1. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-31 15-745 Register Allocation

2. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-32 Intro to Global Register Allocation Problem: Allocation of variables (pseudo-registers) to hardware registers in a procedure One of the most important optimizations Memory accesses are more costly than register accesses –True even with caches –True even with CISC architectures Important for other optimizations –E.g., redundancy elimination assumes old values are kept in registers When it does not work well, the performance impact is noticeable.

3. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-33 Terminology Allocation decision to keep a pseudo-register in a hardware register prior to register allocation, we assume an infinite set of registers –(aka “temps” or “pseudo-registers”). Spilling a pseudo-register is spilled to memory, if not kept in a hardware register Assignment decision to keep a pseudo-register in a specific hardware register

4. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-34 What are the Problems? What is the minimum of registers needed to avoid spilling? Given n registers in a machine, is spilling necessary? Find an assignment for all pseudo-registers, if possible. If there are not enough registers in the machine, how do we spill to memory? IF A goto L1 A =... B =... L1: C =... = A D = B = A D = C

5. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-35 Abstraction for Reg Alloc & Assignment Intuitively: Two pseudo-registers interfere if at some point in the program they cannot both occupy the same register. Interference graph: an undirected graph, where nodes = pseudo-registers there is an edge between two nodes if their corresponding pseudo-registers interfere

6. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-36 Register Allocation and Coloring A graph is n-colorable if every node in the graph can be colored with one of the n colors such that two adjacent nodes do not have the same color. Assigning n registers (without spilling) = Coloring with n colors –assign a node to a register (color) such that no two adjacent nodes are assigned same registers(colors) Is spilling necessary? = Is the graph n-colorable? To determine if a graph is n-colorable is NP-complete, for n>2 –Too expensive –Heuristics

7. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-37 Simple Algorithm Build an interference graph refining notion of a node finding the edges Coloring use heuristics to try to find an n-coloring –Success  colorable and we have an assignment –Failure  graph not colorable, or graph is colorable, but it is too expensive to color

8. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-38 Nodes in an Interference Graph IF A goto L1 A =... B =... L1: C =... = A D = B = A D = C A = 2 = A

9. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-39 Live Ranges & Merged Live Ranges Motivation: to create an interference graph that is easier to color –Eliminate interference in a variable’s “dead” zones. –Increase flexibility in allocation: can allocate same variable to different registers A live range consists of a definition and all the points in a program (e.g. end of an instruction) in which that definition is live. –How to compute a live range? Two overlapping live ranges for the same variable must be merged a = = a a =

10. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-310 Example (Revisited) = A IF A goto L1 A =... B =... L1: C =... = A D = B = A D = C A = D {} {A}{A 1 } {A}{A 1 } {A}{A 1 } {A,C}{A 1,C} {D}{A 1,C,D 1 } {C}{A 1,C} {D}{A 1,B,C,D 1,D 2 } {A}{A 2,B,C,D 1,D 2 } {A}{A 2,B,C,D 1,D 2 } {A}{A 1 } {A,B}{A 1,B} {D}{A 1,B,D 2 } {B}{A 1,B} (A 1 ) 2 ) (D 2 ) 1 ) does not use A, B, C or D {}{A 2,B,C,D 1,D 2 } Reaching DefsLive Vars

11. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-311 Merging Live Ranges Merging definitions into equivalence classes: Start by putting each definition in a different equivalence class For each point in a program –if variable is live, and there are multiple reaching definitions for the variable –merge the equivalence classes of all such definitions into a one equivalence class From now on, refer to merged live ranges simply as live range Merged live ranges are also known as “webs”

12. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-312 Edges of Interference Graph Intuitively: Two live ranges (necessarily of different variables) may interfere if they overlap at some point in the program. Algorithm: –At each point in program, enter an edge for every pair of live ranges at that point An optimized definition & algorithm for edges: For each inst i Let x be live range of definition at inst i For each live range y present at end of inst i insert an edge between x and y Faster Better quality

13. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-313 Example 2 If Q goto L1 A = L1: B = If Q goto L2 = A L2: = B

14. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-314 Coloring Reminder: coloring for n > 2 is NP-complete Observations –a node with degree < n   can always color it successfully, given its neighbors’ colors –a node with degree = n  –a node with degree > n 

15. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-315 Coloring Algorithm Algorithm Iterate until stuck or done –Pick any node with degree < n –Remove the node and its edges from the graph If done (no nodes left) –reverse process and add colors Example (n = 3) Note: degree of a node may drop in iteration Avoids making arbitrary decisions that make coloring fail B E C D A

16. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-316 What Does Coloring Accomplish? Done: colorable also obtained an assignment Stuck: colorable or not? B E C D A

17. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-317 Summary Problems: Given n registers in a machine, is spilling avoided? Find an assignment for all pseudo-registers, whenever possible. Solution: Abstraction: an interference graph –nodes: (merged) live ranges –edges: presence of live range at time of definition Register Allocation and Assignment problems = n-colorability of interference graph  NP-complete Heuristics to find an assignment for n colors –successful: colorable, and finds assignment –unsuccessful:colorability unknown & no assignment

18. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-318 Interference Graph Creation Use liveness information to determine if two temps have overlapping live ranges Nodes in the interference graph represent temps or hard registers in IR (u,v)  G iff u and v interfere What does it mean for two temps to interfere? I.e., What edges should be included in G? What does it mean to be a node in G

19. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-319 Meaning of interference Overlapping live ranges? <- t a <- b <- x <- y <- <- t <- a <- b <- x <- y t a x by Can we do better?

20. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-320 Meaning of interference ( u, v )  G iff u is live at definition point of v t a x by <- t a <- b <- x <- y <- <- t <- a <- b <- x <- y

21. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-321 Nodes of Interference Graph for (i=0; i<10; i++) { … } for (i=0; i<n; i++) { … } One node for i or two? We really don’t want the first register assignment to influence which register the second i is assigned to. How can we automate this? SSA? Reaching Defs?

22. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-322 Building Webs Use reaching defs to create du-chains Use du-chains to create “webs” A web is a collection of du-chains such that for any 2 du- chains in a web they have a stmt in common 1:s <- 2:t <- 3: <- s 9: <- t 10:t <- 11: <- s 4:t <- 5:s <- 6:u <- 7: <- t 8: <- s 12: <- t 13: <- u 1:3,11 2:9 4:7,9 5:8,11 6:13 10:12

23. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-323 Building Webs Use reaching defs to create du-chains Use du-chains to create “webs” A web is a collection of du-chains such that for any 2 du- chains in a web they have a stmt in common 1:s <- 2:t <- 3: <- s 9: <- t 10:t <- 11: <- s 4:t <- 5:s <- 6:u <- 7: <- t 8: <- s 12: <- t 13: <- u 1:3,11 2:9 4:7,9 5:8,11 6:13 10:12

24. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-324 Building Webs Use reaching defs to create du-chains Use du-chains to create “webs” A web is a collection of du-chains such that for any 2 du- chains in a web they have a stmt in common 1:s <- 2:t <- 3: <- s 9: <- t 10:t <- 11: <- s 4:t <- 5:s <- 6:u <- 7: <- t 8: <- s 12: <- t 13: <- u 1:3,11 2:9 4:7,9 5:8,11 6:13 10:12

25. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-325 Building Webs Use reaching defs to create du-chains Use du-chains to create “webs” A web is a collection of du-chains such that for any 2 du- chains in a web they have a stmt in common 1:s <- 2:t <- 3: <- s 9: <- t 10:t <- 11: <- s 4:t <- 5:s <- 6:u <- 7: <- t 8: <- s 12: <- t 13: <- u 1:3,11 2:9 4:7,9 5:8,11 6:13 10:12

26. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-326 Building Webs Use reaching defs to create du-chains Use du-chains to create “webs” A web is a collection of du-chains such that for any 2 du- chains in a web they have a stmt in common 1:s <- 2:t <- 3: <- s 9: <- t 10:t <- 11: <- s 4:t <- 5:s <- 6:u <- 7: <- t 8: <- s 12: <- t 13: <- u 1:3,11 2:9 4:7,9 5:8,11 6:13 10:12

27. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-327 Web Creation Algorithm Compute reaching definitions {{def 0 : i 00, i 01, …, i 0n }, {def 1 : i 10, i 11, …, i 1n }, … } For each def: {d: i 0, i 1, …, i n } if d is not marked create new web, w. Put d into bag, B While B is not empty remove a def, e, from B make e part of W mark e foreach use of d, i, in {i 0, i 1, …, i n } make i part of W for each def, r, that reaches i, add r to B

28. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-328 Creating the Interference Graph Compute liveness Compute webs Foreach instruction, i, that defines T Determine Web, W, for T Foreach live variable at i –Determine Web w, for i –Insert (W,w) into G Almost complete!

29. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-329 K-coloring a graph Lets say we have a node, n, such that n ± < k and let G’ = G – {n}, then: if G’ can be k-colored, then G can be k-colored. Proof? This suggests the following optimistic heuristic: While |G| > 0 choose some n with degree < k push n on stack remove n from G While |S| > 0 pop n from S color with a legal color

30. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-330 Example: K=3 v xw u t v x t u w’w’’w’w’’ w w’ v x

31. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-331 Example: K=3 v xw u t v x t u w’w’’w’w’’ w w’ v x w’’ut

32. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-332 Example: K=3 v xw u t v x u w’w’’w’ w’’ w w’ v x w’’ u t t t u xvw’w w

33. CS745: Register Allocation© Seth Copen Goldstein & Todd C. Mowry 2002-333 Algorithm is not perfect What should we do when there is no node of degree < k?