ECE540S Optimizing Compilers

1
ECE540SOptimizing Compilers

• Register Allocation, March 9, 2005
• Muchnick, Chapter 16

2
Register Allocation

• Weve been using pseudo registers so far
• assumed there was an infinite amount of them
• You need to allocate real space to them
• memory big but slow
• registers only a few but fast
• And, most RISCs only operate on registers

addr x 200 r1 addr x r2 addr a r3
r1 r2 addr y r3 r1 addr y r2 addr
x r3 r1 r2 addr z r3 addr x 10
int x, y , z x 200 y x a z y x x
10
r1 200 r2 r1 r4 r3 r2 r1 r1 10
4 Regs x ? r1 y ? r2 z ? r3 a ? r4
naive
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Register Allocation

• Register allocation improves code
• accessing faster memory
• fewer instructions
• But
• There are a limited number of machine registers
• Some registers can only hold certain types of
  data
• So, which variables do we allocate to registers?
• Register allocation is extremely important
• has huge impact on performance (just look a
  previous slide)
• must be done!
• its NP-Complete (not solvable in polynomial
  time)
• Use heuristics

4
Approaches to Register Allocation

• Global Register Allocation Using Usage Counts
• Assume R registers are available. For each loop
  nest, allocate registers to the R variables which
  show the largest estimated benefit from being
  kept in a register. Little or no cross nest
  allocation is done.
• Register Allocation by Graph Coloring
• our main focus and currently the most common
  method
• known about since 1971 but was impractical in
  early compilers
• Chaitin came up with 1st implementation in 1981
• Briggs proposed an optimistic extension to it
  around 1989
• express overlap of the lifetimes of vars with an
  interference graph
• try to color this graph with R colors
• generate spill code when necessary to make the
  graph R-colorable

5
Global Register Allocation Using Usage Counts

• Look at loops or loop nests independently
• For each of these determine which of the
  variables in it should be allocated to a register
  based on
• netsave(v,i) u usesave d defsave l
  ldcost s stcost
• where usesave is saving for using a value
  in a register vs- memory
• defsave is saving for defining a
  value in a register vs- memory
• ldcost is the execution-time cost
  of a load instruction
• stcost is the execution-time cost of a store
  instruction
• For a loop
• Allocate the R variables that would benefit the
  most

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Global Register Allocation Using Usage Counts
L1 i i 1 j j i k
0 L2 k k j j j k if (
k lt n ) GOTO L2 if ( i lt n ) GOTO L1
Assuming all costs 1, i ? 10, j ? 100, k ?
200, n ? 0
If R 2, allocate j and k If R 3, allocate
i, j and k
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Global Register Allocation Using Usage Counts
L1 i i 1 j j i k
0 L2 k k j j j k if (
k lt n ) GOTO L2 if ( i lt n ) GOTO L1
netsave(i,1) 10 ? (2U 1D 0L 0S)
netsave(j,1) 10 ? (1U 1D 0L 0S)
netsave(k,1) 10 ? (0U 1D 0L 0S)
netsave(k,2) 102 ? (3U 1D 0L 0S)
netsave(j,2) 102 ? (2U 1D 0L 0S)
netsave(n,2) 102 ? (1U 0D 0L 0S)
netsave(i,3) 10 ? (1U 0D 0L 0S)
netsave(n,3) 10 ? (1U 0D 0L 0S)
Just load and store before and after the loop
nest. Assuming all costs 1, i ? 40, j ? 320,
k ? 410, n ? 110
If R 2, allocate j and k If R 3, allocate
n, j and k
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Register Allocation by Graph Coloring

• Observation We cannot allocate two values to the
  same register if they are needed at the same time
  at some point in the program. They are said to
  interfere.

s1 to s6 are called the candidates. We wish to
allocate each in one of 3 machine registers. We
assume they are no longer needed at end.
s1 2 s2 4 s3 s1 s2 s4 s1 1 s5 s1
s2 s6 s4 2
A register holding a variable whose value is no
longer needed may be de-allocated and allocated
to another variable.
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Register Allocation - Overview

• We encode interference information in a graph
  called the interference graph.
• It is an undirected graph in which each vertex is
  a candidate or a machine register.
• There is an edge between two vertices if the
  corresponding two candidates interfere (i.e.,
  they cannot occupy the same register), or if a
  candidate cannot be allocated to a specific
  machine register. Machine registers interfere
  with one another

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Register Allocation - Overview

• The register allocation problem is now to assign
  each vertex in the graph a register such that no
  two vertices that are connected by an edge are
  assigned the same register.
• This problem is an instance of a problem known as
  Graph Coloring Given R colors, is it possible to
  assign each vertex one color such that no two
  connected vertices have the same color?
• If yes, the graph is said to be R-colorable.

s1
s3
s5
r1
3-colorable?
s2
s4
s6
r2
r3
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Register Allocation - Overview

• The register allocation problem is now to assign
  each vertex in the graph a register such that no
  two vertices that are connected by an edge are
  assigned the same register.
• This problem is an instance of a problem known as
  Graph Coloring Given R colors, is it possible to
  assign each vertex one color such that no two
  connected vertices have the same color?
• If yes, the graph is said to be R-colorable.

3-colorable?
Yes!
12
Register Allocation - Overview

• Assigning candidates to registers is now easy
  once the graph has been colored.

r3 2 r2 4 r1 r3 r2 r1 r3 1 r3 r3
r2 r1 r1 2
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Register Allocation - Overview

• When an interference graph cannot be R-colored,
  we cannot assign all candidates to registers.
• We store some candidates in memory, load before
  each use and store after each definition.
• This is referred to as spilling.
• We select the smallest number of candidates whose
  spilling to memory will make the graph
  R-colorable.
• We re-write the code to include spills and
  rebuild the interference graph and color it.

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Register Allocation - Overview

• In our example, assume that we have only 2
  registers. Hence, we want to find out if the
  graph is 2-colorable.

s5
r1
s1
s1
s3
r2
s2
s4
s6
Not 2-colorable!

• Spill s1 to memory.

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Register Allocation - Overview
s1 2 store s1 to Mx s2 4 load s7 from Mx s3
s7 s2 load s8 from Mx s4 s8 1 load s9 from
Mx s5 s9 s2 s6 s4 2
s4
Add edge between s4 and s5
Not 2-colorable!

• Spill s4 to memory.

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Register Allocation - Overview
s1 2 store s1 to Mx s2 4 load s7 from Mx s3
s7 s2 load s8 from Mx s4 s8 1 store s4 to
My load s9 from Mx s5 s9 s2 load s10 from
My s6 s10 2
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Register Allocation - Overview
s1 2 store s1 to Mx s2 4 load s7 from Mx s3
s7 s2 load s8 from Mx s4 s8 1 store s4 to
My load s9 from Mx s5 s9 s2 load s10 from
My s6 s10 2
It is 2-colorable!

• We can now generate register assignments and code.

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Register Allocation - Overview

• Hence, the steps involved in register allocation
  to R registers are
• Identify candidates.
• build interference graph.
• color the interference graph.
• if graph is R-colorable, the doneelse
• select a victim for spilling.
• re-write code.
• repeat.
• We will now examine each of these steps in some
  detail.
• Register allocation is iterative and is
  time-consuming.
• There are other complications (e.g.,
  calling/return conventions, register windows,
  etc) that we will ignore.

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Identifying Candidates

• Using variable names as candidates is not a good
  approach
• all uses of a name will be allocated to the same
  register, but the name may be defined multiple
  times, and hence takes different values. No
  reason to hold in the same register!
• Hence, we define the notion of webs as
  equivalence classes for name uses the same name
  in two different webs implies that the name takes
  an independent value in each web.
• webs separate the life-times of a variable.
• each web may be allocated in a register!

i 0 i i1 i 6
i i-2
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Webs (or Life-time Separation)

• A definition an all its reachable uses are in the
  same web.
• All definitions that reach the same use are in
  the same web.

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Webs (or Life-time Separation)

• A definition an all it reachable uses are in the
  same web.
• All definitions that reach the same use are in
  the same web.

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Life Ranges and Interference
x ...
BB 1
z ...
BB 2
BB 3
BB 4
LR(web 2) BB 2, BB 3)
z
x
y ...
BB 5
BB 6
LR(web 1) BB 1, BB 2, BB 3, BB 4, BB 5, BB
6, BB 7)
y
LR(web 3) BB 5, BB 6)
x
BB 7
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Life Ranges and Interference

• A set S of basic blocks is said to be convex if
  BB a and BB b are in S and c is a BB on a path
  from a to b, then c is in S.
• The live range of a web is the minimal convex set
  of instructions that include all the definitions
  and uses in the web.
• Intuitively, the live range of a web is the
  region of BBs in which the web is live.
• Two webs interfere if their live ranges
  intersect.
• Two webs that interfere must be allocated to
  different registers.
• The interference of the webs is captured using
  the interference graph described earlier.

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Practical Note Intersection of Live Ranges

• Sufficient to include an arc if one of the nodes
  is live at the definition of the other.

ENTRY
?
define a1 an
define b1 bn
?
use a1 an
use b1 bn
EXIT
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Graph Coloring

• Given a graph G(V,E), is it possible to assign
  each vertex in the graph a color such that no two
  adjacent vertices have the same color?
• What is the smallest number of colors?
• Is the graph R-colorable?
• The problem is NP-complete for R ³ 3.
• However, there exists a good heuristic.

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Graph Coloring - Degree lt R rule

• Given a graph G(V,E) with a vertex v with degree
  lt R. G is R-colorable if and only if the graph
  G(V-v,E) is R-colorable.
• If G is R-colorable, then adding the vertex v
  will result in a colorable graph. We pick a color
  for v that different from the colors of vertices
  connected to v. We know we have enough colors
  because D(v) lt R.

D(v) lt R
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Graph Coloring Heuristic

• Remove vertices v (and associated edges) with
  D(v) lt R.
• remove one at a time and push on a stack.
• When graph is empty, start to color
• pop a vertex from the stack.
• assign it a color different from vertices it is
  connected to.
• a color always exists!

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Graph Coloring Heuristic - Example
3-colorable?
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Graph Coloring - Degree lt R rule
3-colorable?
2-colorable?

• Neither is R-colorable using the degree lt R rule!
  
• Yet, they are both R-colorable!

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Graph Coloring - Heuristic (Take 2)

• Remove vertices v (and associated edges) with
  D(v) lt R.
• remove one at a time and push on a stack.
• When all remaining vertices v have D(v) ³ R,
• select a vertex to spill, mark accordingly.
• remove the vertex and push on the stack
• When graph is empty, start to color
• pop a vertex from the stack.
• assign it a color different from vertices it is
  connected to.
• a color always exists if Step 2 was not applied!
• otherwise, there may or may not be a color
  available.

Step 2
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Graph Coloring - Example II
2-colorable?
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Graph Coloring - Example III
3-colorable?
a
b

e
d
c
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When no color is available?

• Re-generate the code, spilling a node (which
  node?)
• or split a candidate into multiple candidates
• Try to color the new code
• Continue to repeat this process until R-colorable

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Spilling

• How to select a web to spill to memory.
• One whose corresponding vertex v has D(v) ³ R.
• One with minimal spill cost.
• The spill cost is the cost of the extra loads and
  stores used to retrieve and store the web to
  memory.

s1 2 store s1 to Mx s2 4 load s7 from Mx s3
s7 s2 load s8 from Mx s4 s8 1 load s9 from
Mx s5 s9 s2 s6 s4 2
s1 2 s2 4 s3 s1 s2 s4 s1 1 s5 s1
s2 s6 s4 2
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Spill Cost

• The spill cost is determined by the dynamic cost
  of the extra loads and stores.
• This is not possible to compute because
• we do not how often branches are taken in the
  CFG,
• we do not know how many times a loop iterates,
  and
• the dynamic cost may be input-dependent and will
  vary from one execution to the next.
• Hence, we statically estimate (reads approximate)
  the spill cost based on the structure of the CFG.
• loops play an important role.
• assume loops execute 10 times.
• may divide by vertex degree to favor vertex with
  high degree.

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Graph Coloring - Heuristic (Take 3)

• Remove vertices v (and associated edges) with
  D(v) lt R.
• remove one at a time and push on a stack.
• When all remaining vertices v have D(v) ³ R,
• select a vertex to spill (smallest cost / D )
• remove the vertex and push on the stack
• When graph is empty, start to color
• pop a vertex from the stack.
• assign it a color different from vertices it is
  connected to.
• a color always exists if Step 2 was not applied!
• otherwise, there may or may not be a color
  available.

Step 2
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Spill Cost - Example
two webs x and i
spill cost (x) 100 101 101 100
22
spill cost (i) 100 101 101 101 100
32
spill x to memory!
only one register! cdef cuse 1
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Splitting

• Break a web into multiple webs to reduce
  interference in the interference graph. This is
  referred to as splitting.
• Insert instructions to spill value to memory and
  load it from memory at point of split.

3
1
2
2-colorable?
39
Representing the Interference Graph

• Usually represented in 2 forms
• Adjacency Matrix
• a lower triangular matrix such that if iltj,
  AdjMatrixi,j true if the ith and jth values
  are adjacent and is false otherwise.
• allows you to quickly identify if 2 nodes are
  adjacent
• good for register coalescing
• Adjacency Lists
• an array of lists of adjacent nodes. Each
  element is a record holding information about the
  node, e.g. color chosen for the node, spill
  location, spill cost,
• used for graph coloring
• is easily built from the Adjacency Matrix
  representation

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Register Allocation Overview

• Hence, the steps involved in register allocation
  to R registers are
• Identify candidates.
• build Adjacency Matrix representation of the
  interference graph
• do register coalescing
• build the Adjacency List representation of the
  interference graph
• color the interference graph.
• if graph is R-colorable, the doneelse
• select a victim for spilling.
• re-write code.
• repeat.

41
Register Coalescing

• Eliminates copies from 1 register to another
• remove unnecessary copies from SSA
• remove moves to required register locations
• Search IR for sj si such that si and sj do not
  interfere with each other or neither si nor sj
  are stored to between the copy assignment and the
  end of the routine.
• find instructions that wrote si and replace si
  with sj.
• update the interference graph
• anything that interfered with si or sj now
  interferes with sj
• however, if using definitions to determine
  interference, definition of sj at sj si is now
  gone. May remove some interference.