Beyond Superlocal: Dominators, Data-Flow Analysis, and DVNT. COMP 506 Rice University Spring target code. source code OpJmizer

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1 COMP 506 Rice University Spring 2017 Beyond Superlocal: Dominators, Data-Flow Analysis, and DVNT source code Front End IR OpJmizer IR Back End target code Copyright 2017, Keith D. Cooper & Linda Torczon, all rights reserved. Students enrolled in Comp 506 at Rice University have explicit permission to make copies of these materials for their personal use. Faculty from other educajonal insjtujons may use these materials for nonprofit educajonal purposes, provided this copyright nojce is preserved SecJon numbers refer to EaC2e.

2 Two Weeks Ago Review Key ideas in opjmizajon Safety Profitability Opportunity Superlocal Value Numbering Control-flow graph construcjon Clever use of renaming and a scoped-hash table Found more opportunijes B A p c + d r c + d m a + b n a + b C q a + b r c + d D e b + 18 E e a + 17 s a + b t c + d u e + f u e + f F v a + b w c + d x e + f To do bexer, we need to understand what informajon we can always use on entry to F and G. G y a + b z c + d We want to find inter-block improvements in F and G COMP 506, Spring

3 Redundancy Elimina?on as an Example An expression x+y is redundant if and only if, along every path from the procedure s entry, it has been evaluated, and its cons7tuent subexpressions (x & y) have not been re-defined. From Lecture 14 Slide 7, with emphasis added We need to understand the proper?es of those paths Specifically, what blocks occur on any path from entry to p If we can find those blocks, we can use their hash tables For example: A is on every path to any other node; and C is on every path to D, E, and F. Only A is on every path to G. B m a + b n a + b COMP 506, Spring A p c + d r c + d G C y a + b z c + d q a + b r c + d D e b + 18 E e a + 17 s a + b t c + d u e + f u e + f F v a + b w c + d x e + f

4 Dominance in a Control-Flow Graph Defini?on In a flow graph, x dominates y if and only if every path from the entry node of the control-flow graph to y includes x By definijon, x dominates x We associate a DOM set with each node DOM(x) contains the set of nodes that dominate x DOM(x ) 1 Immediate dominator For any node x, there must be a y in DOM(x ) closest to x We call this y the immediate dominator of x x cannot be its own immediate dominator, unless x is n 0 As a maxer of notajon, we write this as IDOM(x ) Original idea: R.T. Prosser. ApplicaJons of Boolean matrices to the analysis of flow diagrams, COMP Proceedings 506, Spring of the 2017 Eastern Joint Computer Conference, Spartan Books, New York, pages ,

5 Dominance Dominators have many uses in analysis & transforma?on Finding loops Building SSA form Making code mojon decisions B A is n 0 A p c + d r c + d m a + b n a + b C q a + b r c + d Dominator sets Block DOM IDOM A A B A,B A C A,C A D A,C,D C E A,C,E C F A,C,F C G A,G A Dominator tree A B C G D E F G D e b + 18 E e a + 17 s a + b t c + d u e + f u e + f F y a + b z c + d v a + b w c + d x e + f COMP 506, Spring

6 Compu?ng Dominators To use dominance informa?on, we need to compute DOM sets A node n dominates node m iff n is on every path from n 0 to m Every node dominates itself, by definijon n s immediate dominator is the node in DOM(n ) that is closest to n in the graph 1 Compu?ng DOM We formulate the computajon of DOM sets as a data-flow analysis problem Simultaneous equajons over sets associated with the nodes in the CFG EquaJons relate the set value at a node n to those of its predecessors and successors in the CFG EquaJons must be well-formed We want a unique solujon We want an efficient technique to compute that solujon COMP 1 IDOM(n) 506, Spring n, unless 2017 n is n 0, by convenjon. 6

7 Compu?ng Dominators Equa?ons for Dominance DOM(n 0 ) = { n 0 } DOM(n) = { n } ( p preds(n) DOM(p)) IniJally, DOM(n) = N, n n 0. We can do bexer. Compu?ng DOM These equajons define the data-flow problem DOM has among the simplest equajons of any data-flow problem These equajons have a unique fixed-point solujon That is, an iterajve algorithm will find a unique answer The simplicity of the DOM equajons lead to efficient solujons COMP If necessary, 506, Spring add 2017 a unique entry node n 0 that has no predecessors in the CFG. 7

8 Compu?ng Dominators How do we compute a solu?on to these equa?ons? Build a control-flow graph to idenjfy the blocks & their predecessors IniJalize DOM(n ) appropriately, for all n in the CFG Set DOM(n) to N, for all n n 0 Set DOM(n 0 ) to { n 0 } Repeatedly apply the second equajon, unjl the sets stop changing DOM(n 0 ) { n 0 } for x n 1 to n n DOM(x) { all nodes in graph, N } change true while (change) change false for x n 0 to n n TEMP { x } ( y pred (x) DOM(y )) if DOM(x) TEMP then change true DOM(x) TEMP Round-robin itera?ve algorithm for DOM The round-robin solver is easier to analyze and understand than other solvers. COMP The results 506, Spring with 2017 round-robin easily carry over to other forms of iterajve solvers. 8

9 Compu?ng Dominators Immediate Ques?ons That You Should Ask Termina?on: Does the algorithm halt? Correctness: Does it compute DOM sets correctly? Efficiency: How quickly does it compute DOM? The study of data-flow analysis is a subject unto itself Long history in the literature MathemaJcal bases for answers to our quesjons I will summarize that knowledge DOM(n 0 ) { n 0 } for x n 1 to n n DOM(x) { all nodes in graph, N } change true while (change) change false for x n 0 to n n TEMP { x } ( y pred (x) DOM(y )) if DOM(x) TEMP then change true DOM(x) TEMP Round-robin itera?ve algorithm for DOM COMP 506, Spring

10 Compu?ng Dominators Why do we believe that this algorithm halts? Each DOM set can contain at most N elements a finite number of elements New DOM sets are produced as the intersecjon of old DOM sets (D 1 D 2 ) D 1 (D 1 D 2 ) D 2 New sets are never larger than old sets Since the inijal sets are finite, they can only shrink a finite number of Jmes They must stop changing ater a finite number of updates The algorithm halts ater a finite number of updates DOM(n 0 ) { n 0 } for x n 1 to n n DOM(x) { all nodes in graph, N } change true while (change) change false for x n 0 to n n TEMP { x } ( y pred (x) DOM(y )) if DOM(x) TEMP then change true DOM(x) TEMP Round-robin itera?ve algorithm for DOM COMP 506, Spring

11 Semila_ce for DOM The possible DOM sets fit into a structure called a semila_ce With n = N nodes in the graph, there are 2 n possible DOM sets Think of each n (in binary) as a bit vector of the n nodes, D 0, D 1, D 2, D n One-to-one correspondence between numbers (0 to n-1) and subsets of N IntersecJon imposes an order on the D i s D i D j is no larger than max( D i, D j ) intersec7on is monotonic on the D i s D i D j iff D i D j = D j D i > D j iff D i D j and D i D j Two special elements: and top and bo^om D i = D i, for all D i is D n, the set of all nodes D i =, for all D i is D 0, the empty set Introduces the nojon of a descending chain Sequence of values x 0, x 1, x 2, x k where each x i is a set in D and x i > x i+1, 1 i < n If all descending chains are finite, then the algorithm halts Since D is finite, the chains must be finite COMP 506, Spring

12 Dominators The La_ce for Four Nodes 1111 > > > > COMP 506, Spring

13 Dominators 1110 = 1110 The La_ce for Four Nodes 1111 > > > > COMP 506, Spring

14 Dominators = 0011 The La_ce for Four Nodes 1111 > > > > COMP 506, Spring

15 Dominators = 0000 ( ) The La_ce for Four Nodes 1111 > > > > COMP 506, Spring

16 Dominators And, we can go on and on The La_ce for Four Nodes 1111 > > > > COMP 506, Spring

17 Second Lab: Two Op?ons Moved to middle of class for axendance reasons 1. Generate ILOC code out of your parser Add acjons to your grammar file, and support code, to generate working ILOC code for input programs wrixen in demo This lab builds on your lab 1 experience and code I have a small library of demo programs both small tests and larger programs Demonstrate correctness using an ILOC simulator 2. Take ILOC code generated by my parser & improve it Build a framework that reads in the ILOC code & holds it in some data structure Process that ILOC code to build a control-flow graph Perform value-numbering on the blocks in the ILOC code Demonstrate correctness & measure improvement using an ILOC simulator In either case, I will provide a demo to ILOC compiler and the ILOC simulator as tools to help you test and generate examples. COMP 506, Spring

18 Compu?ng Dominators Why do we believe that this algorithm computes the right answer? With enough math, we can prove that this formulajon has only one fixed point When the while loop halts, the values of the DOM sets are uniquely determined. If there is only one fixed point, then any fixed point we compute is the right one DOM(n 0 ) { n 0 } for x n 1 to n n DOM(x) { all nodes in graph, N } change true while (change) change false for x n 0 to n n TEMP { x } ( y pred (x) DOM(y )) if DOM(x) TEMP then change true DOM(x) TEMP Round-robin itera?ve algorithm for DOM COMP See Kam 506, & Ullman Spring 2017 [210] & [211] in EaC2e 18

19 Compu?ng Dominators How fast can we compute the answers to DOM? Since there is only one answer, we get the same answer no maxer what order we take in the inner loop Order maxers (a lot) A reverse postorder solver will halt in d(g) + 3 iterajons of the while loop d(g) is the maximum number of back edges in any acyclic path in G In pracjce, d(g) is small (< 4) DOM(n 0 ) { n 0 } for x n 1 to n n DOM(x) { all nodes in graph, N } change true while (change) change false for x n 0 to n n TEMP { x } ( y pred (x) DOM(y )) if DOM(x) TEMP then change true DOM(x) TEMP Round-robin itera?ve algorithm for DOM COMP See Kam 506, & Ullman Spring 2017 [210] & [211] in EaC2e 19

20 Ordering the Nodes to Maximize Propaga?on Postorder Numbers 1 Reverse Postorder Numbers RPO ordering visits predecessors before visijng a node (ignoring cycles) For problems that propagate informajon from successors to predecessors, use RPO on the reversed CFG Not the same as reverse preorder (see graph to the right) B 0 B 1 B 2 COMP 506, Spring B 3 B 6 B 4 B 7 Counter example from Exercise 9.4(b) B 5 B 8

21 Example Dominator Calcula?on Added a unique entry node B 0 B 1 B 2 B 3 Itera?on Progress of itera?ve solu?on for DOM DOM(n) N N N N N N N 1 0 0,1 0,1,2 0,1,3 0,1,3,4 0,1,3,5 0,1,3,6 0,1, ,1 0,1,2 0,1,3 0,1,3,4 0,1,3,5 0,1,3,6 0,1,7 B 4 B 5 B 6 B 7 Flow Graph COMP 506, Spring

22 Example Dominator Calcula?on Progress of itera?ve solu?on for DOM B 0 B 1 B 2 B 3 Itera?on DOM(n) N N N N N N N 1 0 0,1 0,1,2 0,1,3 0,1,3,4 0,1,3,5 0,1,3,6 0,1, ,1 0,1,2 0,1,3 0,1,3,4 0,1,3,5 0,1,3,6 0,1,7 B 4 B 5 Results of itera?ve solu?on for DOM B 7 B DOM 0 0,1 0,1,2 0,1,3 0,1,3,4 0,1,3,5 0,1,3,6 0,1,7 IDOM Flow Graph There are asymptojcally faster algorithms. With the right data structures, the iterajve algorithm can be made extremely fast (compejjve out to 10,000 or 20,000 nodes) COMP 506, Spring 2017 See Cooper, Harvey, & Kennedy [100], or in EaC2e. 22

23 Speeding Up DOM No?ce that the IDOM sets encode the dominator tree B 0 B DOM 0 0,1 0,1,2 0,1,3 0,1,3,4 0,1,3,5 0,1,3,6 0,1,7 IDOM B 2 B 3 B 4 B 5 B 6 B 7 Flow Graph Dominator tree We don t need to compute or store DOM We can get away with IDOM and read DOM by walking the tree represented in the IDOM sets This observajon leads to a compact data structure and a very fast algorithm COMP 506, Spring

24 Speeding Up DOM No?ce that the IDOM sets encode the dominator tree DOM 0 0,1 0,1,2 0,1,3 0,1,3,4 0,1,3,5 0,1,3,6 0,1,7 IDOM Dominator tree Key opera?on in the algorithm is intersec?on two DOM sets Number the nodes in the flow graph with their RPO number To intersect DOM(x) and DOM(y), start at x and y and walk up the tree from each of them If RPO( f 1 ) > RPO( f 2 ), move up from f 1 If RPO( f 1 ) < RPO( f 2 ), move up from f 2 If RPO( f 1 ) = RPO( f 2 ), f 1 is the closest common ancestor of x and y, and chain above f 1 dominates both x and y This two-finger algorithm is extremely fast ( in EaC2e) COMP 506, Spring

25 Dominators Can Improve Superlocal Value Numbering SVN did not help with blocks F or G MulJple predecessors Must decide what facts hold in F and in G For G, combine B & F? Merging state is expensive Fall back on what s known Can use table from IDOM(x) to start x Use C for F and A for G Imposes a DOM-based applicajon order B A p 0 c + d r 0 c + d G m 0 a + b n 0 a + b C r 2 φ(r 0,r 1 ) y 0 a + b z 0 c + d q 0 a + b r 1 c + d D e 0 b + 18 E e 1 a + 17 s 0 a + b t 0 c + d u 0 e + f u 1 e + f F e 3 φ(e 0,e 1 ) u 2 φ(u 0,u 1 ) v 0 a + b w 0 c + d x 0 e + f Leads to Dominator VN Technique (DVNT) COMP 506, Spring

26 Dominator-Based Value Numbering The DVNT Algorithm Use superlocal algorithm on extended basic blocks Retain use of scoped hash tables Need to use the SSA name space to avoid bookkeeping headaches Start each node with table from its IDOM DVNT generalizes the superlocal algorithm No values flow along back edges Constant folding, algebraic idenjjes as before (i.e., around loops) Larger scope leads to ( poten7ally) bexer results LVN + SVN + good start for EBBs missed by SVN COMP 506, Spring

27 Dominator-Based Value Numbering B A p c + d r c + d m a + b n a + b C q a + b r c + d DVNT advantages Find more redundancy LiXle addijonal cost Retains online character D e b + 18 E e a + 17 s a + b t c + d u e + f u e + f G F y a + b z c + d v a + b w c + d x e + f DVNT shortcomings Misses some opportunijes No loop-carried redundancies or constants COMP 506, Spring 2017 See [53] or in EaC2e 27