Gradual Relaxation Techniques with Applications to Behavioral Synthesis *

Transcription

1 Gradual Relaxaton Technques wth Applcatons to Behavoral Synthess * Zhru Zhang, Ypng Fan, Modrag Potkonjak, Jason Cong Computer Scence Department, Unversty of Calforna, Los Angeles Los Angeles, CA 90095, USA {zhruz, fanyp, modrag, cong}@cs.ucla.edu ABSTRACT Heurstcs are wdely used for solvng computatonal ntractable synthess problems. However, untl now, there has been lmted effort to systematcally develop heurstcs that can be appled to a varety of synthess tasks. We focus on development of general optmzaton prncples so that they can be appled to a wde range of synthess problems. In partcular, we propose a new way to realze the most constranng prncple where at each step we gradually relax the constrants on the most constraned elements of the soluton. Ths basc optmzaton mechansm s augmented wth several new heurstc prncples: mnmal freedom reducton, negatve thnkng, calbraton, smultaneous step consderaton, and probablstc modelng. We have successfully appled these optmzaton prncples to a number of common behavoral synthess tasks. Specfcally, we demonstrate a systematc way to develop optmzaton algorthms for maxmum ndependent set, tme-constraned schedulng, and soft real-tme system schedulng. The effectveness of the approach and algorthms s valdated on extensve real-lfe benchmarks.. MOTIVATION We have two strategc objectves n ths paper: development of general optmzaton prncples and applcatons to a wde range of synthess problems. The optmzaton goal s to advance the state of the art n the desgn of heurstc algorthms. Heurstcs are wdely used for solvng computatonal ntractable synthess problems. However, untl now, there has been lmted effort to systematcally develop heurstcs that can be easly appled to a varety of synthess tasks. In ths paper, we propose a new heurstc optmzaton paradgm that can be appled on a broad spectrum of computatonally ntractable problems. Whle the tradtonal most constranng prncple always addresses the most constraned part of the problem frst, we employ the most constranng prncple where at each step we make a decson that maxmally relaxes the constrants on the most constraned elements of the soluton. Ths basc optmzaton mechansm s augmented wth several new heurstc nsghts: mnmal freedom reducton, negatve thnkng, calbraton, smultaneous step consderaton, and probablstc modelng. We call them the gradual relaxaton technques. The mnmal freedom reducton prncple ams to make the mnmal possble quantum of decson at each step. The ratonale * Ths research s partally supported by Natonal Scence Foundaton under award CCR s that after a small step s made, one can better evaluate ts mpacts and prevent the heurstc from followng a greedy mode of optmzaton to produce local optmal solutons. The man way to realze ths prncple s to use negatve thnkng,.e., to decde what the optmzaton process wll not do at the next step, nstead of what to do. The optons that stay at the end of ths process form a set of decsons to yeld a hgh qualty soluton. Calbraton s a step where the chances for optmzaton along a partcular drecton are evaluated and compared. If a partcular drecton s not promsng, all efforts along that lne are termnated or assgned a lower prorty. A typcal example s resource allocaton. If the desgn s such that the number of requred adders cannot be reduced below some bound, and there s room for mnmzaton of the number of multplers, the schedulng s conducted so that the operaton that has to be assgned to multplers preserves maxmum slack. Moreover, n some cases, t s advantageous to smultaneously consder several small steps appled on several parts of the problem n order to evaluate ther compound mpact. Fnally, n order to further ncrease the effectveness of the proposed heurstc, we also propose a new, more realstc technque for probablstc modelng durng the optmzaton process. Optmzaton technques can rarely be developed completely out of context. Even more dffcult s to evaluate ther effectveness, unless they are appled on mportant, real-lfe problems and compared wth exstng methods. Therefore, our second and actually man goal s to develop a system of effectve and fast optmzaton technques for common behavoral synthess tasks. Specfcally, we demonstrate a systematc way to develop optmzaton algorthms for maxmum ndependent set, tmeconstraned schedulng, and soft real-tme system schedulng. The problems are selected not only because of ther mportance, but also because they have a very dfferent nature. For example, n statc schedule one has to map all of the operatons to control steps whle n the MIS problem the goal s to select only a subset of nodes. Whle behavoral synthess s stll lookng to establsh tself, the ntuton s that ths wll make algorthms more accurate. Consder the smple motvatonal example shown n Fgure. We lst all the possble confguratons of a CDFG wth operatons scheduled to 5 avalable control steps. We denote prob(, t to be the probablty of assgnng the operaton n control step t. Accordng to tradtonal approaches, unform dstrbuton s assumed. For example, we have prob(, = prob(, = prob(, =0., However, the realstc probabltes should be prob(, = 0.6, prob(, = 0. and prob(, = 0., whch are sgnfcantly dfferent from the prevous ones.

2 c-step c-step c-step c-step4 c-step5 Fgure. A smple example of the non-unform dstrbuton. RELATED WORK In ths secton, we survey the related work along the followng four drectons: maxmally constraned mnmally constranng heurstcs, force-drected optmzaton, statc and soft real-tme system schedulng, and maxmum ndependent set. Btner and Rengold [] were the frst to propose the systematc use of the maxmally constraned varable selecton paradgm. They use t to gude the search wthn a generc backtrackng search for the optmal soluton. The frst to use the maxmally constraned rule as part of fast heurstc program was Brelaz for graph colorng []. Another popular famly of heurstcs s based on noton of slack [7][9]. Slack s usually defned as value proportonal to the cardnalty of a set of stll vable optons for assgnng a specfc varable from the problem formulaton. In CAD lterature, the most constraned least constranng paradgm has been wdely used, mostly under the name of force-drected heurstcs [0]. A more global pcture of the role of a maxmally constraned mnmally constranng approach as an effcent heurstcs s gven n []. Schedulng s a mandatory task n behavoral synthess and complaton [7][8]. In behavoral synthess, schedulng s closely related to resource allocaton, a task where the hardware requred for a specfc applcaton s allocated. We target two mportant and wdely requred types of schedulng: statc and soft real-tme systems. For the former, we assume a synchronous data flow model of computaton [5]. Our goal s to mnmze the requred hardware for a gven samplng rate. The prolferaton of multmeda applcatons and soft real-tme system s ganng rapdly ncreasng attenton. The bass for softreal system schedulng was formed n the work done n the Garca-Molna research group [][]. In multmeda research, soft real-tme system are most often used to model vdeo and WWW servers []. Real tme system communty has placed emphass on a sound formal defnton of requrements assocated wth soft real-tme systems [6]. In CAD and embedded system lterature notable soft real-tme related efforts nclude [8][7][] Maxmum Independent Set (MIS s one of the most popular generc NP-complete problems [8]. For example, t was among the frst set of problems proved to be NP-complete [8]. MIS s manly used n one of ts modfcatons as a set durng graph colorng. For example, t has been expermentally demonstrated that n many domans, fndng an MIS s suffcent to make colorng popular benchmarks both fast and provably optmal [4].. GENERIC TECHNIQUE In ths secton, we ntroduce the seven man new nsghts that can be used to buld a new generaton of effcent heurstcs for a varety of computatonally ntractable problems. The nsghts are as follows: ( Most constraned prncple. The tradtonal most constraned paradgm for developng heurstcs looks to frst resolve the components (e.g., nodes or varable that partcpated n the largest number of constrants or the constrants that are very strct. The ratonale s that t s better to resolve ths stuaton early, whle there s stll enough freedom to resolve them. The components are usually resolved n such a way that the dffculty of stll unresolved constrants s mnmally mpacted. In practce, the effectveness of these heurstcs s manly due to the second nsght. We propose to use a new nsght that resolves the components that partcpate n the same constrants as the most dffcult components. Note that these are not necessarly the most dffcult components themselves. The ratonale s that these components can often be resolved n such a way as to make the most dffcult constrants more relaxed. ( Mnmal freedom reducton. The key mpedment to effectveness of a heurstc s the often greedy behavor of optmzaton, where short term benefts are acheved at the expense of ncreased dffculty and cost n later stages of optmzaton. One way to avod such behavor s to make a small gradual atomc decson and evaluate ts ndvdual mpact before commttng to large decsons. ( Negatve thnkng. Ths dea s a specfc way to realze the prevous paradgm. Essentally, t s often more benefcal to state what wll not be consdered as opton nstead of resolvng a specfc component of the soluton. The smlar deas appeared n [6] for a specal case of schedulng and n [4] for the standard cell global routng (.e., the teratve deleton method. We generalze t other problems. For example, for the MIS problem, t means that we wll elmnate nodes as canddate for MIS one by one, untl the remanng set of nodes does not have any ncdent edges. Ths s n contrast to the standard procedure of selectng nodes for MIS one by one. (4 Compoundng Varables. At frst sght t seems that t s mpossble to apply the prevous two paradgms to problems where varables can be assgned only to bnary values. For example, n the SAT problem, the negatve decson to not assgn a varable to s that the varable s assgned to 0. Ths dffculty can be resolved, f we consder two or more varables smultaneously. For example, we may decde that varable x and x j wll not be smultaneously assgned to value. Therefore, after ths decson, there are stll three optons how these two varables can be smultaneously assgned. (5 Smultaneous steps consderaton. The prevous dea can be further generalzed even when there s no ntrnsc need to create compound varables. The practce ndcates that t s often advantageous to smultaneously consder a small negatve decson on a set of varables. (6 Calbraton. In many stuatons, one can conclude that the way a subset of varables s resolved wll not have an mpact on the qualty of the fnal soluton. In that stuaton, there s no need to preserve the optons for these varables. The soluton to the resource allocaton problem dscussed n Secton s a typcal example of calbraton.

3 (7 Probablstc modelng. Our motvatonal example ndcates ths assumpton s rarely close to realty, even when very small nstances are consdered. Therefore, we propose to model optons on how to resolve each varable n a nonunform way as a functon of all constrants that are mposed on that partcular varable. 4. WHEN IS GRADUAL RELAXATION MOST EFFECTIVE? It s well know that dfferent types of algorthms are best suted for dfferent types of optmzaton problems. For example, greedy algorthms are optmal and the fastest for the problems that have matrod structure [9]. Smlarly, n order to apply dynamc programmng, the problem of nterest has to have the property that the optmal soluton s composed of optmal sub-solutons. At the same tme, t s mportant to realze that the effectveness of a partcular algorthm also greatly depends on the structure of the nstance. For example, t s well known that the graph colorng problem for sparse graphs can be almost always solved optmally [5][5]. In general, however, t s not easy to dentfy classes of problems and nstances that can be solved usng a partcular technque effcently. Most often these nsghts are obtaned by comprehensve studes and statstcal analyss [0][]. Nevertheless, t s possble to obtan some mportant nsghts by consderng specfc characterstcs of the problem or the nstance. We frst focus our attenton on the type of nstances where the system of gradual relaxaton performs well. Force drected approaches work well when there exst sgnfcant dfference n slacks (.e., range of value that a partcular varable can be assgned to and strctness of constrants that are assocated wth each varable. The technque of gradual relaxaton through negatve thnkng provdes addtonal effectveness when a large number (or percentage of varables have sgnfcant slack and when varables have more complex nteractons among a large number of constrants. Ths observaton s a consequence of the fact that n these stuaton gradual relaxaton wll better reveal the mpact of each optmzaton decson. Compoundng varables and smultaneous steps consderaton technques are most effectve when each varable tself has a set of potental values wth small cardnalty. In partcular, t s effectve for bnary varables. Calbraton s most effectve for nstances where the fnal soluton only nvolves relatvely few occurrences of each type of resources. Fnally, probablstc modelng s most relevant for large and complex nstances. Furthermore, we expect that problems where each varable can be assgned to many values (e.g. colorng and schedulng are more amenable for optmzaton usng gradual relaxaton than problems that are defned on bnary varables (e.g. SAT. 5. DRIVER EXAMPLES In ths secton, we demonstrate how a properly selected subset of the proposed heurstc optmzaton prncples can be appled to common behavoral synthess tasks. 5. Maxmum Independent Set and Graph Colorng In ths subsecton, the MIS problem s used to demonstrate the most constranng, mnmal freedom reducton, and negatve thnkng prncples. In behavoral synthess, the resource sharng problem s usually transformed to the graph colorng problem for the correspondng resource conflct graph. Unfortunately, graph colorng s an NPcomplete problem n general [8], defned as: Problem: Graph k-colorng Gven: ( A graph G (V, E wth vertex set V and edge set E, ( A postve nteger K V. Objectve: To determne whether there exsts a colorng of G usng no more than k unque colors so that for every edge (u, v E, u and v have dfferent colors. The graph colorng problem s to fnd the mnmum number of colors needed to color a gven graph. An effcent heurstc to solve ths problem was presented n [4]. Its approach s to dvde the whole problem nto seral ndependent set problems,.e., n every step the algorthm searches a constraned ndependent set and assgns one color to ts vertces. The ndependent set search s bascally an teratve mprovement algorthm, whch randomly generates an ntal soluton and defnes the move as vertex excluson/ncluson. The maxmum ndependent set s also an NP-complete problem stated as: Problem: Maxmum Independent Set Gven: a graph G (V, E wth vertex set V and edge set E. Objectve: To fnd the maxmum-sze ndependent set n G. An ndependent set of G s a subset V V of vertces such that f u V and v V, then u and v are not adjacent. We focus on applyng the gradual relaxaton technque to solve the MIS problem. We observe that typcally n a real-lfe graph, the MIS sze s much smaller than the total graph sze. Thus the decson to choose a vertex nto an ndependent set mposes much more constrants to the other node than the decson to choose a vertex to be out of the ndependent set. Followng the mnmal freedom reducton prncple, we drop off vertces gradually from an ntal vertex set to obtan an ndependent set. The algorthm, descrbed n Fgure, s farly straghtforward. We start wth an ntal soluton set contanng all of the vertces. At each step, we select the maxmally constraned vertex and remove t from the current vertex set untl the soluton s legal. We use force to denote the cost functon. Note that a vertex wth a large number of neghbors s unlkely to be n the resultng ndependent set. Hence, we have the followng Level- heurstc for the force calculaton: force (v = Num_of_Neghbors (v However, by lookng forward one level, we have the observaton that for a vertex, f the number of ts neghbors neghbors s very large, t tends to be n the resultng set. Fgure shows ths stuaton. The black vertex has a maxmum degree, but t should be n the MIS wth the eght leaf vertces.

4 S V whle S s not a ndependent set update force for every vertex select the vertex v wth maxmal force remove v from S return S Fgure. The gradual relaxaton algorthm for MIS To descrbe ths look-ahead force, we defne the Level- heurstc as follows force (v = u Neghbors (v ( / Num_of_Neghbors (u The force update s n O( V tme for the Level- heurstc and O( V for the Level- heurstc. In every step of the algorthm, we remove one vertex. Hence the tme complexty s O( V and O( V. Fgure. An example of the Level- heurstc 5. Tme-Constraned Schedulng for Behavoral Synthess Problem: Tme-Constraned Schedulng Gven: ( A CDFG G (V, E. ( A tmng constrant T. Objectve: to schedule the operatons of V nto T control steps such that the resource usage s mnmzed. In ths subsecton we show how to use the negatve thnkng prncple to mprove the wdely accepted force-drected schedulng (FDS algorthm [0]. 5.. Enhancement by Negatve Thnkng The essence of FDS algorthm s to reduce the resource usage by evenly dstrbutng the operatons to the avalable control steps. The FDS algorthm uses the unform probablty that an operaton wll be scheduled wthn ts ALAP-ASAP range. At each step of FDS algorthm, the tme frame of the most constraned operaton wth least forces wll be reduced to a sngle control step. That s a sgnfcant step wth potentally hgh mpact. Followng the most constranng and negatve thnkng prncples, we frst select the operaton that has optons to be scheduled to the most congested tme slot. After that, we prune only one control step, the most congested one, from the operaton s tme frame. Therefore, we always keep a global pcture before schedulng an operaton by gradually shrnkng the tme frames and postpone the decson to a later stage when more accurate estmates are avalable. 5.. Effcency Improvement Snce the above method tends to ncrease the tmng complexty, we employ the Gradual Tme-Frame Reducton (GTFR technque proposed n [6] to speedup the process. Suppose the tme frame of the operaton n s denoted as [a n, b n ]. In each teraton, GTFR only computes for each node n the forces of assgnng n to a n and b n nstead of all ts feasble control steps. The assgnment ncurrng the most force wll be dentfed and the correspondng control step wll be removed. As ponted out n [6], the tme complexty of ths algorthm s O(T N, where T s the tmng constrant and N s the number of operatons. It s the same as that of the basc force-drected schedulng. Although the removal of each control step ncreases the number of teraton to O(TN, the tentatve assgnments has to be calculated reduce to, nstead of O(T tmes. 5. Soft Real-Tme Schedulng In ths subsecton, we use soft real-tme schedulng to llustrate the effectveness of mnmal freedom reducton and probablstc modelng prncples. Problem: Soft Real-Tme Schedulng Gven: ( A set of Ψ={τ,τ, τ n } tasks and each task τ =(a, d, e s characterzed by an arrval tme a, a deadlne d and an executon tme e. ( A sngle processor P. ( A tmng constrant T. Assumptons: All tasks are non-preemptve and ndependent. Objectve: To schedule a subset of tasks n Ψ on processor P wthn the avalable tme T so that the number of tasks scheduled s maxmzed. 5.. Basc Defntons The tme frame of a task s the tme nterval wthn whch t can be scheduled. Tme frame of task τ s denoted as tme-frame=[a, d ]. The moblty of a task determnes the maxmum number of dfferent schedules for t. We denote the moblty of a task τ as moblty(τ = tme-frame(τ -e +. We denote the probablty of the task τ at a tme slot t as prob(τ, t. It can be derved by the equaton shown n the left sde of Fgure 4. The probablty dstrbuton s non-unform as shown by the trapezum-shape curve n the rght sde of Fgure 4. (t - a + moblty( τ (d - t + prob (τ, t = moblty( τ e moblty( τ t - a + < e d - t + < e Otherwse prob(τ, t e / moblty(τ a a +e d -e Fgure 4. Defnton and the curve of Prob (τ ι, t A dstrbuton graph s defned to capture the lkelhood of the potental compettons among dfferent tasks for a sngle tme slot. The dstrbuton at the tme slot t s computed by DG( t prob( τ, t. Two tasks τ and τ j are sad to be n = conflct when ther tme frames overlap. It s denoted by conflct(τ,τ j =(tme-frame(τ tme-frame(τ j. = n 5.. Soluton Overvew Observe that the less the conflcts exst, the more chances we may have to schedule a larger number of the tasks. We use a two-step heurstc to solve the problem. In the frst step, the tasks are scheduled to mnmze the number of conflcts. Snce ths ntal soluton may not be feasble, the legalzaton s performed n the second step to get the best legal schedule n terms of the number of scheduled tasks. t d

5 Soft-Real-Tme-Schedulng (Ψ /*step: conflct mnmzaton*/ whle there exsts unlocked task n Ψ compute the dstrbuton graph for each unlocked task τ a-force( self-force-arrval(τ d-force( self-force-deadlne(τ choose task τ k whch ncurs the maxmal force //shrnk the tme frame of τ k f a-force(k d-force(k then Tme-Frame(τ k [a k+,d k] else Tme-Frame(τ k [a k,d k-] f Tme-Frame(τ k =e k then set τ k locked /*step: legalzaton*/ sort all τ n the non-decreasng order of a V {v.. v n} for to n for j + to n f all the predcatons hold then add an edge <v, v j> nto G schedule T by traversng the longest path n G Fgure 5. Soft real-tme schedulng algorthm 5.. Conflct Mnmzaton In order to mnmze the conflcts, we agan apply the mnmal freedom reducton technque to gradually shrnk the tme frames. The force concept s also employed to balance the conflcts along the avalable tme slots. We only consder the self-force of relatng a task τ to ts arrval tme and ts deadlne. They are denoted by self-force-arrval(τ and self-force-deadlne(τ, respectvely. self-force-arrval(τ = a + e d prob( τ, t prob( τ, t DG ( t( + DG( t(0 t = a self-force-dealne(τ = d t = d e + tasks( t t = a + e t = a tasks( t d + e prob ( τ, t prob ( τ, t DG ( t ( + DG ( t ( 0 tasks ( t tasks ( t At each step, the assgnment of a task to a tme slot whch results n hghest force wll be rejected, and the tme frame of the correspondng task τ wll be shrunk accordngly. When the sze of a task s tme frame s reduced to ts executon tme, ths task wll be locked (marked as sem-scheduled. The frst step algorthm ends when all tasks are locked wth the hope that they are evenly dstrbuted. The frst part of Fgure 5 llustrates the algorthm Legalzaton After the conflct mnmzaton, all tasks have been semscheduled, namely, the tme frame of each task s shrunk exactly to ts executon tme. A compatblty graph G can be bult from the ntal soluton. Defne G=(V, E where V={v n}. An edge <τ,τ j > E f and only f all the followng three predcatons hold: ( conflct(τ,τ j : τ andτ j are compatble ( a a j : τ arrves earler than τ j ( k <v,v k > E <v k,v j > E: edge <v, v j > s rredundant. It can be easly shown that G s a drected acyclc graph and we can get the best legal schedule by traversng the longest path n G. A task τ s scheduled n ts tme frame f v appears n the longest path, otherwse t s dropped. Note that the longest path n a DAG can be optmally found n lnear tme. The algorthm s llustrated n the second part of Fgure Expermental Results 5.4. Maxmum Independent Set To evaluate our MIS algorthm, we take a set of DIMACS benchmark graphs for the Clque problem challenge. Snce the maxmum clque of a graph s the maxmum ndependent set n the complementary graph, we complemented these benchmarks and apply the algorthm dscussed n Secton 5., as well as Krovsk s algorthm [4] for comparson. Our programs are mplemented wth C++/UNIX envronment. Krovsk s program s the open source from hs personal webste. The experments are run wth a Sun Blade 000 workstaton wth 750 MHz frequency. The expermental results are shown n Table 4. The frst column lsts the name of each benchmark. The second and thrd columns are the correspondng vertex and edge numbers. In column Opt, we lst the optmal soluton f known. The next three columns are ndependent set szes produced by three algorthms. Columns A and B are the results from our algorthm wth the Level- and Level- heurstc respectvely. Column C lsts the results from Krovsk s teratve mprovement algorthm. In most of the cases, our Level- heurstc produced better solutons than the Level- heurstc. Compared wth Krovsk s algorthm, our Level- algorthm acheves only margnally worse result but wth a dramatc mprovement n runtme. As shown n Table 5, on average, our Level- algorthm s more than 50 tmes faster than the Level- algorthm, whch s more than 0 tmes faster than the teratve mprovement algorthm Behavoral Synthess Schedulng Table. Schedulng results comparson under crtcal-path tme constrant Node Cycles ALU ALU* MULT MULT* FFT 7 7 WANG MCM HONDA DIR CHEM_PLANT AIRCRAFT_S _u5ml _u5ml FEIG_DCT GSM AIRCRAFT_L Total Improvement % - 4% We mplemented both the force-drected algorthm and our gradual relaxaton schedulng approach n C++/STL. The selected DFG benchmarks were taken from [] wth several DCT algorthms (WANG, DIR, FEIG_DCT, and a set of real-tme

6 DSP programs. For the expermental results lsted n Table, we used the crtcal path as the tme constrant. The second and the thrd columns lst the sze of DFG and ts crtcal path length, respectvely. The followng columns lst the number of requred functon unt ALU and MULT for dfferent algorthms. Columns named ALU and MULT are the results from the force-drected algorthm, and columns ALU* and MULT* are from our algorthm. For small examples (FFT and WANG, these two algorthms produced the same results. As the DFG sze and parallelsm became larger, our algorthm consstently outperformed the forcedrected algorthm. On average, our algorthm acheved about % resource reducton compared wth the force-drected algorthm. Table. Schedulng results comparson under tme constrant wth.5 tmes of the crtcal path Node Cycles ALU ALU* MULTMULT* FFT 7 0 WANG MCM HONDA DIR CHEM_PLANT AIRCRAFT_S _u5ml _u5ml FEIG_DCT GSM AIRCRAFT_L Total Improvement % - 8% Another set of expermental results n Table s produced when we set the tmng constrant as the.5 tmes of the crtcal path length. Our algorthm acheves 5% ALU reducton and 8% MULT reducton compared wth the force-drected algorthm. Ths ndcates that when we relax the tme constrant, our gradual relaxaton technque performs even better wth more slack on each operaton node Soft Real-Tme Schedulng Table. Soft real-tme schedulng results T Task No. Perod EDF. No. Busy Utl. p_ts p_ts p_ts p4_ts p_ts p_ts p_ts p4_ts Average A set of benchmarks from [6] was appled to evaluate our soft real-tme schedulng algorthm. As shown n Table, every cell n the frst column s a task set executed n a partcular processor. The second and thrd columns are the task numbers and schedulng perods. The fourth column lsts resultng scheduled task numbers from the Earlest-Deadlne Frst (EDF schedulng The full schedulablty s not achevable n our testcases. Otherwse, EDF would generate optmal results. [4] for comparson. The next three columns lst resultng scheduled task number (No., processor busy tme (Busy, and utlzaton rato (Utl. from our algorthm. On average, our algorthm schedules.5 more tasks than EDF, whch s beng used n many real-tme applcatons. Our algorthm acheved about 86% utlzaton rato for ths benchmark set. 6. CONCLUSIONS We ntroduce a sute of new heurstc prncples for desgn of algorthms for computatonal ntractable synthess tasks. The effectveness of these prncples s demonstrated on three mportant behavoral synthess problems: maxmal ndependent set, statc tme-constraned schedulng, and synthess of soft realtme systems. 7. REFERENCES [] B. Adelberg, H. Garca-Molna; and B. Kao, Emulatng Soft Real- Tme Schedulng Usng Tradtonal Operatng System Schedulers, n Proceedngs of Real-Tme Systems Symposum, pp. 9-98, Dec [] J. Btner and E. Rengold, Backtrack Programmng Technques, Communcatons of the ACM, vol. 8(, pp , Nov [] D. Brelaz, New Methods to Color the Vertces of a Graph, Communcatons of the ACM, vol. (4, pp. 5-56, Apr [4] J. Cong and Patrck H. Madden, Performance Drven Global Routng for Standard Cell Desgn, n Proceedngs of Internatonal Symposum on Physcal Desgn, pp. 7-80, Apr [5] O. Coudert, Exact Colorng of Real-Lfe Graphs s Easy, n Proceedngs of the 4th Desgn Automaton Conference, pp. - 6, Jun [6] P. D'Argeno, J.-P. Katoen, and E. Brnksma, Specfcaton and Analyss of Soft Real-Tme Systems: Quantty and Qualty, n Proceedngs of the 0th IEEE Real-Tme Systems Symposum, pp. 04-4, Dec [7] R. I. Davs, K. W. Tndell, and A. Burns, Schedulng Slack Tme n Fxed Prorty Pre-emptve Systems, n Proceedngs of Real-Tme Systems Symposum, pp. -, Dec. 99. [8] M. R. Garey and D. S. Johnson, Computers and Intractablty: A Gude to the Theory of NP-Completeness, San Francsco: Freeman, 979. [9] M. Goldwasser, Patence s a Vrtue: The Effect of Slack on Compettveness for Admsson Control, to appear n Journal of Schedulng, 00. [0] D. S. Johnson, C. R. Aragon, L. A. McGeoch, and C. Schevon, Optmzaton by Smulated Annealng: An Expermental Evaluaton; Part, Graph Parttonng, Operatons Research, vol. 7(6, pp , 989 [] D. S. Johnson, C. R. Aragon, L. A. McGeoch, and C. Schevon, Optmzaton by Smulated Annealng: Part II, Graph Colorng and Number Parttonng, Operatons Research, vol. 9(, pp , 99. [] M. Jones, D. Rosu, and M.-C. Rosu, CPU Reservatons and Tme Constrants: Effcent, Predctable Schedulng of Independent Actvtes, n Proceedngs of the 6th ACM Symposum on Operatng System Prncples, vol. (5, pp. 98-, Dec [] B. Kao and H. Garca-Molna, Subtask Deadlne Assgnment for Complex Dstrbuted Soft Real-Tme Tasks, n Proceedngs of Internatonal Conference on Dstrbuted Computng Systems, pp. 7-8, Jun [4] D. Krovsk and M. Potkonjak, Effcent Colorng of a Large Spectrum of Graphs, n Proceedngs of the 5th Desgn Automaton Conference, pp. 47-4, Jun. 998.

7 [5] E. A. Lee and D. G. Messerschmtt, Synchronous Data Flow, n Proceedngs of the IEEE, vol. 75(9, pp. 5-45, Sep [6] C. Lee, M. Potkonjak, and W. H. Wolf, Synthess of Hard Real- Tme Applcaton Specfc Systems, Desgn Automaton for Embedded Systems, vol. 4(4, pp. 5-4, Oct [7] M. C. McFarland, A. C. Parker, and R. Camposano, The Hgh- Level Synthess of Dgtal Systems, n Proceedngs of the IEEE, vol. 78(, pp. 0-8, Feb [8] G. D. Mchel, "Synthess and Optmzaton of Dgtal Crcuts," McGraw-Hll, 994. [9] C. Papadmtrou and K. Stegltz, Combnatoral Optmzaton: Algorthms and Complexty, Prentce Hall, 98. [0] P. G. Pauln and J. P. Knght, Force-Drected Schedulng for the Behavoral Synthess of ASICs, n IEEE Trans. on Computer-Aded Desgn of Integrated Crcuts and Systems, vol. 8(6, pp , Jun [] J. Pearl, Heurstcs: Intellgent Search Strateges for Computer Problem Solvng, Addson-Wesley, 984. [] K. Rchter, D. Zegenben, M. Jersak, and R. Ernst, Model Composton for Schedulng Analyss n Platform Desgn, n Proceedngs of the 9th Desgn Automaton Conference, pp. 87-9, Jun. 00. [] M. B. Srvastava and M. Potkonjak, Optmum and Heurstc Transformaton Technques for Smultaneous Optmzaton of Latency and Throughput, n IEEE Trans. on Very Large Scale Integraton Systems, vol. (, pp. -9, Mar [4] J. A. Stankovc, M. Spur, M. D. Natale, and G. C. Buttazzo, Implcatons of Classcal Schedulng Results for Real-Tme Systems, IEEE Computer, vol. 8(6, pp. 6-5, Jun [5] J. S. Turner, Almost All k-colorable Graphs are Easy to Color, Journal of Algorthms, vol. 9, pp. 6-8, Mar [6] W. F. J. Verhaegh, P. E. R. Lppens, E. H. L. Aarts, J. H. M. Korst, J. L. van Meerbergen, and A. van der Werf, Improved Force-Drected Schedulng n Hgh-Throughput Dgtal Sgnal Processng, n IEEE Trans. on Computer-Aded Desgn of Integrated Crcuts and Systems, vol. 4(8, pp , Aug [7] D. Verkest, P. Yang, C. Wong, and P. Marchal, Optmsaton Problems for Dynamc Concurrent Task-Based Systems, n Proceedngs of the 00 Internatonal Conference on Computer- Aded Desgn, pp , Nov. 00. [8] D. Zegenben, J. Uerpmann, and R. Ernst, Dynamc Response Tme Optmzaton for SDF Graphs, n Proceedngs of the 000 Internatonal Conference on Computer-Aded Desgn, pp. 5-4, Nov Table 4. Soluton qualty comparson of three MIS algorthms Example V Edges Opt A B C brock00_ ? c-fat c-fat c-fat c-fat c-fat hammng hammng ? 4 8 hammng hammng johnson ? johnson MANN_a ? MANN_a ? MANN_a ? MANN_a ? p_hat ? p_hat ? p_hat ? p_hat ? p_hat ? p_hat p_hat p_hat p_hat ? p_hat ? p_hat ? sanr00_ sanr00_ ? sanr400_ ? 0 sanr400_ ? 7 0 Table 5. Runtme comparson of three MIS algorthms (sec Example V Edges Tme A Tme B Tme C brock00_ c-fat c-fat c-fat c-fat c-fat hammng hammng hammng hammng johnson johnson MANN_a MANN_a MANN_a MANN_a p_hat p_hat p_hat p_hat p_hat p_hat p_hat p_hat p_hat p_hat p_hat sanr00_ sanr00_ sanr400_ sanr400_ Total