1. Global Register Allocation via Graph Coloring Comp 412
FALL 2010
Global Register Allocation via Graph Coloring Comp 412
This lecture focuses on the Chaitin-Briggs approach, which EaC calls the bottom-up global algorithm.
Copyright 2010, Keith D. Cooper & Linda Torczon, all rights reserved.
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2. Global Register Allocation
Taking a global approach
Abandon the distinction between local & global
Make systematic use of registers or memory
Adopt a general scheme to approximate a good allocation
Graph coloring paradigm (Lavrov & (later) Chaitin )
Build an interference graph GI for the procedure
Computing LIVE is harder than in the local case
GI is not an interval graph
(try to) construct a k-coloring
Minimal coloring is NP-Complete
Spill placement becomes a critical issue
Map colors onto physical registers
Comp 412, Fall 2010

3. Graph Coloring (A Background Digression)
The problem
A graph G is said to be k-colorable iff the nodes can be labeled with integers 1 … k so that no edge in G connects two nodes with the same label
Examples
Each color can be mapped to a distinct physical register
2-colorable
3-colorable
Comp 412, Fall 2010

4. Building the Interference Graph
What is an “interference” ? (or conflict)
Two values interfere if there exists an operation where both are simultaneously live
If x and y interfere, they cannot occupy the same register
To compute interferences, we must know where values are “live”
The interference graph, GI = (NI,EI)
Nodes in GI represent values, or live ranges
Edges in GI represent individual interferences
For x, y  NI, <x,y>  EI iff x and y interfere
A k-coloring of GI can be mapped into an allocation to k registers
Comp 412, Fall 2010

5. The Worklist Iterative Algorithm
Computing LIVE Sets
The compiler can solve these equations with a simple algorithm
The world’s quickest introduction to data-flow analysis !
WorkList  { all blocks }
while ( WorkList ≠ Ø)
remove a block b from WorkList
Compute LIVEOUT(b)
Compute LIVEIN(b)
if LIVEIN(b) changed
then add pred (b) to WorkList
Why does this work?
LIVEOUT, LIVEIN  2Names
UEVAR, VARKILL are constants for b
Equations are monotone
Finite # of additions to sets
will reach a fixed point !
Speed of convergence depends on the order in which blocks are “removed” & their sets recomputed
The Worklist Iterative Algorithm
Comp 412, Fall 2010

6. Observation on Coloring for Register Allocation
Suppose you have k registers—look for a k coloring
Any vertex n that has fewer than k neighbors in the interference graph (n < k) can always be colored !
Pick any color not used by its neighbors — there must be one
Ideas behind Chaitin’s algorithm:
Pick any vertex n such that n< k and put it on the stack
Remove that vertex and all edges incident from the interference graph
This may make additional nodes have fewer than k neighbors
At the end, if some vertex n still has k or more neighbors, then spill the live range associated with n
Otherwise successively pop vertices off the stack and color them in the lowest color not used by some neighbor
Comp 412, Fall 2010

7. Chaitin’s Algorithm While  vertices with < k neighbors in GI
Pick any vertex n such that n< k and put it on the stack
Remove that vertex and all edges incident to it from GI
If GI is non-empty (all vertices have k or more neighbors) then:
Pick a vertex n (using some heuristic) and spill the live range associated with n
Remove vertex n from GI , along with all edges incident to it and put it on the “spill list”
If this causes some vertex in GI to have fewer than k neighbors, then go to step 1; otherwise, repeat step 2
If the spill list is not empty, insert spill code, then rebuild the interference graph and try to allocate, again
Otherwise, successively pop vertices off the stack and color them in the lowest color not used by some neighbor
Lowers degree of n’s neighbors
Comp 412, Fall 2010

8. Chaitin’s Algorithm in Practice
3 Registers 2 3 1 4 5
Stack
1 is the only node with degree < 3
Comp 412, Fall 2010

9. Chaitin’s Algorithm in Practice
3 Registers 2 4 5 3 1
Stack
Now, 2 & 3 have degree < 3
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10. Chaitin’s Algorithm in Practice
3 Registers 4 5 3 2 1
Stack
Now all nodes have degree < 3
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11. Chaitin’s Algorithm in Practice
3 Registers 5 4 3 2 1
Stack
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12. Chaitin’s Algorithm in Practice
3 Registers
Colors: 1: 5 3 2: 4 2 3: 1
Stack
Comp 412, Fall 2010

13. Chaitin’s Algorithm in Practice
3 Registers
Colors: 1: 5 3 2: 4 2 3: 1
Stack
Comp 412, Fall 2010

14. Chaitin’s Algorithm in Practice
3 Registers
Colors: 1: 5 2: 4 3 2 3: 1
Stack
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15. Chaitin’s Algorithm in Practice
3 Registers
Colors: 1: 4 5 2: 3 2 3: 1
Stack
Comp 412, Fall 2010

16. Chaitin’s Algorithm in Practice
3 Registers
Colors: 2 1: 4 5 2: 3 3: 1
Stack
Comp 412, Fall 2010

17. Chaitin’s Algorithm in Practice
3 Registers
Colors: 2 1: 1 4 5 2: 3 3:
Stack
Comp 412, Fall 2010

18. Improvement in Coloring Scheme
Optimistic Coloring
If Chaitin’s algorithm reaches a state where every node has k or more neighbors, it chooses a node to spill.
Briggs said, take that same node and push it on the stack
When you pop it off, a color might be available for it!
For example, a node n might have k+2 neighbors, but those neighbors might only use 3 (<k) colors
Degree is a loose upper bound on colorability
Chaitin’s algorithm immediately spills one of these nodes
2 Registers:
Comp 412, Fall 2010
Briggs et al, PLDI 89 (Also, TOPLAS 1994)

19. Improvement in Coloring Scheme
Optimistic Coloring
If Chaitin’s algorithm reaches a state where every node has k or more neighbors, it chooses a node to spill.
Briggs said, take that same node and push it on the stack
When you pop it off, a color might be available for it!
For example, a node n might have k+2 neighbors, but those neighbors might only use just one color (or any number < k )
Degree is a loose upper bound on colorability
Briggs algorithm finds an available color
2 Registers:
2-Colorable
Comp 412, Fall 2010

20. Other Improvements to Chaitin-Briggs
Spilling partial live ranges [Bergner Pldi 97]
Bergner introduced interference region spilling
Limits spilling to regions of high demand for registers
Splitting live ranges [Simpson CC 98, Eckhardt Icplc 05]
Simple idea — break up one or more live ranges
Allocator can use different registers for distinct subranges
Allocator can spill subranges independently (use 1 spill location)
Iterative coalescing [George & Appel]
Use conservative coalescing because it is “safe”
Simplify the graph until only non-trivial nodes remain
Coalesce & try again
If coalescing does not reveal trivial nodes, then spill
Comp 412, Fall 2010

21. Chaitin-Briggs Allocator (Bottom-up Global)
Strengths & Weaknesses
Precise interference graph
Strong coalescing mechanism
Handles register assignment well
Runs fairly quickly
Known to overspill in tight cases
Interference graph has no geography
Spills a live range everywhere
Long blocks devolve into spilling by use counts
Is improvement still possible ?
Rising spill costs, aggressive transformations, & long blocks
 yes, but the returns are getting rather small
Comp 412, Fall 2010

22. What about Top-down Coloring?
The Big Picture
Use high-level priorities to rank live ranges
Allocate registers for them in priority order
Use coloring to assign specific registers to live ranges
The Details
Separate constrained from unconstrained live ranges
A live range is constrained if it has ≥ k neighbors in GI
Color constrained live ranges first
Reserve pool of local registers for spilling (or spill & iterate)
Chow split live ranges before spilling them
Split into block-sized pieces
Recombine as long as  k
Use spill costs as priority function !
Unconstrained must receive a color !
Peixotto’s 2007 MS thesis shows that top-down, in general, produces worse results unless we add an (expensive) adaptive feedback loop
Comp 412, Fall 2010

23. What about Top-down Coloring?
The Big Picture
Use high-level priorities to rank live ranges
Allocate registers for them in priority order
Use coloring to assign specific registers to live ranges
More Details
Chow used an imprecise interference graph
<x,y>  GI  x,y  Live(b) for some block b
Cannot coalesce live ranges since xy  <x,y>  GI
Quicker to build imprecise graph
Chow’s allocator may run faster on small codes, where demand for registers is also likely to be lower
Comp 412, Fall 2010

24. Linear Scan Allocation
Approximate Global Allocation
Linear Scan Allocation
Coloring allocators are often viewed as too expensive for use in JIT environments, where compile time occurs at runtime
Linear scan allocators use an approximate interference graph and a version of the bottom-up local algorithm
Interference graph is an interval graph
Optimal coloring (without spilling) in linear time
Spilling handled well by bottom-up local allocator
Algorithm does allocation in a “linear”
scan of the graph
Linear scan produces faster, albeit less
precise, allocations
Linear scan allocators hit a different point
on the curve of cost versus performance
Live Ranges in LS
Interference graph of a set of intervals is an interval graph.
Comp 412, Fall 2010
Sun’s HotSpot server compiler uses a complete Chaitin-Briggs allocator.

25. Linear Scan Allocation
Building the Interval Graph
Consider the procedure as a linear list of operations
A live range for some name is an interval (x,y)
x and y are the indices of two operations in the list, with x < y
Every operation where name is live falls between x & y, inclusive
Precision of live computation can vary with cost
Interval graph overestimates interference
The Algorithm
Use Best’s algorithm — bottom-up local
Distance to next use is well defined
Algorithm is fast & produces reasonable allocations
Variations have been proposed that build on this scheme
Comp 412, Fall 2010

26. Global Coloring from SSA Form
Observation: The interference graph of a program in SSA form is a chordal graph.
Observation: Chordal graphs can be colored
in O(N ) time.
These two facts suggest allocation using an
interference graph built from SSA Form
Chaitin-Briggs works from live ranges that
qre a coalesced version of SSA names
SSA allocators use raw SSA names as live ranges
Allocate live ranges, then insert copies for φ-functions
SSA-based allocation has created a lot of excitement in the last couple of years.
Chordal Graph
Every cycle of length > 3 has a chord
Comp 412, Fall 2010

27. SSA Name Space
SSA encodes facts about flow of values into the name space
Two principles
Each name is defined by exactly one operation
Each operand refers to exactly one definition
To reconcile these principles with real code
Add subscripts to variable names for uniqueness
Insert -functions at merge points to reconcile name space
x0  ...
x1  ...
x2 (x0,x1)
 x
x  ...
...  x + ...
becomes
Comp 412, Fall 2010

28. SSA Name Space These -functions are unusual constructs …
A -function only occurs at the start of a block
A -function has one argument for each CFG edge entering the block
A -function returns the argument that corresponds to the edge along which control flow entered the block
All -functions in the block execute concurrently
Since machines do not support -functions, must translate back out of SSA form before we produce executable code
All -functions in a block execute concurrently
All read their argument, all perform assignment in parallel
Using SSA form leads to simpler or better formulations of many optimizations (alternative to global data-flow analysis )
Comp 412, Fall 2010

29. Building SSA SSA Form Each name is defined exactly once
Each use refers to exactly one name
What’s Hard?
Straight-line code is easy
Split points are easy
Merge points are hard
(Sloppy) Construction Algorithm
Insert a -function for each variable at each merge point
Rename all values for uniqueness (using subscripts )
This approach
Inserts too many -functions
Inserts -functions in too many places
The rest, however, is optimization & beyond the scope of today’s lecture. (See §9 in EaC)
Back
Comp 412, Fall 2010

30. Global Coloring from SSA Form
Coloring from SSA Names has its advantages
If graph is k-colorable, it finds the coloring
(Opinion ) An SSA-based allocator will find more k-colorable graphs than a live-range based allocator because SSA names are shorter and, thus, have fewer interferences.
Allocator should be faster than a live-range allocator
Cost of live analysis folded into SSA construction, where it is amortized over other passes
Biggest expense in Chaitin-Briggs is the Build-Coalesce phase, which SSA allocator avoids, as it destroys the chordal graph
Comp 412, Fall 2010

31. Global Coloring from SSA Form
Coloring from SSA Names has its disadvantages
Coloring is rarely the problem
Most non-trivial codes spill; on trivial codes, both SSA allocator and classic Chaitin-Briggs are overkill. (Try linear scan?)
SSA form provides no obvious help on spilling
Shorter live ranges will produce local spilling (good & bad)
May increase spills inside loops
After allocation, code is still in SSA form
Need out-of-SSA translation
Introduce copies after allocation
Swap problem may require and extra register
Must run a post-allocation coalescing phase
Algorithms exist that do not use an interference graph
They are not as powerful as the Chaitin-Briggs coalescing phase
Loop-carried value cannot spill before the loop, since its name is only live inside the loop and after the loop.
Comp 412, Fall 2010

32. Hybrid Approach ?
How can the compiler attain both speed and precision?
Observation: lots of procedures are small & do not spill
Observation: some procedures are hard to allocate
Possible solution:
Try different algorithms
First, try linear scan
It is cheap and it may work
If linear scan fails, try heavyweight allocator of choice
Might be Chaitin-Briggs, SSA, or some other algorithm
Use expensive allocator only when cheap one spills
This approach would not help with the speed of a complex compilation, but it might compensate on simple compilations
Comp 412, Fall 2010

33. An Even Stronger Global Allocator
Hierarchical Register Allocation (Koblenz & Callahan)
Analyze control-flow graph to find hierarchy of tiles
Perform allocation on individual tiles, innermost to outermost
Use summary of tile to allocate surrounding tile
Insert compensation code at tile boundaries (LRxLRy)
Anecdotes suggest it is fairly effective
Target machine is multi-threaded multiprocessor (Tera MTA)
Strengths
Decisions are largely local
Use specialized methods on individual tiles
Allocator runs in parallel
Weaknesses
Decisions are made on local information
May insert too many copies
Still, a promising idea
Eckhardt’s MS (Rice, 2005) shows that K&C produces better allocations than C&B, but is much slower
Comp 412, Fall 2010

34. Regional Approaches to Allocation
Probabilistic Register Allocation (Proebsting & Fischer)
Attempt to generalize from Best’s algorithm (bottom-up, local )
Generalizes “furthest next use” to a probability
Perform an initial local allocation using estimated probabilities
Follow this with a global phase
Compute a merit score for each LR as
(benefit from x in a register = probability it stays in a register)
Allocate registers to LRs in priority order, by merit score, working from inner loops to outer loops
Use coloring to perform assignment among allocated LRs
Little direct experience (either anecdotal or experimental)
Combines top-down global with bottom-up local
This idea predated Linear Scan and tried to achieve many of the same benefits.
Comp 412, Fall 2010

35. Regional Approaches to Allocation
Register Allocation via Fusion (Lueh, Adl-Tabatabi, Gross)
Use regional information to drive global allocation
Partition CFGs into regions & build interference graphs
Ensure that each region is k-colorable
Merge regions by fusing them along CFG edges
Maintain k-colorability by splitting along fused edge
Fuse in priority order computed during the graph partition
Assign registers using interference graphs
i.e., execution frequency
Strengths
Flexibility
Fusion operator splits on low-frequency edges
Weaknesses
Choice of regions is critical
Breaks down many values are live across region boundaries
Comp 412, Fall 2010

36. Extra Slides Start Here
Comp 412, Fall 2010