ILP-Based Scheduling with Time and Resource Constraints in High ...

Step 1 (initialization):p k st = number of operators of type kwith [ASAP, ALAP] =[s; t ]Ps;t k =0Step 2:for s = jSj :::1 dotmp t =0for t = s:::jSj dotmp t = p s;t + tmp tPs;t k = Ps;t,1 k + tmp tn ? = k maxfn? ; d P s;tk t,s+1 egendendFigure 2: Algorithm LBND1operator to exactly one control step within its scheduleinterval. Our problem is P to nd the FA that requiresthe minimum FU area, a k2K kn k . It can be easilyseen that each n k can be minimized independently, becausethe operators can be executed on only one typeof FU and the precedence relation between operatorshave been relaxed.The algorithm LBND1, presented above, can be usedto compute the minimum values of n k ;k2 K. Let p k stbe the number of operators with ASAP time s andALAP time t. We dene another quantity Ps;t k to denotethe number of operators whose ASAP and ALAPtimes are within the closed interval [s; t ]. The valuesof p k st can be computed while nding the ASAP andALAP schedules. The concentration of operators ininterval [s; t] is indicated by Ps;t=(t k , s + 1). It will beshown that the minimum number of FU's, n ? k, is givenby the maximum operator concentration of all the intervals.The algorithm to compute the values of Ps;tkand n ? kis presented in Figure 2.To see how the algorithm works, consider the data-ow graph in Figure 3 (a). The schedule interval ofeach operator for a total schedule length of 4 controlsteps is shown in Figure 3 (b). The values of p k stfor thedata ow graph are given in Figure 3 (c), and the correspondingvalues of Ps;t k are given in Figure 3 (d). Maximumoperator concentration occurs in the shaded box,and the corresponding value of n ? 10kis found as d3 e =4.Although the algorithm is intuitively plausible, thecorrectness proof is somewhat long, and will be omittedin the interest of space. From Figure 2 it can be easilyseen that the complexity of LBND1 is independent ofthe number of operators, and is given by O(jSj 2 ), wherejSj is the numberofcontrol steps.4 Analysis of the Structure of the TRCSProblemThe previous section has shown how wecovert an instanceof the TCS problem to a TRCS (time- andresource-constrained scheduling) problem by using thealgorithm LBND1. In the rest of the paper we willconsider the ILP formulation of TRCS, for which bothn k and S have been specied. The eciency of ansctbf g h i1234k(a)1 2 3 41 11 4 20 21(c)daejlscontrol stepoperatorabcde fgh i j k l1234t1234(b)1 2 3 4t-s+11 3 7 121 501031Figure 3: Execution of algorithm LBND1. (a) Data ow graph(b) Schedule intervals of operators (c) Values of p st (d) Valuesof P st . Maximum operator concentration occurs in the interval[2,4] as indicated by the shaded box.ILP algorithm depends on how tightly we can deneP I (Q) without using the integrality constraints. InSection 2, we rst dened P F (Q) in terms of the assignment,precedence, and resource constraints, and thenobtained P I (Q) by adding the integrality constraints.The purpose of this section is to examine how closeP F (Q) istoP I (Q). Although a thorough examinationis as hard as solving the scheduling problem itself, wecan get some useful information by selectively droppingsome of the constraints.First we drop the precedence constraints, andconsider the subset of P F (Q), called the resourceassignmentpolytope P F (R), that satisfy the resourceand the assignment constraints, and is described as:P F (R) =fx 2 R jV j+ j M a x =1; M r x ngNext we drop the resource constraints and consider thesubset of P F (Q) called the precedence-assignment polytope,that satisfy the assignment and precedence constraints,and is described as:P F (N )=fx 2 R jV j+ j M a x =1; M p x 1gWe can show that the polytopes P F (R) and P F (N ) areintegral polytopes. The proofs of these properties involveextensive use of polyhedral theory and graph theory,and are given in [1]. The signicance of the theseresults is that, as long as the resource constraints andthe precedence constraints are considered independentof each other, the constraints presented in our formulationare the tightest constraints possible.The original scheduling polytope P F (Q) is the intersectiontwo integral polytopes P F (R) and P F (I).However, this does not necessarily imply P F (Q) isintegral.It can be easily demonstrated with a counterexample[1] that P F (Q) can have fractional extremepoints (i.e. P I (Q) P F (Q)), so an LP-relaxation ofthe problem could lead to fractional solutions, and wewill have to use branch-and-bound to nd the integraloptimal solution. In order for the branch-and-boundapproach to be successful, it is important to nd a(d)43213

sharp bound on the objective function, so that branchescan be pruned eciently.The structure of P F (Q) presented above can be interpretedusing duality theory [6] to prove that thebounds produced by the LP-relaxation are as goodas the bounds from the Lagrangian relaxation. Lagrangianbounds are tight and have led to the successof other combinatorial optimization problems. Suchtight bounds increase the likelihood that the optimumsolution can be found in a small number of branches,as will be illustrated through experimental results.In order to further improve the formulation we haveto tighten the description of P F (Q) so that it approximatesP I (Q) more closely. This can be done by introducingnew valid inequalities which take into accountthe eect of the precedence and resource constraintsupon one another. We will present a class of valid inequalitiesin the following:Valid Inequality Let jx Vk;s j n k be a resourceconstraint S of Q. Consider a minimal clique coverpV k;s = l=1 V l where each V l represents a clique madeby precedence edges. If, for each v 2 V k;s , p v gives thenumber of cliques that contain v, then the following expressionis a valid inequality X of Q,c v x v n k (3)v2Vk;swhere c v = maxf1;n k + p v , pg5 ResultsThe analysis of the ILP formulation presented in theprevious section provides us with a theoretical groundto expect optimal solutions in a relatively few numberof branches. In this section we will demonstrate thevalidity of this prediction using two benchmark examples:the 34-operator elliptical wave lter (EWF), andthe 48-operator discrete cosine transform (DCT).It should be noted here that any ILP approach producesoptimal results, so we can not expect our schedulesto be better than other ILP solutions. Instead,our objective was to oer a theoretical foundation forevaluating the ILP formulation. Thus for our purposes,we will use the number of branches taken by the ILPas the indicator of performance. We will demonstratethat the number of branches are small, as we predictedin the previous section.The scheduling results are shown in Tables 1 and 2;we used an objective function that tries to minimizethe number of registers. First we solved LBND1 to ndlower bound on resources and then solved the ILP toconstruct the schedule. In a couple of cases, the boundsgiven by LBND1 were too tight for a feasible schedule;in those cases we specied a larger number of FU's untila feasible schedule could be found. The \LV" columnindicate the maximum number of live variables thatcross a control step boundary.We also solved the TCS problems for the abovebenchmarks to observe their performance. These formulationsare less structured, and are expected to requiregreater computation time. For EWF, the TCSproblems could be solved to optimality; however, theytook a larger number of branches. For DCT, the ILPsolver failed to produce the optimal results in somecases even after hundreds of branches. This indicatesNo. of Non-Pipelined Pipelinedcsteps Mult MultTotal Loop ALU Mul LV Branch ALU Mul LV Branch17 17 3 3 10 0 3 2 10 018 18 2 2 9 0 3 1 10 02 2 9 018 16 3 2 10 0 3 1 10 019 19 2 2 9 0 2 1 9 019 17 2 2 9 0 2 1 9 221 21 2 1 9 0 2 1 9 121 19 2 1 9 0 2 1 9 0Table 1: Scheduling Results for the Elliptic Wave FilterNo. of Non-Pipelined Pipelinedcsteps Mult MultALU Mul LV Branch ALU Mul LV Branch7 6 5 12 1 6 8 11 18 5 4 12 1 5 6 13 49 4 3 13 2 4 6 13 19 4 4 13 1 5 6 13 09 5 4 12 1 5 7 0Table 2: Scheduling Results for the Discrete Cosine TransformExamplethat although any ILP formulation theoretically leadsto optimal results, a careful choice should be made inorder solve it eciently.6 ConclusionIn this paper, we have presented an ILP formulationof the scheduling problem, and have formally evaluatedthe structure of the formulation in the presenceof time and resource constraints. Formal analysis hasbeen performed to indicate that the eciency of theILP formulation on the benchmark examples is notan arbitrary event {wehave given a theoretical basisfor expecting ecient solutions from our ILP basedscheduling algorithm. To further increase the eciencyof solving a TCS problem, a methodology has been presentedto add resource constraints by optimally solvinga relaxation of TCS.References[1] Samit Chaudhuri, Robert A. Walker, and John Mitchell. TheStructure of Assignment, Precedence and Resource Constraintsin the ILP Approach to the Scheduling Problem. To appear inProc. of ICCD, 1993.[2] M. R. Garey and D. S. Johnson, editors. Computers and Intractability:A Guide to the Theory of NP-Completeness. W.H. Freeman, 1979.[3] C.H. Gebotys and M.I. Elmasry. Simultaneous Scheduling andAllocation for Cost Constrained Optimal Architectural Synthesis.In Proc. of 28th DAC, pages 2{7, 1991.[4] Cheng-Tsung Hwang, Jiahn-Hurng Lee, and Yu-Chin Hsu. AFormal Approach to the Scheduling Problem in High Level Synthesis.IEEE Trans. on CAD, 10(4):464{475, 1991.[5] G. L. Nemhauser and L.A. Wolsey. Optimization, volume 1 ofHandbooks in Operations Research and Management Science,chapter 6. Elsevier Science Publishers B. V., 1989.[6] G. L. Nemhauser and L. A. Wolsey. Integer and CombinatorialOptimization. John Wiley & Sons, 1988.[7] H. Shin and N. S. Woo. A Cost Function Based OptimizationTechnique for Scheduling in Data Path Synthesis. In Proc. ofICCD, pages 424{427, 1989.[8] J. D. Ullman. NP-Complete Scheduling Problems. J. Comput.System Sci, 10(10):384{393, 1975.[9] T. C. Wilson, N. Mukherjee, M. K. Garg, and D. K. Banerjee. AnIntegrated and Accelerated ILP Solution for Scheduling, ModuleAllocation, and Binding in Datapath Synthesis. In 6th InternationalConference on VLSI Design, pages 192{197, Bombay,India, Jan 1993.4