1. Optimal Polynomial-Time Interprocedural Register Allocation for High-Level Synthesis Using SSA Form
Philip Brisk
Ajay K. Verma
Paolo Ienne
csda

2. Outline Register Allocation Overview
Interprocedural Register Allocation
Related Work
SSA Form With Launch and Landing Pads
Optimal Solution
Experimental Results
Conclusion

3. Modeling Register Allocation
For Procedure Pi…
Build interference graph Gi = (Vi, Ei)
Vi – One vertex for each variable
Ei – Edge between each pair of interfering variables
Two variables interfere if their lifetimes overlap
Compute the chromatic number χ(Gi)
Color assignment = Register assignment
NP-Complete in general

4. Local Interferences Local Interferences – Single Procedure
Overlapping lifetimes
Static Single Assignment (SSA) Form
Interference graph is chordal
X 
Y 
 X
Z 
 Y
 Z Y Z X Y Z X

5. Global Interferences Global Interferences
Variable V is live across a call to procedure P
V interferes with EVERY local variable in P
And all variables in all procedures reachable from P
Must consider all paths through the Call Graph
Main:
V 
Call P
 V P: …
Call Q Q: …
Main P Q

6. Global Interferences and Recursion
Fact:
No register can hold a local variable across a recursive function call
Runtime stack is required
Some exceptions (e.g. static local variables)
Ignored here
Call Graph
Compute strongly connected components (SCCs)
Collapse each SCC into a single node
Resulting “Augmented Component Graph” is acyclic

7. Interprocedural Register Allocation
Interprocedural Interference Graph (IIG)
Undirected graph G = (V, E)
V – All variables in all procedures
E – Local AND global interferences
Compute chromatic number χ(G)

8. Related Work Interprocedural Register Allocation in HLS
Color IIG with heuristic [Vemuri et al., TODAES ’02]
IIG is large
Polynomial heuristics are still slow
Scalable Approach [Beidas and Zhu, ASP-DAC ’05]
Color each procedure individually
Use any heuristic you want
Use any intermediate representation you want
Propagate global interferences at call points
IIG is never built

9. Contribution Interprocedural register allocation
Optimal, polynomial-time algorithm
Scalable
IIG is never built
If built, it would be chordal
Each Procedure colored individually
SSA Form – interference graph is chordal
Special case of [Beidas and Zhu, ASP-DAC ’05]
Top-down color propagation
Novel SSA-based intermediate representation
Chordal color assignment (with offset)

10. Preallocation of Global Registers
Global registers hold variables that are live across procedure calls
How many do we need?
Pi – Procedure
ck – Call Point Pi ck Pj
Procedure Call
P – Set of Procedures in App.
L(ck) – Set of variables live across ck
ck : Call Pj …

11. Preallocation of Global Registers
Compute: δ – Number of variables live…
At the entry of a procedure
Across a call point
Procedure: Pi
ck: Call … δ2 …
(δi is known) δ1 δm Pi
L(ck) …
δi = MAX {δk}
1 ≤ k ≤ m
δk = δi + |L(ck)|
(i.e. Over all points that call Pi)

12. Example δ6 = MAX{δ11, δ14} δ6 = MAX{5, 4} = 5 δ5 = MAX{δ12, δ13}i δi
Example P1 P1 P1 P1 P1 P1 P1 P2 P2 2 P2 2 P2 2 ci
|L(ci)| P3 P3 3 P3 3 P3 3 c7 1 1 P4 2 P4 2 P4 P4 2 c8 2 2 c7 c7 c7 c8 P2 c8 c8 c9 P3 c9 c9
c10
c10
c10 P4
c11
c14 P6
c11
c11 P5 6 P5 6 P5 c9 3 3 P6 P6 5 P6 5 P2
c12 P2 P3
c13 P3 P4
c14 P4
c10 2 2 c7 1 c8 2 c7 1 c7 1 c7
c11 5 5 c8 2 c8 2 c8
c12 3 3
c12
c12
c13 P5
c13
c14 c9 c9 3 c9 3 c9 3
c13 3 3
c10 2
c10 2
c10 2
c10 P5 P6
c14 2 2
c11 5
c11 5
c14 4
c11 5
c11
δ6 = MAX{δ11, δ14}
δ6 = MAX{5, 4} = 5
δ5 = MAX{δ12, δ13}
δ5 = MAX{5, 6} = 6
δ14 = |L(c14)| + δ4
δ14 = = 4
δ13 = |L(c13)| + δ3
δ13 = = 6
δ4 = MAX{δ10}
δ4 = MAX{2} = 2
δ8 = |L(c8)| + δ1
δ8 = = 2
δ11 = |L(c11)| + δ1
δ11 = = 5
δ9 = |L(c9)| + δ1
δ9 = = 3
δ10 = |L(c10)| + δ1
δ10 = = 2
δ7 = |L(c7)| + δ1
δ7 = = 1
δ1 = 0
δ12 = |L(c12)| + δ2
δ12 = = 5
δ3 = MAX{δ9}
δ3 = MAX{3} = 3
δ2 = MAX{δ7, δ8}
δ2 = MAX{1, 2} = 2
c12 5
c12
c12 5
c12 5
c13 6
c13
c13 6
c13 6
c14
c14 4
c14 4

13. Preallocation of Global Registers
N = MAX {δi} – Number of global registers allocated Pi P
T = {T1, …., TN}
When Procedure Pi is called..
At most δi variables live across calls leading to Pi
Holds for every path in the call graph
How to ensure that all variables live across calls leading to Pi are assigned to the right register?

14. Launch and Landing Pads
Procedure Pi calls Pj; (m = δi)
Assign variables live across calls leading to Pi to T1…Tm
Let ck be the call point; n = |L(ck)|
Launch Pad
Parallel copy placed before the call
(Tm+1…Tm+n)  ψ(L(ck))
Landing Pad
Copy the values back after the call
L(ck)  ψ((Tm+1…Tm+n))

15. Theoretical Consequences of Launch and Landing Pads
Theorem:
All global interferences involve at least one global register
Corollary:
Local variables in distinct procedures do not interfere
No local variable in “main” has a global interference
Every variable defined locally in Pi (m = δi)
Interferes with global registers T1…Tm
Does NOT interfere with global registers Tm+1, … TN
=> Can assign local vars in Pi to global registers Tm+1, … TN

16. Reducing the Chromatic Number
Procedure: A
V  …
Call B
W  …
…  V
X  …
…  W
Y  …
…  X
…  Y
Procedure: B
Z  …
…  Z V W V W X Y Z X Y
Chromatic Number = 3

17. Reducing the Chromatic Number
Procedure: A
V  …
T1  Ψ(V)
Call B
V  Ψ-1(T1)
W  …
…  V
X  …
…  W
Y  …
…  X
T1  Ψ(Y)
Y  Ψ-1(T1)
…  Y
Procedure: B
Z  …
…  Z V T1 V W X W V Y X Y T1 Z T1
Chromatic Number = 2

18. Characterizing the IIG
Theorem:
T is a clique in the IIG
IIG is chordal
Chromatic Number of the IIG is: R = MAX{δi + χ(Gi)} Pi P

19. Example CLIQUE Global interference
δ1 = 0
δ2 = 2
δ3 = 3
δ4 = 2
δ3 = 6
δ6 = 5
Global interference
Tj interferes with each local variable in Gi

20. Coloring Algorithm Use SSA+LLP Form, but DON’T build the IIG
For Pi colors in the range 1..δi are unavailable
Color the local (chordal) interference graph Gi of Pi
Complexity: O(Vi + Ei)
For each vertex in Pi, replace color c with c + δi
Complexity: O(Vi)

21. Experiments Applications taken from Mediabench and MiBench
Written in C
Compiled Using Machine SUIF
Optimal color assignment
Compare to heuristics
Color Palette Propagation
Top-Down, Bottom-Up [Beidas and Zhu, ASP-DAC’05]
Heuristic Color Assignment [Matula and Beck, JACM ’83]

22. Registers Allocated (Normalized to Optimal)

23. Runtime (Normalized to Optimal)

24. Runtime of Pegwit (Normalized to Optimal)

25. Limitations Global Variables Static Local Variables Function Pointers
Interfere with all variables in the program
Lifetime can still be analyzed
Static Local Variables
Initialized on first access
Hold their values across function calls
Function Pointers
Resolution is NP-Complete

26. Conclusion Inteprocedural register allocation in HLS A few limitations
Optimal, polynomial-time algorithm
Uses SSA Form + Launch/Landing Pads
IIG is a chordal graph
Scalable – no need to build IIG
Significantly faster than sub-optimal heuristics
A few limitations
Global variables, local static variables
Function pointers
Resolution is NP-Complete

27. Related Work Register Allocation in HLS
Clique Partitioning/Coloring Problem
[Tseng and Siewiorek, ’86]
Scheduled DFGs – Interval Graphs
[Kurdahi and Parker, ’87]
Scheduled Cyclic DFGs – Circular Arc Graphs
(NP-Complete)
[Stok, ’92]
Restrictions on Variable Lifetimes – Chordal Graphs
[Springer and Thomas, ’94]
Static Single Assignment Form – Chordal Graphs
[Brisk et al. 2005/6], [Hack and Goos, 2005/6],
[Bouchez et al. 2005]