Scheduling Contract Algorithms on Multiple Processors

Transcription

1 Fro: AAAI Technical Report FS Copilation copyright 200, AAAI ( All rights reserved. Scheduling Contract Algoriths on Multiple Processors Daniel S. Bernstein, Theodore. Perkins, Shloo Zilberstein Departent of Coputer Science University of Massachusetts Aherst, MA 0003 Lev Finkelstein Coputer Science Departent Technion--Israel Institute of Technology Haifa 32000, Israel Abstract Anytie algoriths offer a tradeoff between coputation tie and the quality of the result returned. They can be divided into two classes: contract algoriths, for which the total run tie ust be specified in advance, and interruptible algoriths, which can be queried at any tie for a solution. An interruptible algorith can be constructed fro a contract algorith by repeatedly activating the contract algorith with increasing run ties. The "acceleration ratio" of a schedule is a worst-case easure of how inefficient the constructed interruptible algorith is copared to the contract algorith. When the contracts are executed serially, i.e., on one processor, it is known how to choose contract lengths to iniize the acceleration ratio. We study the proble of scheduling contracts to n on processors in parallel. We derive an upper bound on the best possible acceleration ratio for processors, providing a siple exponential scheduling strategy that achieves this acceleration ratio. Further, we show that no schedule can yield a better acceleration ratio. Introduction In solving optiization probles, we are often faced with situations in which there is not enough tie to deterine an optial solution. We desire approxiation algoriths that can trade off coputation tie for quality of results. Algoriths with this property have been called anytie algoriths, and have been studied by researchers in artificial intelligence concerned with designing real-tie systes (Horvitz, 987; Russell & Zilberstein, 99). Anytie algoriths have been designed for a range of probles, including planning (Dean& Boddy, 988) and Bayesian inference (Wellan& Liu, 994). Also, general-purpose search algoriths such as local search and siulated annealing are naturally viewed as anytie algoriths. A useful distinction has been ade between two types of anytie algoriths: contract algoriths and interruptible algoriths. Contract algoriths require that the total coputation tie be given in advance. Once activated, a contract algorith ay not produce a useful result until the prespecified aount of tie has elapsed. This characteristic distinguishes the fro interruptible algoriths, which do not need to know the deadline a priori. Contract algoriths can be easier to design because they have access to ore inforation. Soe proble-solving techniques that can be viewed as contract algoriths include depth-bounded heuristic search and solving continuous control probles by discretizing the state space. What is coon to these techniques is that for a given contract tie they can select paraeters (e.g., the depth bound or the coarseness of the discretization) that liit the aount of coputation so as to guarantee returning a solution within the available tie. However, if a contract algorith is given ore tie than it expects, it ay have to be started fro scratch with new paraeters in order to iprove upon its current result. Interruptible algoriths are generally ore flexible and widely applicable than contract algoriths. An interruptible algorith can be fored by repeatedly running a contract algorith with increasing contract lengths, returning the last result produced in the case of an interruption. In the case of serial execution of contracts, (Russell & Zilberstein, 99) suggested the sequence contract lengths:, 2, 4, 8,... They showed that for any interruption tie t >, the last contract copleted is always of length at least t/4. This factor of four is the acceleration ratio of the schedule. In (Zilberstein et al., 999), it was shown that no sequence of contracts on a single processor can reduce the acceleration to below four. By scheduling the contract algorith on parallel processors, it is possible to achieve an acceleration ratio of less than four. In this paper, we describe a siple exponential strategy for scheduling a contract algorith on processors. By analyzing this strategy, we derive an explicit forula for an upper bound on the optial acceleration ratio in ters of ra. This bound approaches as approaches infinity. Furtherore, we show that no schedule yields a better acceleration ratio, and thus the bound is tight. Finally, we discuss extensions to this work and the connection between our proble and a proble involving ultiple robots searching for a point on ultiple rays. Scheduling a contract algorith on ultiple processors An anytie algorith A, when applied to an optiization proble instance for tie t, produces a solution of soe quality QA(t). The function QA is called the perforance profile of the algorith A on the instance. In general, one

2 does not know the perforance profile of an algorith on a proble instance. But the concept of a perforance profile is useful in reasoning about anytie algoriths. We assue that the perforance profile of an anytie algorith on any proble instance is defined for all t > 0 and is a onotonically non-decreasing function of t. We wish to construct an interruptible algorith fro a contract algorith by scheduling a sequence of contracts on processors in parallel. A schedule is a function X : {,...,} x N --+ IL where X(i,j) is the length of the jth contract run on processor i. We assue, without loss of generality, that X(, ) = and that X(i,j) > I for all i and j. A contract algorith A along with a schedule X defines an interruptible algorith B. When B is interrupted, it returns the best solution found by any of the contracts that have copleted. Since we assue perforance profiles are onotonic, this is equivalent to returning the solution of the longest contract that has copleted. This is illustrated in Figure. The algorith B has a perforance profile which depends on the profile of A and the schedule X. Before describing B s perforance profile, we need to ake a few definitions. We define the total tie spent by processor i executing its first j contracts as: = ~ X(i, For a given tie t, we define a function that specifies which contracts finish before that tie: x(t) = {(i,j)l We take the view that when a contract copletes at tie t, its solution is available to be returned upon interruption at any tie r > t. The length of the longest contract to coplete before tie t is: ] ax(i,j)e~x(t) X(i,j) if(i)x(t) Lx(t) I 0 if (I x (t) Thus, the perforance profile for the interruptible algorith B is QB(t) = QA(Lx(t)). We wish to find the schedule X that is optial for a given nuber of processors, independent of the particular contract algorith being used or the proble being solved. We copare schedules based on their acceleration ratios, which is a easure siilar to the copetitive ratio for on-line algoriths (Sleator & Tarjan, 985). Definition The acceleration ratio, R(X), for a given schedule X on processors is the sallest constant r for which QB(t) >_ QA ( t ) for all t > and any contract gorith A. The acceleration ratio tells us how uch longer the interruptible algorith has to work to ensure the sae quality as the contract algorith. The following lea will be useful in the later proofs. Lea Forall X, R(X) = supt> Z--x-x(t)" Proof: By the definitions above, QB(t) = QA(Lx(t)) QA t ) for all t >. Since this holds for any algorith A, we can suppose an algorith A with perforance profile QA(t) = t. Thus Lx(t) >_ R.~(X) >>_ t for all t >. This iplies R(X) > supt> Lx-~" t To show that equality holds, assue the contrary and derive a contradiction with the fact that R.~(X) is defined as the sallest constant enforcing the inequality between QB and QA. [] We define the inial acceleration ratio for processors to be = i{ R(X). In (Zilberstein et al., 999), it was shown that R~ = 4. the following sections, we provide tight bounds on this value for arbitrary re. Upper bound We first prove a lea foralizing the idea that the worst tie to interrupt the schedule is just as a contract ends. Lea 2 For all X, t sup - sup t>l Lx(t) (i,j)#(,) ax(i,j) Lx() Proof: Lx(t) is left-continuous everywhere and piecewise constant, with the pieces deliited by the tie points. For t >, ~ is piecewise linear, increasing, and left-continuous. Thus, the extrea of Lx--~(t can only occur at the points Gx (i, j), (i, j) # (, ); no other in tie ay play a role in the supreu. [] Theore R* < (~+)~-~-. Proof: Consider the schedule X(i, j) (+ ) ~-~+~/s-~). Note that in the one-processor case this reduces to X (i, j) 2 j-. It is straightforward to show that for (i, j) ~ (, X(i-l,j) ifi~l Lx(ax(i,j)) = X(,j - ) if/= Also, the following is true for all (i,j) (, ) = ~ X(i,k) k---=l = + Y = ( + ) -~- ~--~.(rn + k = (+l)i-l~-" (+l)j+l-(+l))-- < ( + I) -~"~ f/% t

3 Perforance profile of the interruptible algorith a" o~ _= o I, Processor X(l,2) Processor 2 x(2,0 X(2,2) [ Processor 3 x(3a) ] x(3.2) I tie Figure : Constructing interruptible algorith B by scheduling contract algorith A on three processors. So for all i,j such that i #, Lx() X(i, j) ( + ) - + < ( + ) -2+~- and for all i, j such that i = i and j #, Lx(Cx(i,j)) X(,j ) + 7 < ( + ) ~i--x -- ( + ) ( + ) Therefore R;, _< R(X) = ( + )~ + sup Lx() - (ij)#0,) o Lower bound In this section, it will be convenient to index contracts by their relative finish ties. The following function counts how any contracts finish no later than the jth contract on the ith processor finishes. For a schedule X, let q~x(i,j) = I{(i,j )lgx(i,/) <_ }l. We assue w.l.o.g, that no two contracts can finish at exactly the sae tie---it is straightforward to show that any schedule that doesn t satisfy this condition is doinated by a schedule that does. This assuption guarantees that ~x is one-to-one; it is also onto and thus an isoorphis. We refer to ~I x (i, j) as the global index of the jth contract run on processor i. We introduce a contract length function that takes as input a global index. For all i, j, let Yx(q~x(i,j)) = X(i,j). For notational siplicity, we will hereafter write Y in place of Yx. We further define a finish tie function that takes as input a global index: Gg(g~x(i,j)) =. Given this definition and the definition of acceleration ratio, it follows that Gy(k + ) _< R(Y)Y(k) for all k. Finally, we define a quantity to represent the su of the lengths of all the contracts finishing no later than contract k finishes: k a~y(k) y~y(). Lea 3 For an arbitrary schedule, for all k >, V y(k ) < R(Y)(a y(k -4- ) - G y(k) Proof: We first relate Gy and G~. Consider the contract with global index k + +. ~ --- Gy (k + l + ) is the su of the finishing ties for the last contracts to finish no later than contract k + + finishes. G~(k + + ) is the su over all processors of the finish tie for the last contract to finish on that processor no later than contract k + + finishes. It is straightforward to show that G~, (k + ) < ~ Gy(k + l + ) (and they are equal if the last contracts to finish include one fro each processor). Furtherore l= y ag(k+l+l) <_ R(Y) ) ZY(k+l /= /= = R(Y) (a r(k + ) - a y(k)). [] 2

4 (-I-l) Theore 2 R~n = Proof: Let us define H(k) = G~(k + )/G~y(k) for all k >. Fro Lea 3, we have G~(k + + ) < R(Y)(G~(k + ) - G~y(k)), and thus SO G y(k) ) G~(k++l) R(Y) a y(k + > G y (k + ) ( R(Y) (- We denote ~ > H(k + ). H(k)...H(k +- )] H*(k) = ax{h(k),..., H(k + )}. There are two cases to consider. In the first case, there exists soe k > such that H*(k ) = H(k + ). Then we have H(k )... H(k + ) < H(k + re,and Thus R~(Y) H(k + >-H(k + ). ( ) R(Y) H(k +l + ) H(k + ) - " We are interested in how sall R(Y) can be. Let C = H(k ~ + ). Suppose we iniize the right-hand side with respect to the only free variable, C, over the region C >. Setting the derivative to zero, we find d +l C (+ )C C+lC - dc C ra - C - (C TM - ) 2 -- =0 ( + )C(C "~- ) - e2~ = o =~ C2 - ( + l)c = The only solution is C = ( + ) ~. At the boundaries C and C = oo, the value goes to infinity, so this solution is the one and only iniu. Substituting into the inequality, we find R(Y) (+l)~ =-(+l)~+l ( + )~ - In the second case, we have H*(k) ~ H(k + for all k _>. Thus g*(k + ) = ax{g(k + ),...,g(k +),H(k ++ )} = ax{h(k + ),...,H(k +)} <_ H*(k), which eans that the H*(k) for a non-increasing sequence. This sequence ust be liited by, so li H*(k) = k--+~ for soe D >. Therefore lik~ H* (k) H* (k + ) = lik-~oo H*(k) "~ = D. Then = R(Y)(- lik-~/-/*(k- )..-H*(k- ( ( ) = R(g) - lisupk_,~h(k- )...H(k- ) = li supr(y)k~ >_ li sup H(k) = li H*(k) k.--~oo ----D. - H(k-)- H(k - ) Using the sae analysis as in the previous case, we have that R(Y) (+ )~+ Cobining this with Theore, we get the desired result. [] Discussion We described a siple exponential strategy for scheduling contract algoriths on ultiple processors to for an interruptible algorith. In addition, we proved that this schedule achieves the inial acceleration ratio aong the set of all schedules. In this work, we assued no knowledge of the deadline or of the contract algorith s perforance profile. In (Zilberstein et al., 999), the authors study the proble where the perforance profile is known and the deadline is drawn fro a known distribution. In this case, the proble of sequencing runs of the contract algorith on one processor to axiize the expected quality of results at the deadline can be fraed as a Markov decision process. It still reains to extend this work to the ultiple processor case. We note that the results presented in this paper are also applicable to a proble involving ultiple robots searching for a goal on ultiple rays. In this proble, k robots start at the intersection of rays and ove along the rays until the goal is found. An optial search strategy is defined to be one that iniizes the copetitive ratio, which is the worst-case ratio of the tie spent searching to the tie that would have been spent if the goal location was known initially. For k, the proble is trivial; the strategy that siply assigns one robot to each ray achieves a ratio of one. If k <, however, robots ay have to return to the origin so as not to neglect rays. The proble with k = and = 2 is studied in (Ricardo et al., 993), and it is shown that the optial copetitive ratio is 9. The general proble is briefly entioned in (Kao et al., 998), where a related proble is studied. It turns out that the analysis in this paper applies to the restricted case where k = -. A sequence of contract lengths for a ) 3

5 processor is analogous to a sequence of search extents for a robot, where a search extent is the distance a robot goes out on a ray before returning to the origin. It can be shown that the copetitive ratio for a ulti-robot schedule of search extents is I + 2r, where r is the acceleration ratio for the schedule. Acknowledgeents This work was supported in part by the National Science Foundation under grants IRI and INT , and by NASA under grants NAG and NAG Daniel Bcrnstein was also supported by a National Science Foundation Graduate Fellowship. Theodore Perkins was supported by a graduate fellowship fro the University of Massachusetts. Any opinions, findings, and conclusions or recoendations expressed in this aterial are those of the authors and do not reflect the views of the NSF or NASA. References Dean, T. & Boddy, M. (988). An analysis of tiedependent planning. In Proceedings of the Seventh National Conference on Artificial Intelligence (pp ). Horvitz, E. (987). Reasoning about beliefs and actions under coputational resource constraints. In Workshop on Uncertainty in Artificial Intelligence. Kao, M.-Y., Ma, Y., Sipser, M. & Yin, Y. (998). Optial constructions of hybrid algoriths. ournal of Algoriths, 29, Ricardo, B.-Y., Culberson,. & Rawlins, G. (993). Searching in the plane. Inforation and Coputation, 06, Russell, S.. & Zilberstein, S. (99). Coposing real-tie systes. Proceedings of the Twelth International oint Conference on Artificial Intelligence (pp ). Sleator, D. D. & Tarjan, R. E. (985). Aortized efficiency of list update and paging rules. Counications of the ACM, 28, Wellan, M. & Liu, C.-L. (994). State-space abstraction for anytie evaluation of probabilistic networks. In Proceedings of the Tenth Conference on Uncertainty in Artificial Intelligence. Zilberstein, S., Charpillet, E & Chassaing, P. (999). Realtie proble-solving with contract algoriths. In Proceedings of the Sixteenth International oint Conference on Artificial Intelligence. 4