The Power of Belady's Algorithm in Register Allocation for Long ...

quently. In the next section, we explain the compiler we designedto perform the register allocation on the FFT and MM codes.3 A Compiler for Long Straight-line CodeBelady’s MIN [2] is a replacement algorithm for pages in virtualmemory. On a replacement, the page to replace is the one withthe farthest next use. The MIN algorithm is optimal because itgenerates the minimal number of physical memory block replacements.However, in this context of virtual memory the MIN algorithmis impractical in most of the applications, because it isusually not known which memory block will be referenced in thefuture.Belady’s MIN algorithm has been proposed to use for registerallocation in long basic blocks, where the compiler knows exactlythe values that will be used in the future. In this context, the MINalgorithm is also known as Farthest First(FF) [6] since, on a registerreplacement, the register to replace first is the one holding thevalue with the farthest next use.The MIN algorithm is not optimal for register allocation sincethe replacement decision is simply based on the distance and noton whether the register has been modified. When a register holdsa value that is not consistent with the value in memory, we say thatthe register is dirty. Otherwise, we say that the register is clean. Ifthe register to be replaced is dirty, the value of the register needs tobe stored back to memory before a new value can be loaded into it.Thus, for a given instruction scheduling the MIN algorithm guaranteesthe minimum number of register replacements, but it doesnot guarantee the minimum traffic with memory, that is, the minimumnumber of load/stores. In our implementation, when thereare several candidates for replacement with the same distance, ourcompiler chooses the one with the clean state to avoid an extrastore.In order to further reduce the number of stores, another simpleheuristic called Clean First (CF) was proposed in [6]. Withthis heuristic, when a live register needs to be replaced, CF firstsearches in the clean registers. The clean register which containsthe value with the farthest next use is chosen. If there are no cleanregister, the most distant dirty one is chosen.We implemented a back-end compiler that uses the MIN andthe CF algorithms for register allocation. Next, we describe theimplementation details of our compiler.3.1 Implementations DetailsWe built a simple compiler that translates high-level code intoassembly code. Our compiler assumes that all the optimizations,including instruction scheduling, have been applied before thehigh-level code is generated. It only performs register allocationusing Belady’s MIN or the CF heuristic explained above.The compiler has two steps. In the first step we transform thelong straight-line code into a static single-assignment (SSA) formand build the definition-use chain for all the variables. At the secondstep we do register allocation as shown in Figure 2. The secondstep is fairly much along the lines of the simple algorithmdescribed in [1].The algorithm can be implemented more efficiently. The registerfile can be a priority queue implemented with a binary heap,where the higher priority is given to the farther next use. Operationssuch as extracting the register with the farthest next use canbe executed in Ç´ÐÓʵ, where R is the number of registers. Sothe time complexity for MIN algorithm is × ¢ Ç´ÐÓʵ, where ×is the number of references to the variables in the program.DATA STRUCTURES:register file:Array of registers. Each register r has 3 fields:- state: Indicates whether the register is clean.- var: The current variable in the register.- addr: The address of the variable in the register.next use list:List used to keep the stmt where each variable is used/defined.Each node in the list has 2 fields:- stmt: The statement number.- status: Definition or use.FUNCTIONS:NEXT USE (reg, stmt):Returns the statement number where reg.var is used next after stmt.MIN REG ALLOC (var, stmt):Returns the register that is going to be used for the variable var at stmt.MAINfor each stmt in the program do beginfor each v on RHS do beginreg MIN REG ALLOC (v, stmt)generate instruction ”load reg, reg.addr”endfor v on LHS do beginreg MIN REG ALLOC (v, stmt)endendgenerate assembly instruction “op r1 r2 r3”MIN REG ALLOC (v, stmt)if visinregreturn regfarthestUse 0regToUse 0for each reg in register file do beginif reg is emptyreturn regif variable in reg is deadreturn regnextUse NEXT USE (reg, stmt)if farthestUse nextUsefarthestUse nextUseregToUse regelse if farthestUse=nextUse and reg.state = CLEANregToUse regendif regToUse.state = DIRTYgenerate register spill “store regToUse, regToUse.addr”return regToUseFigure 2. Pseudocode for Belady’s MIN algorithm.