A Flow Graph Formulation of Optimal Software Pipelining

A Flow Graph Formulation of Optimal Software PipeliningDirk FimmelDresden University of Technology/IEE, 01062Dresden,Germanyfimmel@iee1.et.tu-dresden.deJan MüllerDresden University of Technology/IEE, 01062Dresden,Germanymuellerj@iee1.et.tu-dresden.deAbstractWe present a new approach to the loop scheduling problem, which excels previous solutions in two important aspects: The resource constraints are formulated usingflow graphs, and the initiation interval is treated as a rational vari-able.The approach supports heterogeneous processor ar-chitectures and pipelined functional units,and the Integer Linear Programming implementation produces an optimum loop schedule,whereby a minimum is achieved.Ourflow graph model facilitates the cyclic binding of loop operations to functional pared to previous research results, the solution can provide faster loop schedules and a signifi-cant reduction of the problem complexity and solution time.1.IntroductionRecent developments in processor design indicate that pro-cessor architectures become more and more heterogeneous. The addition of new functional units to processor kernels (e.g.MMX,SSE,and SSE2extensions of Intel Pentium processors)allows to operate more units in parallel. These developments raise the challenge to maximally ex-ploit available resources,and tofind operating schedules with minimum latencies.Current compilers,using heuris-tics for resource exploitation and loop scheduling,offer good solutions to these tasks[1,18].However,for the production of program libraries and for computationally in-tensive algorithms,the best solutions are needed,even if the compilation may take longer than with heuristic ap-proaches.This substantiates the use of Integer Linear Pro-gramming formulations[2,4,12,13,14],which always produce optimal solutions.The usual loop scheduling method,called Software Pipelin-ing[15,18],handles program loops with single assignments of operation results to loop variables.The result is an execu-tion schedule periodic in,the so-called initiation interval. It is well-known that a minimization of results in a min-imum loop latency[11].Despite the discrete time steps in a schedule,the initiation interval may be a rational number2.Problem descriptionIn this section we introduce our way of representing the pro-gram loops,the processor model,the objective of the loop scheduling,and the constraints resulting from the loop and the processor.Here we present a"conventional"resource constraint formulation which is given for the sake of com-pleteness and comparison with the formulations usingflow graphs(Section3).2.1.Loop and schedule representationIn this paper we restrict our attention to single loops:for toend for,(1)where are single valued operations,and are state-ments(assignments),are dependence distances be-tween statements;we use indices for iterations.We suppose that the loop has a single assignment form,i.e. each instance of a variable is assigned only once in the loop.The single assignment form may result from the appli-cation of an array dataflow analysis and a renaming of vari-ables[5].Then all data dependences areflow dependences which can be represented in a reduced dependence graph (RDG).In this RDG,the statements form the vertices ;directed edges are data depen-dences weighted by the dependence distances. The source of an edge is called and the sink.In our loop scheduling,the aim is tofind an-periodic schedule,i.e.a set of starting times:is called initiation interval. The expressionsare individual time offsets for each statement. Since we are bound to discrete time steps(clocks),we have to adapt fractional times using the operator.Example1[Loop Representation]As an example for the loop representation and the RDG,we choose the second or-der wave digitalfilter[7]shown in Figure1.The operation of statement,for example,is an addition;depends only on through variable(with a dependence distance of),since is not being assigned in the loop.2.2.Functional unitsAtfirst we have to introduce a bunch of variables describing functional units and relationships between them.A set of functional units(types)available for the computationfor toFigure1.Program code and reduced depen-dence graph of Example1of the operations of statements is supposed to be given.Theset of units that are able to execute a particular statement,is denoted as.Each functional unit is associated with a latency,defining the time between the start of the computation and the availability of the result,and with anoffset time,defining the minimum time between the start of two computations if has a pipeline architecture (otherwise).The set of statements that can be executed in is called .To implement the binding of statements to resources, wefinally introduce a function:,expressing that statement is evaluated by the functional unit if,and a function:,expressing that statement is evaluated by the instance of unit type.The number of instances of is denoted as.For convenience,we use the abbreviationsand.Example2[Functional units]Since this might have beena bit confusing,we explain the variables in a constructed example.Imagine a processor having four parts,two ALUs,one multiplier,and one adder(all non-pipelined). An ALU can execute multiplications,subtractions,and additions,the multiplier can only multiply,the Adder can add and subtract,i.e.with the statements of Ex-ample1:.Statement could then be exe-cuted by functional units.A multiplication takes three cycles in the multiplier andfour cycles in the ALU,the addition/subtraction takes al-ways one cycle:If then statement is executed in an ALU; if in addition then it is executed in the second (instance of the)ALU.2.3.Objective functionThe aim of loop scheduling is usually to minimize the looplatency:being the difference between thefirst start of a statement’sexecution and the last ending of an execution.In[11]we de-duced that the minimization of is equivalent to the mini-(5) mization of the initiation intervalwith being the cardinality of set,i.e.the number ofinstances being busy at a time must not exceed the numberof available instances.We introduce the decomposition1Modulo operator:The following conditions have to be satisfied:1.,i.e.each vertex has exactly one positive(incoming)edge and exactly one negative (outgoing)edge,2.,i.e.statements con-nected by an edge are bound to the same func-tional unit.The edges are weighted by,and the vertices by such that:(8) with4.Integer Linear ProgramIn this section we construct an Integer Linear Program to determine a scheduling with a minimum initiation interval .In the following analyses we change from a rational as a fraction of two integer variables and,to a rationalvariable.This difference was less important for thederivations in the previous sections(since a rational num-ber is the fraction of two integers),however,it brings about some advantages for the linear programming implementa-tion of our approach:We have to cope with fewer inequal-ities and(now only binary and rational)variables.We can renounce the binary decomposition of,or the treatment of as a constant,as in[6,11].Furthermore,a rational variable improves the condition of the linear program. We start at introducing an equivalent scheduling function ,which is followed by re-formulations of the causality constraints and the resource constraints for both static and cyclic binding,the latter using theflow graph approach(the "conventional"approach of cyclic binding has been dis-cussed extensively in[11]).We prove the validity of the program by showing that the constraints of Section2follow from the constraints of that program.4.1.Scheduling functionWe introduce a new scheduling function,as in[6]:(11) Theorem3With,the causality constraints(3)follow from(12).4.3.Resource constraints with static bindingFor the implementation of resource constraints with static binding(Section 2.5.1)the original formulation is lin-earized,and appropriate expressions for the binding func-tions and in(4)and for the resource constraints are found.This derivation is discussed in detail in[9].4.4.Resource constraints with aflow graph formu-lation of cyclic bindingFirst we show that a formulation of edge and vertex weights using leads to the formulation in Definition3.1. Theorem5WithWhen searching for the minimum of the objective function, will become zero if,as demanded in (13).At last we implement the edge weights.Here we use variables to express,since from(13),if,andif:(21)Similarly to(20),will become zero if,as demanded in(13).This concludes the re-formulation of the resource con-straints as a set of linear inequalities.4.5.Estimation of boundsWe have used bounds on several variables in the preceeding sections,which are now to be estimated.An upper bound of is given when the loop body of one iteration is executed sequentially,i.e.;however,this is very pessimistic. The maximum utilization of resources gives rise to a lower bound,where is a cycle in,and.The upper bound of[11],and of the flow graph implementation with a rational(Section4.4). Example4[Program:WDF](Examples1,3continued). For the scheduling of the loop shown in Figure1,a target processor having one adder(Add)and one multiplier(Mul), with latencies of one and three cycles,respectively,is sup-posed to be available.We determine the following schedul-ing functions,represented graphically in Figure2:The respectiveflow graph is shown in Figure3.Example5[Program:Calland[3]]In this example(Fig-ure4),two identical ALUs are given.The latencies for for toend forFigure4.Program code&RDG of Example5 the operations are,,, ,,and.We get the follow-ing scheduling functions,shown graphically in Figure5:Resources Iteration Iteration IterationFigure5.Schedule of Example5 Example6[Program:van Dongen[21]]In this example (Figure6)we consider two given adders having a pipeline architecture.The evaluation of an addition takes two cycles; an adder can start the next evaluation after one cycle.The resulting scheduling functions,as shown in Figure7,are:With the rational we have a schedule speed-up of 9.1%compared to the best integral solution.Let us now discuss the technical results of the three exam-ples’scheduling.We compare parameters of the linear pro-gramming solution for cyclic binding.In the case of static binding,the solution times are shorter,of course.However, the schedule of Example5has a greater latency than with cyclic binding,since idle sections are added to comply with the dependences.The results of static binding solutions are discussed in detail in[11].Example()() 4(WDF)664193(72)640.2125(Calland)1098110(144)540.1246(van Dongen)720 4.3(64)130 1.710Theflow graph formulation is better suited for a linear pro-gramming implementation than previous formulations,such that the number of variables and constraints is greatly re-duced,and the optimum solution is obtained significantly faster.The second major improvement was achieved by the treat-ment of the initiation interval as a rational variable,re-placing the integer representation of,or two separate vari-ables and,。