Scheduling Contract Algorithms on Multiple Processors

Transcription

1 Shedling Contrat Algorithms on Mltiple Proessors Daniel S. Bernstein, Theodore J. Perkins, Shlomo Zilberstein Department of Compter Siene University of Massahsetts Amherst, MA 0100 Lev Finkelstein Compter Siene Department Tehnion Israel Institte of Tehnology aifa 2000, Israel Abstrat Anytime algorithms offer a tradeoff between omptation time and the qality of the reslt retrned. They an be divided into two lasses: ontrat algorithms, for whih the total rn time mst be speified in advane, and interrptible algorithms, whih an be qeried at any time for a soltion. An interrptible algorithm an be onstrted from a ontrat algorithm by repeatedly ativating the ontrat algorithm with inreasing rn times. The aeleration ratio of a shedle is a worstase measre of how ineffiient the onstrted interrptible algorithm is ompared to the ontrat algorithm. When the ontrats are exeted serially, i.e., on one proessor, it is known how to hoose ontrat lengths to minimize the aeleration ratio. We stdy the problem of shedling ontrats to rn on proessors in parallel. We derive an pper bond on the best possible aeleration ratio for proessors, providing a simple exponential shedling strategy that ahieves this aeleration ratio. Frther, we show that no shedle an yield a better aeleration ratio. Introdtion In solving optimization problems, we are often faed with sitations in whih there is not enogh time to determine an optimal soltion. We desire approximation algorithms that an trade off omptation time for qality of reslts. Algorithms with this property have been alled anytime algorithms, and have been stdied by researhers in artifiial intelligene onerned with designing realtime systems (orvitz, 197; Rssell Zilberstein, Anytime algorithms have been designed for a range of problems, inlding planning (Dean Boddy, 19 and Bayesian inferene (Wellman Li, Also, generalprpose searh algorithms sh as loal searh and simlated annealing are natrally viewed as anytime algorithms. A sefl distintion has been made between two types of anytime algorithms: ontrat algorithms and interrptible algorithms. Contrat algorithms reqire that the total omptation time be given in advane. One ativated, a ontrat algorithm may not prode a sefl reslt ntil the prespeified amont of time has elapsed. This harateristi distingishes them from interrptible algorithms, whih do not need to know the deadline a priori. Contrat algorithms an be easier to design bease they have aess to more information. Some problemsolving tehniqes that an be viewed as ontrat algorithms inlde depthbonded heristi searh and solving ontinos ontrol problems by disretizing the state spae. What is ommon to these tehniqes is that for a given ontrat time they an selet parameters (e.g., the depth bond or the oarseness of the disretization that limit the amont of omptation so as to garantee retrning a soltion within the available time. owever, if a ontrat algorithm is given more time than it expets, it may have to be started from srath with new parameters in order to improve pon its rrent reslt. Interrptible algorithms are generally more flexible and widely appliable than ontrat algorithms. An interrptible algorithm an be formed by repeatedly rnning a ontrat algorithm with inreasing ontrat lengths, retrning the last reslt proded in the ase of an interrption. In the ase of serial exetion of ontrats, (Rssell Zilberstein, 1991 sggested the seqene of ontrat lengths:. They showed that for any interrption time, the last ontrat ompleted is always of length at least. This fator of for is the aeleration ratio of the shedle. In (Zilberstein et al., 1999, it was shown that no seqene of ontrats on a single proessor an rede the aeleration to below for. By shedling the ontrat algorithm on parallel proessors, it is possible to ahieve an aeleration ratio of less than for. In this paper, we desribe a simple exponential strategy for shedling a ontrat algorithm on proessors. By analyzing this strategy, we derive an expliit formla for an pper bond on the optimal aeleration ratio in terms of. This bond approahes 1 as approahes infinity. Frthermore, we show that no shedle yields a better aeleration ratio, and ths the bond is tight. Finally, we disss extensions to this work and the onnetion between or problem and a problem involving mltiple robots searhing for a point on mltiple rays. Shedling a ontrat algorithm on mltiple proessors An anytime algorithm, when applied to an optimization problem instane for time, prodes a soltion of some qality The fntion! is alled the performane profile of the algorithm on the instane. In general, one

2 > o o does not know the performane profile of an algorithm on a problem instane. Bt the onept of a performane profile is sefl in reasoning abot anytime algorithms. We assme that the performane profile of an anytime algorithm on any problem instane is defined for all and is a monotonially nondereasing fntion of. We wish to onstrt an interrptible algorithm from a ontrat algorithm by shedling a seqene of ontrats on proessors in parallel. A shedle is a fntion, where is the length of the th ontrat rn on proessor. We assme, withot loss of generality, that and that for all and. A ontrat algorithm along with a shedle defines an interrptible algorithm. When is interrpted, it retrns the best soltion fond by any of the ontrats that have ompleted. Sine we assme performane profiles are monotoni, this is eqivalent to retrning the soltion of the longest ontrat that has ompleted. This is illstrated in Figre 1. The algorithm has a performane profile whih depends on the profile of and the shedle. Before desribing s performane profile, we need to make a few definitions. We define the total time spent by proessor exeting its first ontrats as:! For a given time, we define a fntion that speifies whih ontrats finish before that time: " # %$ # ' We take the view that when a ontrat ompletes at time, its soltion is available to be retrned pon interrption at any time ( The length of the longest ontrat to omplete before time is: 9:<;4= *,+.0/ " if " A@ CB if DCB Ths, the performane profile for the interrptible algorithm is FE We wish to find the shedle that is optimal for a given nmber of proessors, independent of the partilar ontrat algorithm being sed or the problem being solved. We ompare shedles based on their aeleration ratios, whih is a measre similar to the ompetitive ratio for online algorithms (Sleator Tarjan, 195. Definition 1 The aeleration ratio, GA, for a given shedle on proessors is the smallest onstant I for whih E KJ<> LNM for all and any ontrat algorithm. The aeleration ratio tells s how mh longer the interrptible algorithm has to work to ensre the same qality as the ontrat algorithm. The following lemma will be sefl in the later proofs. Lemma 1 For all, # O9PRQ T = > Proof: By the definitions above, E U XZ \[ for all! WV > Sine this holds for any algorithm, we an sppose an algorithm with performane profile! ]. Ths XZ > _^ T = for all. This implies O9PRQ T = `> >. `> To show that eqality holds, assme the ontrary and derive a ontradition with the fat that is defined as the smallest onstant enforing the ineqality between E and. a We define the minimal aeleration ratio for to be GFb 6dRe In (Zilberstein et al., 1999, it was shown that G b. proessors. In the following setions, we provide tight bonds on this vale for arbitrary. Upper bond We first prove a lemma formalizing the idea that the worst time to interrpt the shedle is jst as a ontrat ends. Lemma 2 For all, OfPRQ 99g O9PRQ Proof: is leftontinos everywhere and pieewise onstant, with the piees delimited by the time points #. For!, T = > is pieewise linear, inreasing, and leftontinos. Ths, the extrema of T = > an only or at the points # h@ ; no other points in time may play a role in the spremm. a Theorem 1 ji b lk nmpo. o#mnvxw o#y Proof: Consider the shedle # q sr <t Note \{ that in the oneproessor ase this redes to # z. It is straightforward to show that for h@ *.+ #n if }@ if z Also, the following is tre for all h@ : # ~ \ \ #! r t.r t.r t.r t o#mn w om<v? ƒ oy r r k r.

3 a o i Performane profile of the interrptible algorithm soltion qality... Proessor 1 Proessor 2 Proessor X(1,1 X(2,1 X(,1 X(1,2 X(2,2 X(,2 X(1,... time Figre 1: Constrting interrptible algorithm by shedling ontrat algorithm on three proessors. So for all sh o#mn w,r t m< w.r t and for all sh that q #.r w Therefore GFb ji Gh 99g O9PRQ `5#7 r 7,.r r nmpo nmpo r Lower bond In this setion, it will be onvenient to index ontrats by their relative finish times. The following fntion onts how many ontrats finish no later than the th ontrat on the th proessor finishes. For a shedle, let # $ # %$ i m o # $ We assme w.l.o.g. that no two ontrats an finish at exatly the same time it is straightforward to show that any shedle that doesn t satisfy this ondition is dominated by a shedle that does. This assmption garantees that is onetoone; it is also onto and ths an isomorphism. We refer to # as the global index of the th ontrat rn on proessor. We introde a ontrat length fntion that takes as inpt a global index. For all, let # * For notational simpliity, we will hereafter write in plae of. We frther define a finish time fntion that takes as inpt a global index: # * # Given this definition and the definition of aeleration ratio, it follows that Fr i Gh for all. Finally, we define a qantity to represent the sm of the lengths of all the ontrats finishing no later than ontrat finishes:? Lemma For an arbitrary shedle, for all, r r i Ar q Proof: We first relate and. Consider the ontrat with global index ]r r. ` UrZr is the sm of the finishing times for the last ontrats to finish no later than ontrat r r finishes. r,r is the sm over all proessors of the finish time for the last ontrat to finish on that proessor no later than ontrat r r finishes. It is straightforward to show that r r i? r r (and they are eqal if the last ontrats to finish inlde one from eah proessor. Frthermore ` a Fr r i Gh?ƒ r Fr ƒ

4 i ^ ^ o Theorem 2 b 'k m o Proof: Let s define and ths so.. From Lemma, we have Fr r i r r Fr ƒ Fr for all Fr r Fr r b 1 b r r i r We denote D 0/ r There are two ases to onsider. In the first ase, there exists some sh that. Then we have, and Ths r r r lk r We are interested in how small an be. Let, r. Sppose we minimize the righthand side with respet to the only free variable,, over the region. Setting the derivative to zero, we find.r lk lk {.r ƒ r The only soltion is r At the bondaries and, the vale goes to infinity, so this soltion is the one and only minimm. Sbstitting into the ineqality, we find,r r nmpo r In the seond ase, we have ]r for all. Ths b r 0/ 1 0/ 1 b nm o r Fr r r r Fr whih means that the b form a noninreasing seqene. This seqene mst be limited by 1, so b for some z Gh C CGh O9PRQ O9PRQ. Therefore b b. Then b b qr b b O9PRQ ] 2 ] ] Using the same analysis as in the previos ase, we have that,r Combining this with Theorem 1, we get the desired reslt. a Disssion We desribed a simple exponential strategy for shedling ontrat algorithms on mltiple proessors to form an interrptible algorithm. In addition, we proved that this shedle ahieves the minimal aeleration ratio among the set of all shedles. In this work, we assmed no knowledge of the deadline or of the ontrat algorithm s performane profile. In (Zilberstein et al., 1999, the athors stdy the problem where the performane profile is known and the deadline is drawn from a known distribtion. In this ase, the problem of seqening rns of the ontrat algorithm on one proessor to maximize the expeted qality of reslts at the deadline an be framed as a Markov deision proess. It still remains to extend this work to the mltiple proessor ase. We note that the reslts presented in this paper are also appliable to a problem involving mltiple robots searhing for a goal on mltiple rays. In this problem, robots start at the intersetion of rays and move along the rays ntil the goal is fond. An optimal searh strategy is defined to be one that minimizes the ompetitive ratio, whih is the worstase ratio of the time spent searhing to the time that wold have been spent if the goal loation was known initially. For, the problem is trivial; the strategy that simply assigns one robot to eah ray ahieves a ratio of one. If _, however, robots may have to retrn to the origin so as not to neglet rays. The problem with ] and is stdied in (Riardo et al., 199, and it is shown that the optimal ompetitive ratio is 9. The general problem is briefly mentioned in (Kao et al., 199, where a related problem is stdied. It trns ot that the analysis in this paper applies to the restrited ase wherec. A seqene of ontrat lengths for a nmpo

5 proessor is analogos to a seqene of searh extents for a robot, where a searh extent is the distane a robot goes ot on a ray before retrning to the origin. It an be shown that the ompetitive ratio for a mltirobot shedle of searh extents is r I, where I is the aeleration ratio for the shedle. Aknowledgements This work was spported in part by the National Siene Fondation nder grants IRI and INT , and by NASA nder grants NAG2194 and NAG2146. Daniel Bernstein was also spported by a National Siene Fondation Gradate Fellowship. Theodore Perkins was spported by a gradate fellowship from the University of Massahsetts. Any opinions, findings, and onlsions or reommendations expressed in this material are those of the athors and do not reflet the views of the NSF or NASA. Referenes Dean, T. Boddy, M. (19. An analysis of timedependent planning. In Proeedings of the Seventh National Conferene on Artifiial Intelligene (pp orvitz, E. (197. Reasoning abot beliefs and ations nder omptational resore onstraints. In Workshop on Unertainty in Artifiial Intelligene. Kao, M.., Ma,., Sipser, M. in,. (199. Optimal onstrtions of hybrid algorithms. Jornal of Algorithms, 29, Riardo, B.., Clberson, J. Rawlins, G. (199. Searhing in the plane. Information and Comptation, 106, Rssell, S. J. Zilberstein, S. (1991. Composing realtime systems. Proeedings of the Twelth International Joint Conferene on Artifiial Intelligene (pp Sleator, D. D. Tarjan, R. E. (195. Amortized effiieny of list pdate and paging rles. Commniations of the ACM, 2, Wellman, M. Li, C.L. (1994. Statespae abstration for anytime evalation of probabilisti networks. In Proeedings of the Tenth Conferene on Unertainty in Artifiial Intelligene. Zilberstein, S., Charpillet, F. Chassaing, P. (1999. Realtime problemsolving with ontrat algorithms. In Proeedings of the Sixteenth International Joint Conferene on Artifiial Intelligene.