Lecture 7 March 20, 2002

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Lecture 7 March 20, 2002"

Transcription

1 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 problems and formulaton of such problems as nstances of network flow problems and stable marrage problems. 7. Introducton/Motvaton Server Data Center Clent Fgure 7.. Smple model of the Internet. In our model, the Internet conssts of a collecton of clents, servers and data centers. A data center s smply a group of servers all stuated at the same place. Our goal s to map the clents to the servers (or data centers) gven a set of constrants. In our model, a clent can be servced by any server (ths s n contrast to the clent-cache model present the prevous week where data s avalable only on selected servers. Our goal s to ensure that localty s enforced and clents are served by servers whch are located near them. 7.. Obectves In makng assgnments, the followng are the obectves that we would lke to meet:. Assgn clents to close servers. We can defne dstance accordng to varous measures (.e., png latency or AS hops). 2. Map clents to servers wth low utlzaton. As shown n Fgure 7.2, we wll consder 2 ways of modelng the cost of utlzaton. In one model, there s no cost untl the load on a server 7-

2 MIT Lecture 7 March 20, 2002 Sprng 2002 reaches a hard capacty Ñ Ü, and thereafter the cost s nfnte; n the second model, cost ncreases as a (typcally convex) functon of utlzaton. 3. Mnmze the bandwdth cost. The cost of bandwdth s typcally a concave functon wth respect to bandwdth utlzaton as shown n Fgure 7.3. We note that obectves and 2 are desrable from the perspectve of the clents/users whle obectve 3 s desrable from the perspectve of the servers/provders. Also, the frst two obectves usuallly act n opposton to the thrd. For the purposes of ths lecture, we wll focus on obectves and 2. Cost Cost +oo 2 C max Load Bandwdth Utlzaton Fgure 7.2. Possble cost models. Fgure 7.3. Graph of cost versus bandwdth What Happens n Practce We ll focus n ths talk on ways to model and solve abstract load balancng problems; however, actual load balancng systems typcally nvolve much more complex machnery than ust a core algorthm. In practce, our load balancng system may need to do qute a bt of work ust to mantan the problem nstance (clent nformaton, server nformaton, and localty nformaton between the two) whch s perodcally fed to the load balancng algorthm. In practce, the number of clents s too large for us to assgn them ndvdually to servers, so we usually aggregate clents n some fashon (.e., by network or by IP address). Servers need to montored to make sure that f a server crashes, we remove t from the problem nstance. Fnally, one must somehow determne and mantan a measure of localty (e.g. AS hops) between clents and servers. Our load balancng algorthm wll perodcally re-compute assgnments from clents to servers, whch then need to be publshed (e.g., usng DNS). In terms of tmescale, any tme a clent s reassgned to a new server t takes some tme for traffc from that clent to mgrate fully to the new server, especally for long connectons such as streams. So we are lookng at a system that wll probably be updatng ts assgnments every few seconds or few mnutes. 7.2 Network Flows In ths secton we descrbe how to formulate ncreasngly-sophstcated varants of the clent-server assgnment problem each as network flow problems. 7-2

3 MIT Lecture 7 March 20, 2002 Sprng Fndng a Feasble Assgnment We start wth the followng basc formulaton of the problem: we have Ò clents (whch are actually aggregates of ndvdual clents as descrbed above) and Ñ servers. Each clent has an assocated Ñ Ò µ and each server has an assocated Ô ØÝ µ. Each clent can be mapped to a subset of the servers (presumably those close to t). We can represent ths problem nstance usng a drected bpartte graph as shown n Fgure 7.4. We allow the demand for a clent to be splt between varous servers, and we want to determne f a feasble assgnment exsts whch satsfes the clent demands and respects the server capactes. demand() capacty() s hgh capacty u = capacty() t t u = demand() s n clents n clents Fgure 7.4. Basc model. Fgure 7.5. As a max-flow problem. To solve ths problem, we transform t nto an equvalent max-flow problem by addng a source node,, and a snk node, Ø. We also add an edge between and all the clents and also an edge between all the servers and Ø. The capactes for the source-clent edges are set as Ñ Ò µ whle that for the server-snk edges are set as Ô ØÝ µ. A feasble assgnment exsts (gven by the flows on the nteror edges) only f the max -Ø flow saturates all of the edges leavng the source. The well-known max-flow problem s defned as follows: Gven a drected graph Î µ, source and snk nodes and Ø, and non-negatve capactes Ù for all edges µ ¾, fnd the maxmum flow Ü ¾Ê one may send from the source to the snk. As a mathematcal program, Maxmze Ü µ¾ nodes ¾ Î Ò Ø Ü Ü ¼ (Flow Conservaton) edges µ ¾ ¼ Ü Ù (Lower and Upper Capactes) The max-flow problem has been extremely well-studed and many effcent soluton algorthms exst for ts soluton Mnmzng Maxmum Utlzaton We next consder an extenson of our problem n whch we wsh to mnmze the maxmum utlzaton among the servers. To do so, we set the capacty of the edge onng each server to the snk as Ô ØÝ µ, as shown n Fgure 7.6. We now want to mnmze such that there exsts a 7-3

4 MIT Lecture 7 March 20, 2002 Sprng 2002 max flow saturatng all of the edges leavng the source. Ths turns out to be a well studed problem called the parametrc max-flow problem. Interestng note: algorthms exst whch can solve ths type of parametrc max flow problem n essentally the same amount of tme as a standard max flow problem (ths s a result of Gallo, Grgorads, and Taran). s hgh capacty u = λcapacty() t t cost c hgh capacty u t = capacty() zero cost t u = demand() s n clents n clents Fgure 7.6. Mnmzng Max Utlzaton. Fgure 7.7. Addng Mappng Costs Addng a Cost to the Mappng Between Clents and Servers Next, we add a cost for each connecton between the clent and the server as shown n Fgure 7.7 and attempt to fnd a mnmal-cost assgnment of the demands. The problem now becomes a mn-cost flow problem: Mnmze nodes ¾ Î edges µ ¾ Ü µ¾ Ü Ü ¼ Ü Ù We assgn to each clent, Ñ Ò µ, to each server, ¼, and to Ø, Ø The mn-cost flow problem s also well-studed and effcently solvable Replacng Capacty wth a Cost Functon Ñ Ò µ. All of the prevous formulatons adhere to our frst model of utlzaton cost (Fgure 7.2) n whch utlzaton of a server s free up untl the server s load reaches some hard capacty lmt. We now consder the second model of utlzaton cost, n whch we replace the capactes, Ô ØÝ µ, wth load-dependent cost functons as shown n Fgure 7.8. Ths changes only the obectve functon of our mn-cost flow problem: Mnmze nodes ¾ Î edges µ ¾ Ü 7-4 µ¾ Ü µ Ü ¼ Ü Ù

5 MIT Lecture 7 March 20, 2002 Sprng 2002 The server-snk edges may contnue to have upper capactes Ù Ø, although ths s no longer necessary as our cost functons can express an upper capacty by umpng to nfnty when load reaches a certan level. Also, by adustng the costs assocated wth the server-snk edges versus the costs assocated wth the clent-server edges, we can select whch obectve we wsh to emphasze more: mappng wth good localty or achevng low utlzaton. Assumng the cost functons are all convex (as s typcally the case), the entre obectve functon wll be convex, so we are mnmzng a convex functon over a convex polyhedral set of feasble solutons; such a problem stll les wthn the realm of nce optmzaton problems. Moreover, f we approxmate the convex cost functons as pecewse lnear functons, we can reduce ths problem back to the smpler prevous (lnear) mn-cost flow problem. To do so, we replace each server-snk lnk wth several parallel lnks, each representng one lnear pece of the pecewse lnear cost functon as shown n Fgure 7.9. Cost A hgh capacty (convex) cost c (x ) t t 2 A (lnear) cost c x hgh capacty t Load 4 A 2 2 2, A 3 4 Server t 4,A n clents 4,2A (Capacty, Cost) Fgure 7.8. Replacng capactes wth costs. Fgure 7.9. Pecewse lnear approxmaton. There are no known good technques for solvng such problems wth non-convex cost functons; moreover, t was shown n a prevous lecture that such problems wth concave cost functons are NP-Hard and also very dffcult to approxmate. 7.3 Advanced Topcs for Network Flows We now proceed to dscuss some nterestng research topcs related to the above network flow problems Unsplttable Flows For the unsplttable flow problem, we assume for a gven nstance of our basc assgnment problem wth costs (Fgure 7.7) that a feasble fractonal assgnment exsts. The queston that we would lke to ask s whether an assgnment exsts for whch the demand for each clent cannot be splt between servers and s hence satsfed by only one server. The approach that we wll take to solve 7-5

6 MIT Lecture 7 March 20, 2002 Sprng 2002 ths problem s to take the fractonal soluton and round t to gve us an unsplttable soluton wth no worse cost, allowng the capacty constrants to be slghtly volated n the process (we shall refer to the excess capacty as congeston ). Congeston may be defned two dfferent ways: () the addtve amount we ncrease capacty, or () the multplcatve factor by whch we must scale capacty. The problem then becomes that of fnd the unsplt assgnment wth mnmum congeston,.e. that mnmzes the maxmum congeston experenced over all edges, assumng that a feasble fractonal assgnment ntally exsts. There s a lower bound for ths problem whch states that n the worst case, one experences congeston of at least Ñ Ü Ñ Ü Ñ Ò µ. Ths s addtve congeston, as opposed to multplcatve congeston. The example gven n Fgure 7.0 satsfes ths bound. In ths example, we ve reversed the locaton of the clents and servers, as s often done wth unspltable flow problems, snce we now want to route flow on dsont paths from the source (whch was formerly the snk) to each of the clents. Here, the total demand s equal to total capacty, and a feasble fractonal assgnment exsts. However, because the demand of each clent s but each server has Ñ Ñ ½ only capacty ½, each server can satsfy only one clent ntegrally wthout experencng congeston. Snce there s one clent more than the number of servers, one of the servers wll have to servce 2 clents n an unsplt soluton, and the edge lnkng the source to that server wll be congested by an addtve amount whch gets arbtrarly close to Ñ Ü as Ñ gets large. d m = m+ s complete bpartte graph d m = m+ m+ clents d = m+ m m+ Fgure 7.0. Example whch satsfes lower bound. It can be shown that n the bpartte case, there exst smple polynomal-tme approxmaton algorthms whch acheve an addtve congeston of Ñ Ü. It s an open queston as to when we can meet ths bound wthout the bpartte constrant Vector (Mult-resource) Load Balancng The vector load balancng problem s formulated by generalzng the demand and capacty functons from scalars to vectors. An example s gven n Fgure 7.. Clearly a problem that allows for fractonal assgnment can be solved as a lnear program by addng more constrants. The nterestng queston s thus whether there s a feasble ntegral assgnment for a gven problem. We know that such a problem s NP-Complete, snce t s NP-Complete for scalars (t s the unsplttable flow problem). It mght be nterestng to know f the problem can be solved more effcently than 7-6

7 MIT Lecture 7 March 20, 2002 Sprng 2002 wth lnear programmng (ponted out by Bobby). It mght also be nterestng to see f any results for the unsplttable flow problem generalze to the vector problem. demand() cost c capacty() = bandwdth CPU tme dsk memory = bandwdth CPU tme dsk memory n clents Unknown Demand Fgure 7.. An example of a vector load balancng problem Oftentmes, we know the number of requests made by a clent, but we do not know beforehand the actual load whch wll be mposed on the servers by the requests. Thus, we may be nterested to fgure out how we can do load assgnment wthout full knowledge of the demand. One smple approach s to derve a scalng factor based on the total system-wde load and requests measured n some hstorcal nterval of tme: total load total # of requests. We then estmate the demand for each clent based on the number of requests on the assumpton that the rato s a constant across all clents. Although ths approach seems to provde us wth a reasonable approxmaton, the assumpton that the rato s a constant across all clents s questonable. In partcular, f we consder the aggregaton of clents by geographc locaton, then due to dfferent tme zones, t s clear that the nature of the actvtes of dfferent clents would be dfferent for dfferent tme zone (.e. clents n areas where t s nghttme may have slow dalup access whle those n areas where t s daytme may be usng hgh speed Internet access n ther offces). The challenge s thus to come up wth a scheme that can handle unknown demand wthout assumng that s system-wde constant. The mappng through DNS s prmarly responsble for ths problem, snce the clent requests can be dentfed wth IP address of the clent s DNS server, but the load nduced on servers s dentfed wth the actual IP address of the clent (not the clent s DNS server) Dynamc Stablty Clearly, the demands of clents are not constant, but they change n tme. Hence, we want an algorthm that wll not make changes n assgnment f there are only small changes n the demand. We would lke to have persstent mappngs because ths s requred n cachng and state preservaton. For example, t would be very annoyng for a user f hs shoppng cart gets lost frequently because hs browser s drected to another server. Ths s also to make the system mmune to dynamc nstablty and oscllatons. One smple approach for ths problem s to slghtly ncrease the cost of adacent lnks once a stable soluton s found for a gven lnk. 7-7

8 MIT Lecture 7 March 20, 2002 Sprng Stable Marrage Technques In the second half of ths talk, we dscuss methods for formulatng and solvng load balancng problems usng stable marrage technques. In the basc stable marrage problem, we have Ò men and Ò women, whch we can model as nodes n a bpartte graph. Assocated wth each man s a preference lst of all the women and assocated wth each woman s a preference lst of all the men. We then compute a marrage between the men and women, whch s nothng more than a matchng n our bpartte graph. w3, w2, w m, m2, m3 w3, w, w2 m2, m, m3 w, w3, w2 m, m3, m2 n men n women Fgure 7.2. A stable marrage. Gven some marrage, we call a man-woman couple Ñ Ûµ a rogue couple f they are not pared together but both of them prefer each-other to ther currently-assgned mates. A marrage s sad to be stable f t has no rogue couples, and unstable otherwse Gale-Shapley Algorthm Stable marrages are typcally computed usng the well-known Gale-Shapley algorthm. The algorthm s actually qute smple: each man proceeds through hs preference lst, startng from hs preferred mate, and he proposes to each of the women on the lst n sequence untl he s accepted. Each woman smply wats and accepts the best proposal ssued to her so far. If she s currently engaged to a man Ñ and then receves a proposal for a more favorable man, she wll drop Ñ and accept the new proposal, n whch case Ñ wll smply contnue down hs preference lst untl he s agan accepted. The algorthm s actually completely symmetrc, n that ether the men or the women may do the proposng, but for smplcty we ll assume here that the men do the proposng. More precsely, the algorthm repeats the followng operatons untl all men and women are pared: Fnd an un-pared man Ñ A proposal s ssued by Ñ to the next canddate on hs lst, Û If Ñ s more preferable to Û than her currently-assgned mate (f any), then she accepts Ñ and drops her currently-assgned mate (f any). Ths algorthm runs n tme Ç Ò ¾ µ n the worst case, snce we consder at most once each entry n each man s preference lst, and spend Ç ½µ tme per entry. Theorem 7.. At the end of the Gale-Shapley algorthm, everyone s engaged. 7-8

9 MIT Lecture 7 March 20, 2002 Sprng 2002 Proof: Note that the number of engaged men at the end wll be equal to the number of engaged women at the end. It could not be the case that there remans an un-pared couple Ñ Ûµ, snce at some pont Ñ would have proposed to Û, and she would have accepted. Theorem 7.2. The soluton obtaned by the Gale-Shapley Algorthm s a stable matchng. Proof: Assume that the fnal matchng s not stable,.e. that there exsts parngs Ñ ½ Û ½ µ and Ñ ¾ Û ¾ µ but Ñ ½ Û ¾ µ s a rogue couple. Snce Ñ ½ prefers Û ¾ to Û ½, Ñ ½ would have proposed to Û ¾ frst, who must have reected hm, so Û ¾ s current mate must be preferred to Ñ ½. Ths s a contradcton, snce Ñ ½ s hgher than Ñ ¾ n Û ¾ s preference lst. Hence, the matchng must be stable. A stable par Ñ Ûµ s a par whch s matched n some stable marrage. If Ñ Ûµ s a stable par, we say that Û s a stable partner of Ñ. Theorem 7.3. The outcome of the Gale-Shapley Algorthm s a man-optmal assgnment. That s, each man s pared wth hs most-preferred stable partner. Proof: Suppose not. Consder some executon order and consder the frst pont n tme when a stable par, Ñ ½ Û ¾ µ, s reected...., w2,..., w,... m Stable par..., w2,..., w3,... m2 Frst reecton of a stable par w w2..., m2,..., m,... Another stable par w3 Ths reecton can ether occur f Û ¾ s already engaged to a preferred man (Ñ ¾ ), or f Ñ ¾ comes along and proposes to Û ¾ whle she s engaged to Ñ ½. Snce we re presumng that Ñ ½ Û ¾ µ s a stable par, there s some stable matchng Å n whch Ñ ½ and Û ¾ are pared together let Û be the woman wth whom Ñ ¾ s pared n Å, meanng that Ñ ¾ Û µ s also a stable par. It cannot be the case that Û s hgher than Û ¾ on Ñ ¾ s preference lst, as that would mply that Ñ ¾ would have already proposed to Û and been reected, and we are assumng that Ñ ½ s reecton by Û ¾ s the frst reecton of a stable partner so far. Thus Û ¾ appears before Û on Ñ ¾ s preference lst. Ths means that Ñ ¾ Û ¾ µ s a rogue couple n Å, whch contradcts the stablty of Å. The precedng theorem tells us that remarkably, the order of the proposals durng the algorthm s rrelevant. The algorthm always arrves at the same stable matchng, even though there s not necessarly a unque stable matchng. Usng a smlar lne of reasonng, we can show that the matchng computed by the Gale-Shapley algorthm s also woman-pessmal, n the every woman ends up assgned to her least-preferred 7-9

10 MIT Lecture 7 March 20, 2002 Sprng 2002 stable partner. Rememberng the symmetry of the problem, however, one should note that f the women were to do the proposng, they would receve an optmal assgment whle the men would receve a pessmal assgnment. Some researchers have attempted to address the queston of computng stable marrages whch are far to both partes Truncatng the Preference Lsts In order to model the load balancng problem usng stable marrages, we need to consder a few extensons to the basc stable marrage problem above. Frst, we consder a class of problems where we truncate the preference lsts of the men and/or, snce for large load balancng problems, we shall see that mantanng complete preference lsts may lead to mpractcal storage requrements. For the purposes of determnng whom to reect, women can consder the men not on ther preference lsts as beng arbtrarly-ordered at the bottom of ther preference lsts. We now redefne Ñ Ûµ as a rogue couple f:. Both Ñ and Û are on each other s preference lsts, 2. Ñ s ether unassgned or prefers Û to hs current mate, and 3. Û s ether unassgned or prefers Ñ to her current mate The Gale-Shapley algorthm runs essentally the same on problem nstances wth truncated preference lsts. The outcome wll be a stable assgnment (.e. there wll be no rogue couples, as defned above), although t s possble that some men and women may end up unassgned. The followng theorem then holds. Theorem 7.4. Applyng the Gale-Shapley Algorthm wth truncated preference lsts stll gves a stable assgnment. Furthermore, each man who ends up assgned (unassgned) wll be assgned (unassgned) n every stable assgnment, and each assgned (unassgned) woman wll be end up beng assgned (unassgned) n every stable assgnment. Proof of ths theorem and the remanng theorems s omtted. For a complete treatment of these topcs, refer to the book The Stable Marrage Problem by Gusfeld and Irvng Dfferent Numbers of Men and Women Next, we can consder a class of problems where the numbers of men and women are not the same but the men and women all have complete preference lsts. The Gale-Shapley algorthm stll works n ths settng, and stll computes a stable matchng wth some nce propertes. Theorem 7.5. If there are fewer men than women, and everyone has a complete preference lst, the Gale-Shapley algorthm computes a stable matchng whch s stll man-optmal and womanpossble. Moreover, the men wll all be assgned, and the women wll be parttoned nto 2 sets: () those who are always assgned n every stable matchng, and () those who are never assgned n any stable matchng. If there are more men than women, then the women wll all be assgned and the men wll undergo a partton nto 2 sets. 7-0

11 MIT Lecture 7 March 20, 2002 Sprng 2002 Theorem 7.6. In the event that there are dfferent numbers of men and women and that preference lsts are ncomplete, there wll be a partton nto 2 sets nduced n both the men and n the women. Members of one set wll be assgned n every stable matchng, and members n the other set wll never be assgned n any stable matchng Many-to-One Assgnments: Addng Capactes To avod any awkward stuatons gong forward, we wll alter our dscusson to talk about assgnng medcal students to hosptals (ths was actually one of the orgnal applcatons of stable marrage technques). The dfference here s that hosptals can accept more than one medcal student; to each hosptal À we assgn an ntegral capacty...., H,... S C nodes, all wth dentcal preference lsts H..., S,... Capacty C Medcal Students Hosptals Fgure 7.3. Assgnng medcal students to hosptals (wth capactes). Ths generalzaton of the stable marrage problem s easy to accomodate f we smply splt every hosptal node À nto nodes wth unt capacty, all wth equvalent preference lsts. Ths transforms the problem nto an uncapactated stable marrage problem, whch we can solve usng the Gale-Shapely algorthm as descrbed above. One should note that t s not necessary to explctly perform ths transformaton. All we need to do s mantan, for each hosptal, a lst of medcal students currently assgned to t. If the hosptal ever receves a proposal when t has already reached capacty, t smply reects the least-preferred of these students (ncludng possbly the proposng student). If the algorthm stores these lsts of assgned students n heaps, then we wll only spend Ç ÐÓ Òµ tme per proposal (here, Ò s the number of students), for a total runnng tme whch s Ç Ä ÐÓ Òµ, where Ä s the sum of the lengths of all of the medcal students preference lsts (snce Ä s an upper bound on the total number of proposals ssued). 7-

12 MIT Lecture 7 March 20, 2002 Sprng Dealng wth Demands and Capactes We now take the fnal steps necessary to model our load balancng problem. Consder the problem of assgnng clents to servers, where each server has an ntegral capacty and each clent has an ntegral demand. The clents rank the servers accordng to localty, and the servers must choose some method of rankng the clents one potental way to do ths s to have a server Ë prefer clent to clent f clent B wll suffer less from a reecton (.e., clent B may have other choces n ts preference lst whch are nearly as good as Ë). We reduce the problem to an uncapactated problem by splttng each clent wth demand nto unt-szed nodes wth equal preference lsts, and by (mplctly) splttng each server wth capacty nto unt-szed nodes wth equal preference lsts. To avod utlzng excessve space, we typcally truncate each clent s preference lst to contan only ts frst few preferred servers. We may not need to explctly generate and store preference lsts for the servers at all, for example f we base these on the clent preference lsts as dscussed above. The Gale-Shapley Algorthm gves us a stable assgnment, but t may be fractonal n the sense that a clent may have some unts of ts demand assgned to dfferent servers. It s also possble that f there s not enough capacty, there may not be a matchng whch assgns all of the demand. However, due to the stable marrage propertes prevously dscussed, we at least know that f a clent has only an «fracton of ts demand satsfed by our stable marrage soluton, then t wll have exactly an «fracton of ts demand satsfed n every stable matchng. Smlarly, we utlze the same amount of capacty n each server n every stable marrage soluton Improvng the Runnng Tme Recall that when we added capactes to the servers, we could mplctly blow up server nodes nto clouds of unt nodes wthout affectng runnng tme much. Ths s not qute possble wth the clent nodes, however, so the runnng tme of our algorthm wll now be roughly lnear n the total demand over all clents, whch could be qute neffcent. To reduce ths runnng tme, we make requests n batches. That s, an aggregate clent node wll propose to a server wth a block of several unts of demand all at once. If the server can accept all of these, then we contnue to treat the clent node as an aggregate quantty. If the server accepts only part of the request, then we splt the clent node nto two smaller aggregate nodes, one of whch s assgned to the server and the other of whch contnues to ssue aggregate proposals. The ssuance of aggregate proposals by tself does not mprove our worst-case runnng tme, so we use the followng trck: a clent node wll only ssue an aggregate request f at least some fracton of ts demand remans unassgned. If ths s the case, there wll be Ç Ä µ total requests, and one may conclude after a bt of analyss that ths yelds a total runnng tme of Ç Ä ÐÓ Òµ. The algorthm may termnate wth some demand unassgned, but at most an fracton of every clent s demand wll reman unassgned, whch s perhaps a reasonable trade-off for a runnng tme whch s nearly lnear n the szes of the clents preference lsts. 7-2

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

1 Approximation Algorithms

1 Approximation Algorithms 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

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

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

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

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

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

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

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

What is Candidate Sampling

What is Candidate Sampling What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

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

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

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

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

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

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

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

Nonlinear data mapping by neural networks

Nonlinear data mapping by neural networks Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

PAS: A Packet Accounting System to Limit the Effects of DoS & DDoS. Debish Fesehaye & Klara Naherstedt University of Illinois-Urbana Champaign

PAS: A Packet Accounting System to Limit the Effects of DoS & DDoS. Debish Fesehaye & Klara Naherstedt University of Illinois-Urbana Champaign PAS: A Packet Accountng System to Lmt the Effects of DoS & DDoS Debsh Fesehaye & Klara Naherstedt Unversty of Illnos-Urbana Champagn DoS and DDoS DDoS attacks are ncreasng threats to our dgtal world. Exstng

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

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

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

L10: Linear discriminants analysis

L10: Linear discriminants analysis L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss

More information

Enabling P2P One-view Multi-party Video Conferencing

Enabling P2P One-view Multi-party Video Conferencing Enablng P2P One-vew Mult-party Vdeo Conferencng Yongxang Zhao, Yong Lu, Changja Chen, and JanYn Zhang Abstract Mult-Party Vdeo Conferencng (MPVC) facltates realtme group nteracton between users. Whle P2P

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

Energy Efficient Routing in Ad Hoc Disaster Recovery Networks

Energy Efficient Routing in Ad Hoc Disaster Recovery Networks Energy Effcent Routng n Ad Hoc Dsaster Recovery Networks Gl Zussman and Adran Segall Department of Electrcal Engneerng Technon Israel Insttute of Technology Hafa 32000, Israel {glz@tx, segall@ee}.technon.ac.l

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

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

To Fill or not to Fill: The Gas Station Problem

To Fill or not to Fill: The Gas Station Problem To Fll or not to Fll: The Gas Staton Problem Samr Khuller Azarakhsh Malekan Julán Mestre Abstract In ths paper we study several routng problems that generalze shortest paths and the Travelng Salesman Problem.

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

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

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

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.

More information

Section 5.3 Annuities, Future Value, and Sinking Funds

Section 5.3 Annuities, Future Value, and Sinking Funds Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme

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

7.5. Present Value of an Annuity. Investigate

7.5. Present Value of an Annuity. Investigate 7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on

More information

Hollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA )

Hollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA ) February 17, 2011 Andrew J. Hatnay ahatnay@kmlaw.ca Dear Sr/Madam: Re: Re: Hollnger Canadan Publshng Holdngs Co. ( HCPH ) proceedng under the Companes Credtors Arrangement Act ( CCAA ) Update on CCAA Proceedngs

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

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

Traffic State Estimation in the Traffic Management Center of Berlin

Traffic State Estimation in the Traffic Management Center of Berlin Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D-763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,

More information

Calculating the high frequency transmission line parameters of power cables

Calculating the high frequency transmission line parameters of power cables < ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,

More information

On the Optimal Control of a Cascade of Hydro-Electric Power Stations

On the Optimal Control of a Cascade of Hydro-Electric Power Stations On the Optmal Control of a Cascade of Hydro-Electrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;

More information

Simple Interest Loans (Section 5.1) :

Simple Interest Loans (Section 5.1) : Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

More information

ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING

ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 610-519-4390,

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

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

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

Compiling for Parallelism & Locality. Dependence Testing in General. Algorithms for Solving the Dependence Problem. Dependence Testing

Compiling for Parallelism & Locality. Dependence Testing in General. Algorithms for Solving the Dependence Problem. Dependence Testing Complng for Parallelsm & Localty Dependence Testng n General Assgnments Deadlne for proect 4 extended to Dec 1 Last tme Data dependences and loops Today Fnsh data dependence analyss for loops General code

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

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

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

More information

Power-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts

Power-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts Power-of-wo Polces for Sngle- Warehouse Mult-Retaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)

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

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

SDN: Systemic Risks due to Dynamic Load Balancing

SDN: Systemic Risks due to Dynamic Load Balancing SDN: Systemc Rsks due to Dynamc Load Balancng Vladmr Marbukh IRTF SDN Abstract SDN acltates dynamc load balancng Systemc benets o dynamc load balancng: - economc: hgher resource utlzaton, hgher revenue,..

More information

Period and Deadline Selection for Schedulability in Real-Time Systems

Period and Deadline Selection for Schedulability in Real-Time Systems Perod and Deadlne Selecton for Schedulablty n Real-Tme Systems Thdapat Chantem, Xaofeng Wang, M.D. Lemmon, and X. Sharon Hu Department of Computer Scence and Engneerng, Department of Electrcal Engneerng

More information

Time Value of Money Module

Time Value of Money Module Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the

More information

denote the location of a node, and suppose node X . This transmission causes a successful reception by node X for any other node

denote the location of a node, and suppose node X . This transmission causes a successful reception by node X for any other node Fnal Report of EE359 Class Proect Throughput and Delay n Wreless Ad Hoc Networs Changhua He changhua@stanford.edu Abstract: Networ throughput and pacet delay are the two most mportant parameters to evaluate

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

I. SCOPE, APPLICABILITY AND PARAMETERS Scope

I. SCOPE, APPLICABILITY AND PARAMETERS Scope D Executve Board Annex 9 Page A/R ethodologcal Tool alculaton of the number of sample plots for measurements wthn A/R D project actvtes (Verson 0) I. SOPE, PIABIITY AD PARAETERS Scope. Ths tool s applcable

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

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

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

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

More information

Network Aware Load-Balancing via Parallel VM Migration for Data Centers

Network Aware Load-Balancing via Parallel VM Migration for Data Centers Network Aware Load-Balancng va Parallel VM Mgraton for Data Centers Kun-Tng Chen 2, Chen Chen 12, Po-Hsang Wang 2 1 Informaton Technology Servce Center, 2 Department of Computer Scence Natonal Chao Tung

More information

Financial Mathemetics

Financial Mathemetics Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,

More information

Week 6 Market Failure due to Externalities

Week 6 Market Failure due to Externalities Week 6 Market Falure due to Externaltes 1. Externaltes n externalty exsts when the acton of one agent unavodably affects the welfare of another agent. The affected agent may be a consumer, gvng rse to

More information

Optimization of network mesh topologies and link capacities for congestion relief

Optimization of network mesh topologies and link capacities for congestion relief Optmzaton of networ mesh topologes and ln capactes for congeston relef D. de Vllers * J.M. Hattngh School of Computer-, Statstcal- and Mathematcal Scences Potchefstroom Unversty for CHE * E-mal: rwddv@pu.ac.za

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

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

A Resource-trading Mechanism for Efficient Distribution of Large-volume Contents on Peer-to-Peer Networks

A Resource-trading Mechanism for Efficient Distribution of Large-volume Contents on Peer-to-Peer Networks A Resource-tradng Mechansm for Effcent Dstrbuton of Large-volume Contents on Peer-to-Peer Networks SmonG.M.Koo,C.S.GeorgeLee, Karthk Kannan School of Electrcal and Computer Engneerng Krannet School of

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

J. Parallel Distrib. Comput. Environment-conscious scheduling of HPC applications on distributed Cloud-oriented data centers

J. Parallel Distrib. Comput. Environment-conscious scheduling of HPC applications on distributed Cloud-oriented data centers J. Parallel Dstrb. Comput. 71 (2011) 732 749 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. ournal homepage: www.elsever.com/locate/pdc Envronment-conscous schedulng of HPC applcatons

More information

"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *

Research Note APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES * Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789-794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC

More information

Joint Resource Allocation and Base-Station. Assignment for the Downlink in CDMA Networks

Joint Resource Allocation and Base-Station. Assignment for the Downlink in CDMA Networks Jont Resource Allocaton and Base-Staton 1 Assgnment for the Downlnk n CDMA Networks Jang Won Lee, Rav R. Mazumdar, and Ness B. Shroff School of Electrcal and Computer Engneerng Purdue Unversty West Lafayette,

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

Robust Design of Public Storage Warehouses. Yeming (Yale) Gong EMLYON Business School

Robust Design of Public Storage Warehouses. Yeming (Yale) Gong EMLYON Business School Robust Desgn of Publc Storage Warehouses Yemng (Yale) Gong EMLYON Busness School Rene de Koster Rotterdam school of management, Erasmus Unversty Abstract We apply robust optmzaton and revenue management

More information

) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance

) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance Calbraton Method Instances of the Cell class (one nstance for each FMS cell) contan ADC raw data and methods assocated wth each partcular FMS cell. The calbraton method ncludes event selecton (Class Cell

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

AN APPOINTMENT ORDER OUTPATIENT SCHEDULING SYSTEM THAT IMPROVES OUTPATIENT EXPERIENCE

AN APPOINTMENT ORDER OUTPATIENT SCHEDULING SYSTEM THAT IMPROVES OUTPATIENT EXPERIENCE AN APPOINTMENT ORDER OUTPATIENT SCHEDULING SYSTEM THAT IMPROVES OUTPATIENT EXPERIENCE Yu-L Huang Industral Engneerng Department New Mexco State Unversty Las Cruces, New Mexco 88003, U.S.A. Abstract Patent

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

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

Forecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network

Forecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network 700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School

More information

RequIn, a tool for fast web traffic inference

RequIn, a tool for fast web traffic inference RequIn, a tool for fast web traffc nference Olver aul, Jean Etenne Kba GET/INT, LOR Department 9 rue Charles Fourer 90 Evry, France Olver.aul@nt-evry.fr, Jean-Etenne.Kba@nt-evry.fr Abstract As networked

More information

1.1 The University may award Higher Doctorate degrees as specified from time-to-time in UPR AS11 1.

1.1 The University may award Higher Doctorate degrees as specified from time-to-time in UPR AS11 1. HIGHER DOCTORATE DEGREES SUMMARY OF PRINCIPAL CHANGES General changes None Secton 3.2 Refer to text (Amendments to verson 03.0, UPR AS02 are shown n talcs.) 1 INTRODUCTION 1.1 The Unversty may award Hgher

More information

Multi-Resource Fair Allocation in Heterogeneous Cloud Computing Systems

Multi-Resource Fair Allocation in Heterogeneous Cloud Computing Systems 1 Mult-Resource Far Allocaton n Heterogeneous Cloud Computng Systems We Wang, Student Member, IEEE, Ben Lang, Senor Member, IEEE, Baochun L, Senor Member, IEEE Abstract We study the mult-resource allocaton

More information

The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets

The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets . The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely

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

Efficient Striping Techniques for Variable Bit Rate Continuous Media File Servers æ

Efficient Striping Techniques for Variable Bit Rate Continuous Media File Servers æ Effcent Strpng Technques for Varable Bt Rate Contnuous Meda Fle Servers æ Prashant J. Shenoy Harrck M. Vn Department of Computer Scence, Department of Computer Scences, Unversty of Massachusetts at Amherst

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

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

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

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

A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATION-BASED OPTIMIZATION. Michael E. Kuhl Radhamés A. Tolentino-Peña

A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATION-BASED OPTIMIZATION. Michael E. Kuhl Radhamés A. Tolentino-Peña Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATION-BASED OPTIMIZATION

More information

Energy Conserving Routing in Wireless Ad-hoc Networks

Energy Conserving Routing in Wireless Ad-hoc Networks Energy Conservng Routng n Wreless Ad-hoc Networks Jae-Hwan Chang and Leandros Tassulas Department of Electrcal and Computer Engneerng & Insttute for Systems Research Unversty of Maryland at College ark

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

A generalized hierarchical fair service curve algorithm for high network utilization and link-sharing

A generalized hierarchical fair service curve algorithm for high network utilization and link-sharing Computer Networks 43 (2003) 669 694 www.elsever.com/locate/comnet A generalzed herarchcal far servce curve algorthm for hgh network utlzaton and lnk-sharng Khyun Pyun *, Junehwa Song, Heung-Kyu Lee Department

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