Concept: Types of algorithms


 Martha Mathews
 4 years ago
 Views:
Transcription
1 Discrete Math for Bioiformatics WS 10/11:, by A. Bockmayr/K. Reiert, 18. Oktober 2010, 21: Cocept: Types of algorithms The expositio is based o the followig sources, which are all required readig: 1. Corma, Leiserso, Rivest: Chapter 1 ad Motwai, Raghava: Chapter 1. I this lecture we will discuss differet ways to categorize classes of algorithms. There is o oe correct classificatio. Oe should regard the task of categorizig algorithms more as givig them certai attributes. After discussig how a give algorithm ca be labeled (e.g. as a radomized, divideadcoquer algorithm) we will discuss differet techiques to aalyze algorithms. Usually the labels with which we categorized a algorithm are quite helpful i choosig the appropriate type of aalysis. Determiistic vs. Radomized Oe importat (ad exclusive) distictio oe ca make is, whether the algorithm is determiistic or radomized. Determiistic algorithms produce o a give iput the same results followig the same computatio steps. Radomized algorithms throw cois durig executio. Hece either the order of executio or the result of the algorithm might be differet for each ru o the same iput. There are subclasses for radomized algorithms. Mote Carlo type algorithms ad Las Vegas type algorithms. A Las Vegas algorithm will always produce the same result o a give iput. Radomizatio will oly affect the order of the iteral executios. I the case of Mote Carlo algorithms, the result may might chage, eve be wrog. However, a Mote Carlo algorithm will produce the correct result with a certai probability. So of course the questio arises: What are radomized algorithms good for? The computatio might chage depedig o coi throws. Mote Carlo algorithms do ot eve have to produce the correct result. Why would that be desirable? The aswer is twofold: Radomized algorithms usually have the effect of perturbig the iput. Or put it differetly, the iput looks radom, which makes bad cases very seldom. Radomized algorithms are ofte coceptually very easy to implemet. At the same time they are i ru time ofte superior to their determiistic couterparts. Ca you thik of a obvious example? We will come to the example later o i more detail. Offlie vs. Olie Aother importat (ad exclusive) distictio oe ca make is, whether the algorithm is offlie or olie. Olie algorithms are algorithms that do ot kow their iput at the begiig. It is give to them olie, whereas ormally algorithms kow their iput beforehad. What seems like a mior detail has profoud effects o the desig of algorithms ad o their aalysis. Olie algorithms are usually aalyzed by usig the cocept of competitiveess, that is the worst case factor they take loger compared to the best algorithm with complete iformatio. Oe example for a olie problem is the ski problem.
2 1002 Cocept: Types of algorithms ad algorithm aalyses, by Kut Reiert, 18. Oktober 2010, 21:22 A skier must decide every day she goes skiig, whether to ret or to buy skis, uless or util she decides to buy them. The skier does ot kow how may days she ca ski, because the whether is upredictable. Call the umber of days she will ski T. The cost to ret skis is 1 uit, while the cost of buyig skis is B. What is the optimal offlie algorithm miimizig the worst case cost? Ad what would be the optimal strategy i the olie case? Exact vs approximate vs. heuristic vs. operatioal Usually algorithms have a optimizatio goal i mid, e.g. compute the shortest path or the aligmet or miimal edit distace. Exact algorithms aim at computig the optimal solutio give such a goal. Ofte this is quite expesive i terms of ru time or memory ad hece ot possible for large iput. I such cases oe tries other strategies. Approximatio algorithms aim at computig a solutio which is for example oly a certai, guarateed factor worse tha the optimal solutio, that meas a algorithm yields a c approximatio, if it ca guaratee that its solutio is ever worse tha a factor c compared to the optimal solutio. Alteratively, heuristic algorithms try to reach the optimal solutio without givig a guaratee that they always do. Ofte it is easy to costruct a couter example. A good heuristics is almost always ear or at the optimal value. Fially there are algorithms which do ot aim at optimizig a objective fuctio. I call them operatioal sice they chai a series of computatioal operatios guided by expert kowledge but ot i cojuctio with a specific objective fuctio (e.g. ClustalW). Example: Approximatio algorithm As a example thik of the Travelig Salema Problem with triagle iequality for cities. This is a NPhard problem (o polyomialtime algorithm is kow). The followig greedy, determiistic algorithm yields a 2 approximatio for the TSP with triagle iequality i time O( 2 ). 1. Compute a miimum spaig tree T for the complete graph implied by the cities. 2. Duplicate all edges of T yieldig a Euleria graph T ad the fid a Euleria path i T. 3. Covert the Euleria cycle ito a Hamiltoia cycle by takig shortcuts. Ca you ow argue why this is a 2 approximatio? Categorizatio accordig to mai cocept Aother way which you have ofte heard util ow is to use the mai algorithmic paradigm to categorize a algorithm, such as: Simple recursive algorithms Backtrackig algorithms Divideadcoquer algorithms Dyamic programmig algorithms Greedy algorithms Brachadboud algorithms
3 Cocept: Types of algorithms ad algorithm aalyses, by Kut Reiert, 18. Oktober 2010, 21: Brute force algorithms ad others... Simple recursive algorithms A simple recursive algorithm Solves the base cases directly Recurs with a simpler subproblem Does some extra work to covert the solutio to the simpler subproblem ito a solutio to the give problem Examples are: To cout the umber of elemets i a list: If the list is empty, retur zero; otherwise, Step past the first elemet, ad cout the remaiig elemets i the list Add oe to the result To test if a value occurs i a list: If the list is empty, retur false; otherwise, If the first thig i the list is the give value, retur true; otherwise Step past the first elemet, ad test whether the value occurs i the remaider of the list Backtrackig algorithms A backtrackig algorithm is based o a depthfirst recursive search. It Tests to see if a solutio has bee foud, ad if so, returs it; otherwise For each choice that ca be made at this poit, Make that choice Recur If the recursio returs a solutio, retur it If o choices remai, retur failure For example color a map with o more tha four colors: color(coutry ) If all coutries have bee colored ( > umber of coutries) retur success; otherwise, For each color c of four colors, If coutry is ot adjacet to a coutry that has bee colored c Color coutry with color c recursivly color coutry + 1 If successful, retur success Retur failure (if loop exits)
4 1004 Cocept: Types of algorithms ad algorithm aalyses, by Kut Reiert, 18. Oktober 2010, 21:22 Divideadcoquer algorithms A divideadcoquer algorithm cosists of two parts. Divide the problem ito smaller subproblems of the same type ad solve these subproblems recursively Combie the solutios to the subproblems ito a solutio to the origial problem Traditioally, a algorithm is oly called divideadcoquer if it cotais two or more recursive calls. Two examples: Quicksort: Partitio the array ito two parts, ad quicksort each of the parts No additioal work is required to combie the two sorted parts Mergesort: Cut the array i half, ad mergesort each half Combie the two sorted arrays ito a sigle sorted array by mergig them Dyamic programmig algorithms A dyamic programmig algorithm remembers past results ad uses them to fid ew results. Dyamic programmig is geerally used for optimizatio problems i which: Multiple solutios exist, eed to fid the best oe Requires optimal substructure ad overlappig subproblem Optimal substructure: Optimal solutio cotais optimal solutios to subproblems Overlappig subproblems: Solutios to subproblems ca be stored ad reused i a bottomup fashio This differs from DivideadCoquer, where subproblems geerally eed ot overlap. There are may examples i bioiformatics. For example: Compute a optimal pairwise aligmet Optimal substructure: the aligmet of two prefixes cotais solutios for the optimal aligmets of smaller prefixes. Overlappig subproblems: The solutio for the optimal aligmet of two prefixes ca be costructed usig the stored solutios of the aligmet of three subproblems (i the liear gap model). Compute a Viterbi path i a HMM Optimal substructure: the Viterbi path for a iput prefix edig i a state of a HMM cotais shorter Viterbi paths for smaller parts of the iput ad other HMM states. Overlappig subproblems: The solutio for the Viterbi path for a iput prefix edig i a state of a HMM ca be costructed usig the stored solutios of Viterbi paths for a shorter iput prefix ad all HMM states. Greedy algorithms
5 Cocept: Types of algorithms ad algorithm aalyses, by Kut Reiert, 18. Oktober 2010, 21: A greedy algorithm sometimes works well for optimizatio problems. A greedy algorithm works i phases. At each phase: You take the best you ca get right ow, without regard for future cosequeces You hope that by choosig a local optimum at each step, you will ed up at a global optimum This strategy actually ofte works quite well ad for some class of problems it always yields a optimal solutio. Do you kow a simple graph problem which is solved greedily to optimality? Aother example would be the followig. Suppose you wat to cout out a certai amout of moey, usig the fewest possible bills ad cois. A greedy algorithm would do this would be to take the largest possible bill or coi that does ot overshoot. For example: To make $6.39, you ca choose: a $5 bill a $1 bill, to make $6 a 25c coi, to make $6.25 A 10c coi, to make $6.35 four 1c cois, to make $6.39 For US moey, the greedy algorithm always gives the optimum solutio (cautio: for other moey systems ot (imagie a currecy with uits of 1, 7, ad 10 ad try the algorithm for 15 uits). Brachadboud algorithms Brachadboud algorithms are geerally used for optimizatio problems. As the algorithm progresses, a tree of subproblems is formed. The origial problem is cosidered the root problem. A method is used to costruct a upper ad lower boud for a give problem. At each ode, apply the boudig methods. If the bouds match, it is deemed a feasible solutio to that particular subproblem. If bouds do ot match, partitio the problem represeted by that ode, ad make the two subproblems ito childre odes. Cotiue, usig the best kow feasible solutio to trim sectios of the tree, util all odes have bee solved or trimmed. A example of a brachadboud algorithms would be the folloig for the Travellig salesma problem (TSP). A salesma has to visit each of cities (at least) oce each, ad wats to miimize total distace travelled. Cosider the root problem to be the problem of fidig the shortest route through a set of cities visitig each city oce Split the ode ito two child problems: Shortest route visitig city A first Shortest route ot visitig city A first Cotiue subdividig similarly as the tree grows Brute force algorithms
6 1006 Cocept: Amortized aalysis, by Kut Reiert, 18. Oktober 2010, 21:22 A brute force algorithm simply tries all possibilities util a satisfactory solutio is foud. Such a algorithm ca be: Optimizig: Fid the best solutio. This may require fidig all solutios, or if a value for the best solutio is kow, it may stop whe ay best solutio is foud (Example: Fidig the best path for a travellig salesma) Satisficig: Stop as soo as a solutio is foud that is good eough (Example: Fidig a travellig salesma path that is withi 10% of optimal) Coclusio I preseted to you may categories with which you ca classify or label algorihms. Such a classificatio gives you a clear uderstadig about how a algorithm works ad a idicatio how to aalyze it. What algorithms do you kow? ad what labels would they get? I the followig we wil talk about how to aalyze the differet kid of algorithms with appropriate techiques. Cocept: Ru time aalysis The expositio is based o the followig sources, which are all required readig: 1. Corma, Leiserso, Rivest: Itroductio to algorithms, Chapter 18 I this sectio we recall the basic otatios for ru time aalyses ad the describe the differet cocepts of worstcase ru time, average case ru time, expected ru time, amortized ru time, ad the aalysis of competitiveess. Lets start by recallig the defiitios of the Ladau symbols (O,Ω,Ω,Θ,o,ω). O(f ) := {g : N R + : c R > 0, 0 N : N, 0 : g() c f ()} (1.1) Ω(f ) := {g : N R + : c R > 0, 0 N : N, 0 : g() c f ()} (1.2) Ω (f ) := {g : N R + : c R > 0 : m N : N, > m : g() c f ()} (1.3) Θ(f ) := {g : N R + : g O(f ) ad g Ω(f )} (1.4) o(f ) := g() {g : N R + : lim = 0} f () (1.5) ω(f ) := f () {g : N R + : lim = 0} g() (1.6) I the followig we list some commoly used adjectives describig classes of fuctios. Mid that we use the more commo = sig istead of the (more correct) sig. We say af fuctio f : is costat, if f () = Θ(1) grows logarithmically, if f () = O(log ) grows polylogarithmically, if f () = O(log k ()) for a k N. grows liearly, if f () = O() grows quadraticly, if f () = O( 2 ) We say af fuctio f :
7 Cocept: Amortized aalysis, by Kut Reiert, 18. Oktober 2010, 21: grows polyomially, if f () = O( k ), for a k N. grows superpolyomially, if f () = ω( k ), k N. grows subexpoetially, if f () = o(2 c ), 0 < c R. grows expoetially, if f () = O(2 c ) for a 0 < c R. After that remider lets itroduce the differet ru time defiitios ad explai them usig a example. worst case aalysis: We assume for both, the iput ad the executio of the algorithm the worst case. The latter is of course oly applicable for odetermiistic algorithms. best case aalysis: We assume for both, the iput ad the executio of the algorithm the best case. The latter is of course oly applicable for odetermiistic algorithms. average case aalysis: We average over all possible iput the ru time of our (determiistic) algorithm. expected ru time aalysis: Our algorithm rus depedig o the value of some radom variables for which we kow their distributios. Hece we try to estimate the expected ru time of the algorithm. amortized aalysis: Sometimes, a algorithm (usually a operatio o a data structure) eeds a log time to ru, but chages the data structure such that subsequet operatios are ot costly. A worst case ru time aalysis would be iappropriate. A amortized aalysis averages over a series of operatios (ot over the iput). competitiveess aalysis: For olie algorithms we eed a ew cocept of ru time aalysis. The mai cocept is to compare the ru time a algorithm eeds i the worst case (i.e. forall possible iputs) ot kowig the iput, with the rutime of a optimal offlie algorithm (which kows the iput). The wellkow (determiistic) quicksort algorithm for sortig a array chooses a fixed elemet as its pivot elemet, lets say w.l.o.g. the first oe. It arrages all smaller elemets o the left of the pivot, all larger oes o the right ad recurses o the two halfs. worst case aalysis: I the worst case, the left (or the right) half are always empty. Hece the worst case ru time is the solutio to the recurrece f () = ( 1) + f ( 1),f (0) = 0. Obviously f = O( 2 ). best case aalysis: I the best case, the left ad the right half differ i size by at most oe. Hece the best case ru time is the solutio to the recurrece f () = ( 1) + 2 f (/2),f (0) = 0. Obviously f = O( log). average case aalysis: We average over all possible iputs the ru time of the determiistic quicksort. The result is that o average quicksort eeds O( log ) comparisos. The depedece o the iput ad the bad worst case ru time of quicksort are worrysome. Quicksort (like may other algorithms) ca be made cosiderably more robust by radomizig the algorithm. I radomized quicksort we choose the pivot elemet radomly usig the value of a radom variable uiformly distributed over [1, ]. expected ru time aalysis: radomized quicksort ca be show to ru i expected time O( log) with high probability. We will discuss such a aalysis ow i detail (ad a similar oe usig skiplists later i the lecture). Example: radomized quicksort
8 1008 Cocept: Amortized aalysis, by Kut Reiert, 18. Oktober 2010, 21:22 Quicksort is a perfect example to demostrate the power of radomizatio. First we create a radomized versio of quicksort called RadQS by simply ot choosig the first elemet as pivot elemet but a radom elemet of the sublist we have to sort i each recursive call. Hece RadQs is a LasVegas style radomized, divideadcoquer algorithm. By doig so, we created a algorithm whose outcome depeds o radom choices, or put it differetly, o the values of some radom variables. Hece we have to aalyse its expected ru time or more particularly the expected umer of comparisos i a executio of RadQS. Lets do that. Let S (i) deote the elemet of rak i i the set S which we wat to sort. Now we defie the radom variable X ij to assume the value 1 if S (i) ad S (j) are compared i a executio of RadQS. The variable is 0 otherwise. So obviously the ru time of RadQs is proportioal to X = i=1 j>i X ij. The sum of radom variables is itself a radom variable. I the aalysis of the expected ru time we are hece iterested i E(X) = i=1 j>i E(X ij ). Let p ij be the probability that S (i) ad S (j) are compared i a executio of RadQS. The: E(X ij ) = p ij 1 + (1 p ij ) 0 = p ij So we have to cocetrate o the questio how large p ij is. To aalyze this we view the executio of RadQS as a biary tree T, i which each ode is labeled with a distict elemest y S. The elemets i the left subtree are the all to y, the elemets i the right subtree are all > tha y. Observe that the root of the tree is compared to all elemets i the tree, but there is o compariso betwee a elemet of the left subtree with a elemet of the right subtree. Hece two elemets S (i) ad S (j) are oly compared if oe is a acestor of the other. A iorder traversal of the tree output the elemets of S i sorted order. For the aalysis we focus o the levelorder traversal. This traversal goes levelbylevel ad lefttoright ad yields a permutatio π of the elemets of S. Now we make two key observatios: 1. a compariso betwee S (i) ad S (j) if ad oly if S (i) or S (j) occurs earlier i π tha ay elemet of rak betwee S (i) ad S (j). (Why?) 2. Ay of the elemets S (i),s i+1,...,s (j) is equally likely to be the first oe i the executio of RadQS. Hece the probability of either S (i) or S (j) beig the first oe is exactly From these two observatios follows that p ij = 2 j 1+1. Usig this value i our computatio of E(X) yields: i=1 j>i p ij = 2 i=1 j>i i+1 i=1 k=1 2 j k i=1 k=1 1 k 2 j 1+1. (1.7) (1.8) (1.9) = 2 H. (1.10) where H is the th harmoic umber. give that H l + Θ(1) it follows that the expected ru time of RadQS is O( log).
9 Cocept: Amortized aalysis, by Kut Reiert, 18. Oktober 2010, 21: Comparig this to the worst case time of the determiistic Quicksort which is O( 2 ) shows how powerful a coi throw ca be. As a secod cocept i ru time aalysis we have a look at amortized aalysis. Ofte (o matter whether for radomized or determiistic algorithms) the worst case aalysis is ot appropriate, because the wrost case caot happe ofte. No matter what the iput is. If this is the case, the the ru time averaged over a series of operatios caot be equal to times the worst case ru time. Ideed, the averaged ru time will ofte be much better. It is importat to ote that this is differet form the average ru time aalysis. There oe averages over the distributio of iputs. So it could still be that a algorithm, preseted with a bad iput rus very slowly. Here, we average over all possible excutios of the algorithms for ay give iput. To make this distictio clear, this type of aalysis is called amortized aalysis. Cocept: Amortized aalysis Imagie for example a stack. We have the followig operatios o the stack Pop(S) pops the top elemet of the stack ad returs it. Push(S,x) pushed elemet x o the stack. MultiPop(S,k) returs at most the top k elemets from the stack (it calls Pop k times). Obviously the operatios Pop ad Push have worst case time O(1). However the operatio MultiPop ca be liear i the stack size. So if we assume that at most objects are o the stack a multipop operatio ca have worst case cost of O(). Hece i the worst case a series of stack operatios is bouded by O( 2 ). We will ow use this example to illustrate three differet techiques to fid a more realistic amortized boud for operatios. The three methods to coduct the aalysis which are: The aggregate method The accoutig method The potetial method The aggregate method Here we show that for all, a sequece of operatios takes worstcase time T () i total. Hece, for this worst case, the average cost, or amortized cost is T ()/. Note that this method charges the same amortized cost to each operatio i the sequece of operatios, eve if this sequece cotais differet types of operatios. The other two methods ca assig idividual amortized costs for each type of operatio. We argue as follows. I ay sequece of operatios o a iitially empty stack, each object ca be popped at most oce for each time it is pushed. Therefore, the umber of times Pop ca be called o a oempty stack (icludig calls withi MultiPop), is at most the umber of Push operatios, which is at most. Hece, for ay value of, ay sequece of Push, Pop, ad MultiPop operatios takes a total of O() time. Hece the amortized cost of ay operatio is O()/ = O(1). The accoutig method I the accoutig method we assig differig charges to differet operatios, with some operatios charge more or less tha they actually cost. The amout we charge a operatio is called its amortized cost.
10 1010 Cocept: Amortized aalysis, by Kut Reiert, 18. Oktober 2010, 21:22 Whe a operatio s amortized cost exceeds its actual cost, the differece is assiged to specific objects i the data structure as credit. Credit ca the be used to pay later for operatios whose amortized cost is less tha their actual cost. Oe must choose the amortized costs carefully. If we wat the aalysis with amortized costs to show that i the worst case the average cost per operatio is small, the the total amortized cost of a sequece of operatios must be a upper boud o the total actual cost of the sequece. Moreover, this relatioship must hold for all sequeces of operatios, thus the total credit must be oegative at all times. Let us retur to our stack example. The actual costs are: Pop(S) 1, Push(S,x) 1, MultiPop(S,k) mi(k,s), where s is the size of the stack. Let us assig the followig amortized costs. Pop(S) 0, Push(S,x) 2, MultiPop(S,k) 0. Note that the amortized costs of all operatios is O(1) ad hece the amortized cost of operatios is O(). Note also that the actual cost of MultiPop is variable whereas the amortized cost is costat. Usig the same argumet as i the aggregate method it is easy to see that our accout is always charged startig with a empty stack. Each Push operatio pays 2 credits. Oe for its ow cost ad oe for the cost of poppig the elemet off the stack, either through a ormal Pop or through a MultiPop operatio. Please ote that this method ca assig idividual, differet amortized costs to each operatio. The potetial method Istead of represetig prepaid work as credit stored with specific objects i the data structure, the potetial method represets the prepaid work as potetial eergy or simply potetial that ca be released to pay for future operatios. The potetial is associated with the data structure as a whole rather tha with specific objects withi the data structure. It works as follows. We start with a iitial data structure D 0, o which operatios are performed. For each i = 1,2,..., we let c i be the actual cost of the ith operatio ad D i be the data structure that results after applyig the ith operatio. A potetial fuctio Φ maps each data structure D i to a real umber Φ(D i ), which is the potetial associated with the data structure D i. The amortized cost ĉ i of the ith operatio with respect to the potetial fuctio Φ is defied by ĉ i = c i + Φ(D i ) Φ(D i 1 ). that is, the amortized cost is the actual cost plus the icrease i potetial due to its operatio. Hece the total amortized cost of the operatios is: i=1 ĉ i = i=1 (c i + Φ(D i ) Φ(D i 1 )) = i=1 c i + Φ(D ) Φ(D 0 ). If we ca defie a potetial fuctio Φ so that Φ(D ) Φ(D 0 ), the the total amortized cost i=1 ĉi is a upper boud o the total actual cost. I practice we do ot kow how may operatios might be performed ad therefore
11 = k (1.12) Cocept: Amortized aalysis, by Kut Reiert, 18. Oktober 2010, 21: guaratee the Φ(D i ) Φ(D 0 ), for all i. It is ofte coveiet to defie Φ(D 0 ) = 0 ad the show that all other potetials are o egative. Lets illustrate the method usig our stack example. We defie the potetial fuctio o the stack as the umber of elemets it cotais. Hece the empty stack D 0 has Φ(D 0 ) = 0. Sice the umber of objects o the stack is ever egative we have Φ(D i ) 0 for all stacks D i resultig after the ith operatio. Let us ow compute the amortized costs of the various stack operatios. If the ith operatio o a stack cotaiig s objects is a Push operatio, the the potetial differece is Φ(D i ) Φ(D i 1 ) = (s + 1) s = 1 (1.11) Hece the amortized cost is: ĉ i = c i + Φ(D i ) Φ(D i 1 ) = = 2. If the ith operatio o a stack cotaiig s objects is a MultiPop(S,k) the k = mi(k,s) objects are popped off the stack. The actual cost of the operatio is k ad the potetial differece is: Φ(D i ) Φ(D i 1 ) = (s k ) s Hece the amortized cost is: ĉ i = c i + Φ(D i ) Φ(D i 1 ) = k k = 0. A similar result is obtaied for Pop. This shows that the amortized cost of each operatio is O(1) ad hece the total amortized cost of a sequece of operatios is O().
5 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 informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
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 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 informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationProject Deliverables. CS 361, Lecture 28. Outline. Project Deliverables. Administrative. Project Comments
Project Deliverables CS 361, Lecture 28 Jared Saia Uiversity of New Mexico Each Group should tur i oe group project cosistig of: About 612 pages of text (ca be loger with appedix) 612 figures (please
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 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 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 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 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 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 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 informationRunning Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis
Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a stepbystep procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
More informationLecture 2: Karger s Min Cut Algorithm
priceto uiv. F 3 cos 5: Advaced Algorithm Desig Lecture : Karger s Mi Cut Algorithm Lecturer: Sajeev Arora Scribe:Sajeev Today s topic is simple but gorgeous: Karger s mi cut algorithm ad its extesio.
More informationChatpun Khamyat Department of Industrial Engineering, Kasetsart University, Bangkok, Thailand ocpky@hotmail.com
SOLVING THE OIL DELIVERY TRUCKS ROUTING PROBLEM WITH MODIFY MULTITRAVELING SALESMAN PROBLEM APPROACH CASE STUDY: THE SME'S OIL LOGISTIC COMPANY IN BANGKOK THAILAND Chatpu Khamyat Departmet of Idustrial
More informationPresent Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value
Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig
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 informationAnalyzing Longitudinal Data from Complex Surveys Using SUDAAN
Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationTrading the randomness  Designing an optimal trading strategy under a drifted random walk price model
Tradig the radomess  Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore
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 information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
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 informationSimple Annuities Present Value.
Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX9850GB PLUS to efficietly compute values associated with preset value auities.
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
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 informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
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 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 informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
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 information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More informationA ConstantFactor Approximation Algorithm for the Link Building Problem
A CostatFactor Approximatio Algorithm for the Lik Buildig Problem Marti Olse 1, Aastasios Viglas 2, ad Ilia Zvedeiouk 2 1 Ceter for Iovatio ad Busiess Developmet, Istitute of Busiess ad Techology, Aarhus
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 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 informationThe Stable Marriage Problem
The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More informationThe Power of Free Branching in a General Model of Backtracking and Dynamic Programming Algorithms
The Power of Free Brachig i a Geeral Model of Backtrackig ad Dyamic Programmig Algorithms SASHKA DAVIS IDA/Ceter for Computig Scieces Bowie, MD sashka.davis@gmail.com RUSSELL IMPAGLIAZZO Dept. of Computer
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 informationChair for Network Architectures and Services Institute of Informatics TU München Prof. Carle. Network Security. Chapter 2 Basics
Chair for Network Architectures ad Services Istitute of Iformatics TU Müche Prof. Carle Network Security Chapter 2 Basics 2.4 Radom Number Geeratio for Cryptographic Protocols Motivatio It is crucial to
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationCS100: Introduction to Computer Science
Review: History of Computers CS100: Itroductio to Computer Sciece Maiframes Miicomputers Lecture 2: Data Storage  Bits, their storage ad mai memory Persoal Computers & Workstatios Review: The Role of
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationCHAPTER 7: Central Limit Theorem: CLT for Averages (Means)
CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measuretheoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationNotes on exponential generating functions and structures.
Notes o expoetial geeratig fuctios ad structures. 1. The cocept of a structure. Cosider the followig coutig problems: (1) to fid for each the umber of partitios of a elemet set, (2) to fid for each the
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationExploratory Data Analysis
1 Exploratory Data Aalysis Exploratory data aalysis is ofte the rst step i a statistical aalysis, for it helps uderstadig the mai features of the particular sample that a aalyst is usig. Itelliget descriptios
More informationLecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k.
18.409 A Algorithmist s Toolkit September 17, 009 Lecture 3 Lecturer: Joatha Keler Scribe: Adre Wibisoo 1 Outlie Today s lecture covers three mai parts: CouratFischer formula ad Rayleigh quotiets The
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 information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
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 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 informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
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 informationI. Why is there a time value to money (TVM)?
Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios
More informationA Faster ClauseShortening Algorithm for SAT with No Restriction on Clause Length
Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 4960 A Faster ClauseShorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More information23 The Remainder and Factor Theorems
 The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
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 informationTime Value of Money. First some technical stuff. HP10B II users
Time Value of Moey Basis for the course Power of compoud iterest $3,600 each year ito a 401(k) pla yields $2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationDomain 1: Designing a SQL Server Instance and a Database Solution
Maual SQL Server 2008 Desig, Optimize ad Maitai (70450) 18004186789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a
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 informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
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 informationTHE HEIGHT OF qbinary SEARCH TREES
THE HEIGHT OF qbinary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average
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 informationPage 1. Real Options for Engineering Systems. What are we up to? Today s agenda. J1: Real Options for Engineering Systems. Richard de Neufville
Real Optios for Egieerig Systems J: Real Optios for Egieerig Systems By (MIT) Stefa Scholtes (CU) Course website: http://msl.mit.edu/cmi/ardet_2002 Stefa Scholtes Judge Istitute of Maagemet, CU Slide What
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationSolutions to Exercises Chapter 4: Recurrence relations and generating functions
Solutios to Exercises Chapter 4: Recurrece relatios ad geeratig fuctios 1 (a) There are seatig positios arraged i a lie. Prove that the umber of ways of choosig a subset of these positios, with o two chose
More informationINVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology
Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology
More informationCenter, Spread, and Shape in Inference: Claims, Caveats, and Insights
Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the
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 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 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 informationSolutions to Selected Problems In: Pattern Classification by Duda, Hart, Stork
Solutios to Selected Problems I: Patter Classificatio by Duda, Hart, Stork Joh L. Weatherwax February 4, 008 Problem Solutios Chapter Bayesia Decisio Theory Problem radomized rules Part a: Let Rx be the
More informationSolving Logarithms and Exponential Equations
Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:
More informationChapter 7: Confidence Interval and Sample Size
Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum
More informationA Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design
A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 168040030 haupt@ieee.org Abstract:
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 informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationHeterogeneous Vehicle Routing Problem with profits Dynamic solving by Clustering Genetic Algorithm
IJCSI Iteratioal Joural of Computer Sciece Issues, Vol. 10, Issue 4, No 1, July 2013 ISSN (Prit): 16940814 ISSN (Olie): 16940784 www.ijcsi.org 247 Heterogeeous Vehicle Routig Problem with profits Dyamic
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive
More informationOn the Capacity of Hybrid Wireless Networks
O the Capacity of Hybrid ireless Networks Beyua Liu,ZheLiu +,DoTowsley Departmet of Computer Sciece Uiversity of Massachusetts Amherst, MA 0002 + IBM T.J. atso Research Ceter P.O. Box 704 Yorktow Heights,
More informationBENEFITCOST ANALYSIS Financial and Economic Appraisal using Spreadsheets
BENEITCST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal  Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts
More informationChapter 7  Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7  Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
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
More informationPENSION ANNUITY. Policy Conditions Document reference: PPAS1(7) This is an important document. Please keep it in a safe place.
PENSION ANNUITY Policy Coditios Documet referece: PPAS1(7) This is a importat documet. Please keep it i a safe place. Pesio Auity Policy Coditios Welcome to LV=, ad thak you for choosig our Pesio Auity.
More informationResearch Method (I) Knowledge on Sampling (Simple Random Sampling)
Research Method (I) Kowledge o Samplig (Simple Radom Samplig) 1. Itroductio to samplig 1.1 Defiitio of samplig Samplig ca be defied as selectig part of the elemets i a populatio. It results i the fact
More information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationEscola Federal de Engenharia de Itajubá
Escola Federal de Egeharia de Itajubá Departameto de Egeharia Mecâica PósGraduação em Egeharia Mecâica MPF04 ANÁLISE DE SINAIS E AQUISÇÃO DE DADOS SINAIS E SISTEMAS Trabalho 02 (MATLAB) Prof. Dr. José
More informationCS85: You Can t Do That (Lower Bounds in Computer Science) Lecture Notes, Spring 2008. Amit Chakrabarti Dartmouth College
CS85: You Ca t Do That () Lecture Notes, Sprig 2008 Amit Chakrabarti Dartmouth College Latest Update: May 9, 2008 Lecture 1 Compariso Trees: Sortig ad Selectio Scribe: William Che 1.1 Sortig Defiitio 1.1.1
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More information