrib. Comput. J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n

More information

A Constant Factor Approximation for the Single Sink Edge Installation Problem

A Constant Factor Approximation for the Single Sink Edge Installation Problem A Constant Factor Approxmaton for the Sngle Snk Edge Installaton Problem Sudpto Guha Adam Meyerson Kamesh Munagala Abstract We present the frst constant approxmaton to the sngle snk buy-at-bulk network

More information

A Note on the Decomposition of a Random Sample Size

A Note on the Decomposition of a Random Sample Size A Note on the Decomposton of a Random Sample Sze Klaus Th. Hess Insttut für Mathematsche Stochastk Technsche Unverstät Dresden Abstract Ths note addresses some results of Hess 2000) on the decomposton

More information

Chapter 7. Random-Variate Generation 7.1. Prof. Dr. Mesut Güneş Ch. 7 Random-Variate Generation

Chapter 7. Random-Variate Generation 7.1. Prof. Dr. Mesut Güneş Ch. 7 Random-Variate Generation Chapter 7 Random-Varate Generaton 7. Contents Inverse-transform Technque Acceptance-Rejecton Technque Specal Propertes 7. Purpose & Overvew Develop understandng of generatng samples from a specfed dstrbuton

More information

9.1 The Cumulative Sum Control Chart

9.1 The Cumulative Sum Control Chart Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s

More information

We are now ready to answer the question: What are the possible cardinalities for finite fields?

We are now ready to answer the question: What are the possible cardinalities for finite fields? Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the

More information

Figure 1. Inventory Level vs. Time - EOQ Problem

Figure 1. Inventory Level vs. Time - EOQ Problem IEOR 54 Sprng, 009 rof Leahman otes on Eonom Lot Shedulng and Eonom Rotaton Cyles he Eonom Order Quantty (EOQ) Consder an nventory tem n solaton wth demand rate, holdng ost h per unt per unt tme, and replenshment

More information

2.4 Bivariate distributions

2.4 Bivariate distributions page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

More information

Project Networks With Mixed-Time Constraints

Project Networks With Mixed-Time Constraints Project Networs Wth Mxed-Tme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa

More information

Fisher Markets and Convex Programs

Fisher Markets and Convex Programs Fsher Markets and Convex Programs Nkhl R. Devanur 1 Introducton Convex programmng dualty s usually stated n ts most general form, wth convex objectve functons and convex constrants. (The book by Boyd and

More information

Online Learning from Experts: Minimax Regret

Online Learning from Experts: Minimax Regret E0 370 tatstcal Learnng Theory Lecture 2 Nov 24, 20) Onlne Learnng from Experts: Mn Regret Lecturer: hvan garwal crbe: Nkhl Vdhan Introducton In the last three lectures we have been dscussng the onlne

More information

CS 2750 Machine Learning. Lecture 3. Density estimation. CS 2750 Machine Learning. Announcements

CS 2750 Machine Learning. Lecture 3. Density estimation. CS 2750 Machine Learning. Announcements Lecture 3 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 5329 Sennott Square Next lecture: Matlab tutoral Announcements Rules for attendng the class: Regstered for credt Regstered for audt (only f there

More information

Logistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification

Logistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson

More information

Ring structure of splines on triangulations

Ring structure of splines on triangulations www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAM-Report 2014-48 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon

More information

Non-degenerate Hilbert Cubes in Random Sets

Non-degenerate Hilbert Cubes in Random Sets Journal de Théore des Nombres de Bordeaux 00 (XXXX), 000 000 Non-degenerate Hlbert Cubes n Random Sets par Csaba Sándor Résumé. Une légère modfcaton de la démonstraton du lemme des cubes de Szemeréd donne

More information

Availability-Based Path Selection and Network Vulnerability Assessment

Availability-Based Path Selection and Network Vulnerability Assessment Avalablty-Based Path Selecton and Network Vulnerablty Assessment Song Yang, Stojan Trajanovsk and Fernando A. Kupers Delft Unversty of Technology, The Netherlands {S.Yang, S.Trajanovsk, F.A.Kupers}@tudelft.nl

More information

HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION

HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION Abdul Ghapor Hussn Centre for Foundaton Studes n Scence Unversty of Malaya 563 KUALA LUMPUR E-mal: ghapor@umedumy Abstract Ths paper

More information

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2) MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information

Lesson 2 Chapter Two Three Phase Uncontrolled Rectifier

Lesson 2 Chapter Two Three Phase Uncontrolled Rectifier Lesson 2 Chapter Two Three Phase Uncontrolled Rectfer. Operatng prncple of three phase half wave uncontrolled rectfer The half wave uncontrolled converter s the smplest of all three phase rectfer topologes.

More information

Calculation of Sampling Weights

Calculation of Sampling Weights Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuity Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

More information

IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUARE-ROOT INTERVAL METHOD FOR MULTIPLE ZEROS 1

IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUARE-ROOT INTERVAL METHOD FOR MULTIPLE ZEROS 1 Nov Sad J. Math. Vol. 36, No. 2, 2006, 0-09 IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUARE-ROOT INTERVAL METHOD FOR MULTIPLE ZEROS Modrag S. Petkovć 2, Dušan M. Mloševć 3 Abstract. A new theorem concerned

More information

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

More information

Analysis of Small-signal Transistor Amplifiers

Analysis of Small-signal Transistor Amplifiers Analyss of Small-sgnal Transstor Amplfers On completon of ths chapter you should be able to predct the behaour of gen transstor amplfer crcuts by usng equatons and/or equalent crcuts that represent the

More information

Heuristic Static Load-Balancing Algorithm Applied to CESM

Heuristic Static Load-Balancing Algorithm Applied to CESM Heurstc Statc Load-Balancng Algorthm Appled to CESM 1 Yur Alexeev, 1 Sher Mckelson, 1 Sven Leyffer, 1 Robert Jacob, 2 Anthony Crag 1 Argonne Natonal Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439,

More information

NPAR TESTS. One-Sample Chi-Square Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6

NPAR TESTS. One-Sample Chi-Square Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6 PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has

More information

SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:

SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background: SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and

More information

EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN

EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson - 3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson - 6 Hrs.) Voltage

More information

Formulating & Solving Integer Problems Chapter 11 289

Formulating & Solving Integer Problems Chapter 11 289 Formulatng & Solvng Integer Problems Chapter 11 289 The Optonal Stop TSP If we drop the requrement that every stop must be vsted, we then get the optonal stop TSP. Ths mght correspond to a ob sequencng

More information

Considering manufacturing cost and scheduling performance on a CNC turning machine

Considering manufacturing cost and scheduling performance on a CNC turning machine European Journal of Operatonal Research 77 (2007) 325 343 Producton, Manufacturng and Logstcs Consderng manufacturng cost and schedulng performance on a CNC turnng machne Snan Gurel, M Selm Akturk * Department

More information

Lecture 7 March 20, 2002

Lecture 7 March 20, 2002 MIT 8.996: Topc n TCS: Internet Research Problems Sprng 2002 Lecture 7 March 20, 2002 Lecturer: Bran Dean Global Load Balancng Scrbe: John Kogel, Ben Leong In today s lecture, we dscuss global load balancng

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

Section B9: Zener Diodes

Section B9: Zener Diodes Secton B9: Zener Dodes When we frst talked about practcal dodes, t was mentoned that a parameter assocated wth the dode n the reverse bas regon was the breakdown voltage, BR, also known as the peak-nverse

More information

Research Article Enhanced Two-Step Method via Relaxed Order of α-satisfactory Degrees for Fuzzy Multiobjective Optimization

Research Article Enhanced Two-Step Method via Relaxed Order of α-satisfactory Degrees for Fuzzy Multiobjective Optimization Hndaw Publshng Corporaton Mathematcal Problems n Engneerng Artcle ID 867836 pages http://dxdoorg/055/204/867836 Research Artcle Enhanced Two-Step Method va Relaxed Order of α-satsfactory Degrees for Fuzzy

More information

Aryabhata s Root Extraction Methods. Abhishek Parakh Louisiana State University Aug 31 st 2006

Aryabhata s Root Extraction Methods. Abhishek Parakh Louisiana State University Aug 31 st 2006 Aryabhata s Root Extracton Methods Abhshek Parakh Lousana State Unversty Aug 1 st 1 Introducton Ths artcle presents an analyss of the root extracton algorthms of Aryabhata gven n hs book Āryabhatīya [1,

More information

New bounds in Balog-Szemerédi-Gowers theorem

New bounds in Balog-Szemerédi-Gowers theorem New bounds n Balog-Szemeréd-Gowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A

More information

Lecture 2: Absorbing states in Markov chains. Mean time to absorption. Wright-Fisher Model. Moran Model.

Lecture 2: Absorbing states in Markov chains. Mean time to absorption. Wright-Fisher Model. Moran Model. Lecture 2: Absorbng states n Markov chans. Mean tme to absorpton. Wrght-Fsher Model. Moran Model. Antonna Mtrofanova, NYU, department of Computer Scence December 8, 2007 Hgher Order Transton Probabltes

More information

Online Advertisement, Optimization and Stochastic Networks

Online Advertisement, Optimization and Stochastic Networks Onlne Advertsement, Optmzaton and Stochastc Networks Bo (Rambo) Tan and R. Srkant Department of Electrcal and Computer Engneerng Unversty of Illnos at Urbana-Champagn Urbana, IL, USA 1 arxv:1009.0870v6

More information

Virtual Network Embedding with Coordinated Node and Link Mapping

Virtual Network Embedding with Coordinated Node and Link Mapping Vrtual Network Embeddng wth Coordnated Node and Lnk Mappng N. M. Mosharaf Kabr Chowdhury Cherton School of Computer Scence Unversty of Waterloo Waterloo, Canada Emal: nmmkchow@uwaterloo.ca Muntasr Rahan

More information

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

NON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia

NON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia To appear n Journal o Appled Probablty June 2007 O-COSTAT SUM RED-AD-BLACK GAMES WITH BET-DEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate

More information

An MILP model for planning of batch plants operating in a campaign-mode

An MILP model for planning of batch plants operating in a campaign-mode An MILP model for plannng of batch plants operatng n a campagn-mode Yanna Fumero Insttuto de Desarrollo y Dseño CONICET UTN yfumero@santafe-concet.gov.ar Gabrela Corsano Insttuto de Desarrollo y Dseño

More information

+ + + - - This circuit than can be reduced to a planar circuit

+ + + - - This circuit than can be reduced to a planar circuit MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

More information

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6)

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6) Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called

More information

Real-Time Process Scheduling

Real-Time Process Scheduling Real-Tme Process Schedulng ktw@cse.ntu.edu.tw (Real-Tme and Embedded Systems Laboratory) Independent Process Schedulng Processes share nothng but CPU Papers for dscussons: C.L. Lu and James. W. Layland,

More information

State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness

State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators-> normalzaton, orthogonalty completeness egenvalues and

More information

Math 131: Homework 4 Solutions

Math 131: Homework 4 Solutions Math 3: Homework 4 Solutons Greg Parker, Wyatt Mackey, Chrstan Carrck October 6, 05 Problem (Munkres 3.) Let {A n } be a sequence of connected subspaces of X such that A n \ A n+ 6= ; for all n. Then S

More information

6. EIGENVALUES AND EIGENVECTORS 3 = 3 2

6. EIGENVALUES AND EIGENVECTORS 3 = 3 2 EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a non-zero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :

More information

A Lyapunov Optimization Approach to Repeated Stochastic Games

A Lyapunov Optimization Approach to Repeated Stochastic Games PROC. ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, OCT. 2013 1 A Lyapunov Optmzaton Approach to Repeated Stochastc Games Mchael J. Neely Unversty of Southern Calforna http://www-bcf.usc.edu/

More information

Linear Regression, Regularization Bias-Variance Tradeoff

Linear Regression, Regularization Bias-Variance Tradeoff HTF: Ch3, 7 B: Ch3 Lnear Regresson, Regularzaton Bas-Varance Tradeoff Thanks to C Guestrn, T Detterch, R Parr, N Ray 1 Outlne Lnear Regresson MLE = Least Squares! Bass functons Evaluatng Predctors Tranng

More information

Questions that we may have about the variables

Questions that we may have about the variables Antono Olmos, 01 Multple Regresson Problem: we want to determne the effect of Desre for control, Famly support, Number of frends, and Score on the BDI test on Perceved Support of Latno women. Dependent

More information

Chapter 7: Answers to Questions and Problems

Chapter 7: Answers to Questions and Problems 19. Based on the nformaton contaned n Table 7-3 of the text, the food and apparel ndustres are most compettve and therefore probably represent the best match for the expertse of these managers. Chapter

More information

2008/8. An integrated model for warehouse and inventory planning. Géraldine Strack and Yves Pochet

2008/8. An integrated model for warehouse and inventory planning. Géraldine Strack and Yves Pochet 2008/8 An ntegrated model for warehouse and nventory plannng Géraldne Strack and Yves Pochet CORE Voe du Roman Pays 34 B-1348 Louvan-la-Neuve, Belgum. Tel (32 10) 47 43 04 Fax (32 10) 47 43 01 E-mal: corestat-lbrary@uclouvan.be

More information

Multicomponent Distillation

Multicomponent Distillation Multcomponent Dstllaton need more than one dstllaton tower, for n components, n-1 fractonators are requred Specfcaton Lmtatons The followng are establshed at the begnnng 1. Temperature, pressure, composton,

More information

A Probabilistic Theory of Coherence

A Probabilistic Theory of Coherence A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want

More information

Multivariate EWMA Control Chart

Multivariate EWMA Control Chart Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant

More information

AN OPTIMAL ALGORITHM FOR CONFLICT-FREE COLORING FOR TREE OF RINGS

AN OPTIMAL ALGORITHM FOR CONFLICT-FREE COLORING FOR TREE OF RINGS AN OPTIMAL ALGORITHM FOR CONFLICT-FREE COLORING FOR TREE OF RINGS Enollah Pra The Busness Tranng Center of TabrzIran pra_ep2006@yahoo.com ABSTRACT An optmal algorthm s presented about Conflct-Free Colorng

More information

CS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering

CS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering Lecture 7a Clusterng Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Clusterng Groups together smlar nstances n the data sample Basc clusterng problem: dstrbute data nto k dfferent groups such that

More information

A hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm

A hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):1884-1889 Research Artcle ISSN : 0975-7384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

POLYSA: A Polynomial Algorithm for Non-binary Constraint Satisfaction Problems with and

POLYSA: A Polynomial Algorithm for Non-binary Constraint Satisfaction Problems with and POLYSA: A Polynomal Algorthm for Non-bnary Constrant Satsfacton Problems wth and Mguel A. Saldo, Federco Barber Dpto. Sstemas Informátcos y Computacón Unversdad Poltécnca de Valenca, Camno de Vera s/n

More information

1. Measuring association using correlation and regression

1. Measuring association using correlation and regression How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a

More information

Examensarbete. Rotating Workforce Scheduling. Caroline Granfeldt

Examensarbete. Rotating Workforce Scheduling. Caroline Granfeldt Examensarbete Rotatng Workforce Schedulng Carolne Granfeldt LTH - MAT - EX - - 2015 / 08 - - SE Rotatng Workforce Schedulng Optmerngslära, Lnköpngs Unverstet Carolne Granfeldt LTH - MAT - EX - - 2015

More information

Analysis of Covariance

Analysis of Covariance Chapter 551 Analyss of Covarance Introducton A common tas n research s to compare the averages of two or more populatons (groups). We mght want to compare the ncome level of two regons, the ntrogen content

More information

Nuno Vasconcelos UCSD

Nuno Vasconcelos UCSD Bayesan parameter estmaton Nuno Vasconcelos UCSD 1 Maxmum lkelhood parameter estmaton n three steps: 1 choose a parametrc model for probabltes to make ths clear we denote the vector of parameters by Θ

More information

Solving Factored MDPs with Continuous and Discrete Variables

Solving Factored MDPs with Continuous and Discrete Variables Solvng Factored MPs wth Contnuous and screte Varables Carlos Guestrn Berkeley Research Center Intel Corporaton Mlos Hauskrecht epartment of Computer Scence Unversty of Pttsburgh Branslav Kveton Intellgent

More information

Descriptive Models. Cluster Analysis. Example. General Applications of Clustering. Examples of Clustering Applications

Descriptive Models. Cluster Analysis. Example. General Applications of Clustering. Examples of Clustering Applications CMSC828G Prncples of Data Mnng Lecture #9 Today s Readng: HMS, chapter 9 Today s Lecture: Descrptve Modelng Clusterng Algorthms Descrptve Models model presents the man features of the data, a global summary

More information