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Register Allocation Mooly Sagiv Schrierber 317 03-640-7606 Wed 10:00-12:00 html://www.math.tau.ac.il/~msagiv/courses/wcc.html

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Already Studied Source program (string) lexical analysis syntax analysis semantic analysis Translate Tokens Abstract syntax tree Tree IR Abstract syntax tree Cannon Assem (with many reg) Instruction Selection Cannonical Tree IR

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Register Allocation Input: –Sequence of machine code instructions (assembly) Unbounded number of temporary registers Output –Sequence of machine code instructions (assembly) –Machine registers –Some MOVE instructions removed –Missing prologue and epilogue

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Register Allocation Process Repeat Construct a interference graph Nodes = Machine registers and temporaries Interference-Edges = Temporaries with overlapping lifetimes Move-edges= Source and Target of MOVE Color graph nodes with machine registers Adjacent nodes are not colored by the same register Spill a temporary into activation record Until no more spill

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Running Example /* k, j */ g := mem[j+12] h := k - 1 f := g * h e := mem[j+8] m := mem[j+16] b := mem[f] c := e + 8 d := c k := m + 4 j := b /* d, k, j */

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Challenges The Coloring problem is computationally hard The number of machine registers may be small Avoid too many MOVEs Handle “pre-colored” nodes

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Coloring by Simplification Kempe 1879 Let K be the number of machine Let G(V, E) be the interference graph Consider a node v V which has K-1 or less neighbors –Color G – v in K colors –Color v in a color different that its (colored) neighbors

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Graph Coloring by Simplification Build: Construct the interference graph Simplify: Recursively remove nodes with less than K neighbors ; Push removed nodes into stack Potential-Spill: Spill some nodes and remove nodes Push removed nodes into stack Select: Assign actual registers (from simplify/spill stack) Actual-Spill: Spill some potential spills and repeat the process

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Coalescing MOVs can be removed if the source and the target share the same register The source and the target of the move can be merged into a single node (unifying the sets of neighbors) May require more registers Conservative Coalescing –Merge nodes only if the resulting node has fewer than K neighbors with degree K (in the resulting graph)

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Constrained Moves A instruction T := S is constrained – if S and T interfere May happen after coalescing Assume X and Z interfere and –X := Y –Y := Z After coalescing X and Y we get –XY := Z with interference between XY and Z Constrained MOVs are not coalesced

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Graph Coloring with Coalescing Build: Construct the interference graph Simplify: Recursively remove non MOVE nodes with less than K neighbors; Push removed nodes into stack Potential-Spill: Spill some nodes and remove nodes Push removed nodes into stack Select: Assign actual registers (from simplify/spill stack) Actual-Spill: Spill some potential spills and repeat the process Coalesce: Conservatively merge unconstrained MOV related nodes with fewer that K “heavy” neighbors Freeze: Give-Up Coalescing on some low-degree MOV related nodes

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Spilling Many heuristics exist –Maximal degree –Live-ranges – Number of usages in loops The whole process need to be repeated after an actual spill

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Pre-Colored Nodes Some registers are in the intermediate language are pre-colored –correspond to real registers (stack-pointer, frame-pointer, …) Cannot be Simplified, Coalesced, or Spilled (infinite degree) Interfered with each other But normal temporaries can be coalesced into pre- colored registers Register allocation is completed when all the nodes are pre-colored

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Caller-Save and Callee-Save Registers callee-save-registers (MIPS 16-23) – Saved by the callee caller-save-registers – Saved by the caller Interprocedural analysis may improve results int f(int a){ int b=a+1; g(); h(b); return(b+2); } void h (int y) { int x=y+1; f(y); f(2); }

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Saving Callee-Save Registers enter: def(r 7 ) … exit:use(r 7 ) enter: def(r 7 ) t 231 := r 7 … r 7 := t 231 exit:use(r 7 )

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A Realistic Example enter: c := r3 a := r1 b := r2 d := 0 e := a loop: d := d+b e := e-1 if e>0 goto loop r1 := d r3 := c return /* r1,r3 */ r1, r2caller save r3 callee-save enter: /* r2, r1, r3 */ c := r3 /* c, r2, r1 */ a := r1 /* a, c, r2 */ b := r2 /* a, c, b */ d := 0 /* a, c, b, d */ e := a / * e, c, b, d */ loop: d := d+b /* e, c, b, d */ e := e-1 /* e, c, b, d */ if e>0 goto loop /* c, d */ r1 := d /* r1, c */ r3 := c /* r1, r3 */ return /* r1,r3 */

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A Realistic Example(cont) enter: /* r2, r1, r3 */ c := r3 /* c, r2, r1 */ a := r1 /* a, c, r2 */ b := r2 /* a, c, b */ d := 0 /* a, c, b, d */ e := a / * e, c, b, d */ loop: d := d+b /* e, c, b, d */ e := e-1 /* e, c, b, d */ if e>0 goto loop /* c, d */ r1 := d /* r1, c */ r3 := c /* r1, r3 */ return /* r1,r3 */

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Graph Coloring with Coalescing Build: Construct the interference graph Simplify: Recursively remove non MOVE nodes with less than K neighbors; Push removed nodes into stack Potential-Spill: Spill some nodes and remove nodes Push removed nodes into stack Select: Assign actual registers (from simplify/spill stack) Actual-Spill: Spill some potential spills and repeat the process Coalesce: Conservatively merge unconstrained MOV related nodes with fewer that K “heavy” neighbors Freeze: Give-Up Coalescing on some low-degree MOV related nodes

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enter: /* r2, r1, r3 */ c := r3 /* c, r2, r1 */ a := r1 /* a, c, r2 */ b := r2 /* a, c, b */ d := 0 /* a, c, b, d */ e := a / * e, c, b, d */ loop: d := d+b /* e, c, b, d */ e := e-1 /* e, c, b, d */ if e>0 goto loop /* c, d */ r1 := d /* r1, c */ r3 := c /* r1, r3 */ return /* r1,r3 */ use+ def outsi de loop use+ def withi n loop degspill priority a2040.5 b1112.75 c2060.33 d2245.5 e13310.3 spill = (uo + 10 ui)/deg

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Spilling c c stack

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Coalescing a+e c stack c

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Coalescing b+r2 c stack c

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Coalescing ae+r1 c stack c r1ae and d are constrained

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Simplifying d dcdc stack c

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Pop d dcdc stack c d is assigned to r3

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Pop c c stack actual spill! c

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enter: /* r2, r1, r3 */ c1 := r3 /* c1, r2, r1 */ M[c_loc] := c1 /* r2 */ a := r1 /* a, r2 */ b := r2 /* a, b */ d := 0 /* a, b, d */ e := a / * e, b, d */ loop: d := d+b /* e, b, d */ e := e-1 /* e, b, d */ if e>0 goto loop /* d */ r1 := d /* r1 */ c2 := M[c_loc] /* r1, c2 */ r3 := c2 /* r1, r3 */ return /* r1,r3 */ enter: /* r2, r1, r3 */ c := r3 /* c, r2, r1 */ a := r1 /* a, c, r2 */ b := r2 /* a, c, b */ d := 0 /* a, c, b, d */ e := a / * e, c, b, d */ loop: d := d+b /* e, c, b, d */ e := e-1 /* e, c, b, d */ if e>0 goto loop /* c, d */ r1 := d /* r1, c */ r3 := c /* r1, r3 */ return /* r1,r3 */

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enter: /* r2, r1, r3 */ c1 := r3 /* c1, r2, r1 */ M[c_loc] := c1 /* r2 */ a := r1 /* a, r2 */ b := r2 /* a, b */ d := 0 /* a, b, d */ e := a / * e, b, d */ loop: d := d+b /* e, b, d */ e := e-1 /* e, b, d */ if e>0 goto loop /* d */ r1 := d /* r1 */ c2 := M[c_loc] /* r1, c2 */ r3 := c2 /* r1, r3 */ return /* r1,r3 */

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Coalescing c1+r3; c2+c1r3 stack

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Coalescing a+e; b+r2 stack

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Coalescing ae+r1; Simplify d d stack

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Pop d stack d d a r1 b r2 c1 r3 c2 r3 dr3 er1

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enter: c1 := r3 M[c_loc] := c1 a := r1 b := r2 d := 0 e := a loop: d := d+b e := e-1 if e>0 goto loop r1 := d c2 := M[c_loc] r3 := c2 return /* r1,r3 */ a r1 b r2 c1 r3 c2 r3 dr3 er1 enter: r3 := r3 M[c_loc] := r3 r1 := r1 r2 := r2 r3 := 0 r1 := r1 loop: r3 := r3+r2 r1 := r1-1 if r1>0 goto loop r1 := r3 r3 := M[c_loc] r3 := r3 return /* r1,r3 */

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enter: r3 := r3 M[c_loc] := r3 r1 := r1 r2 := r2 r3 := 0 r1 := r1 loop: r3 := r3+r2 r1 := r1-1 if r1>0 goto loop r1 := r3 r3 := M[c_loc] r3 := r3 return /* r1,r3 */ enter: M[c_loc] := r3 r3 := 0 loop: r3 := r3+r2 r1 := r1-1 if r1>0 goto loop r1 := r3 r3 := M[c_loc] return /* r1,r3 */

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Graph Coloring Implementation Two kinds of queries on interference graph –Get all the nodes adjacent to node X –Is X and Y adjacent? Significant save is obtained by not explicitly representing adjacency lists for machine registers –Need to be traversed when nodes are coalesced Heuristic conditions for coalescing temporary X with machine register R: –For every T that is a neighbor of X any of the following is true (guarantee that the number of neighbors of $T$ are not increased from

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Worklist Management Use work-list to avoid expensive iterative queries Maintain various invariants as data structures are updated Implement work-lists as doubly linked lists to allow element-deletion

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Graph Coloring Expression Trees Drastically simpler No need for data flow analysis Optimal solution can be computed (using dynamic programming)

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Simple Register Allocation int n := 0 function SimpleAlloc(t) { for each non trivial tile u that is a child of t SimpleAlloc(u) for each non trivial tile u that is a child of t n := n - 1 n := n + 1 assign r n to hold the value of t }

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Bad Example PLUS BINOP MEM BINOP NAME a TIMESMEM NAME bNAME c

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Two Phase Solution Dynamic Programming Sethi & Ullman Bottom-up (labeling) –Compute for every subtree The minimal number of registers needed Top-Down –Generate the code using labeling by preferring “heavier” subtrees (larger labeling)

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The Labeling Algorithm void label(t) { for each tile u that is a child of t do label(u) if t is trivial then need[t] := 0 else if t has two children, left and right then if need[left] = need[right] then need[t] := need[left] + 1 else need[t] := max(1, need[left], need[right]) else if t has one child, c then need[t] := max(1, need[c]) else if t has no children then need[t] := 1 }

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int n := 0 void StehiUllman(t) { if t has two children,left and right then if need[left]>K and need[right]>K then StehiUllman (right) ; n := n – 1 ; spill: emit instruction to store reg[right] StehiUllman (left) unspill: emit instruction to fetch reg[right] else if need[left]> need[right] then StehiUllman (left) ; StehiUllman (right) ; n := n - 1 else if need[right]> need[left] then StehiUllman (right) ; StehiUllman (left) ; n := n – 1 reg[t] := r n ; emit(OPER, instruction[t], reg[t], reg[left], reg[right]) else if t has one child c then StehiUllman(c) ; reg[t] : = r n ; emit(OPER, instruction[t], reg[t], reg[c]) else if t is non trivial but has no children n := n + 1 ; reg[t] := r n ; emit (OPER, instruction[t], reg[t]) else if t a trivial node TEMP(r i ) then reg[t] := r i

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Summary Two Register Allocation Methods –Local of every IR tree Simultaneous instruction selection and register allocation Optimality (under certain conditions) –Global of every function Applied after instruction selection Performs well for machines with many registers Can handle instruction level parallelism Missing –Interprocedural allocation

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