CS293S Redundancy Removal Yufei Ding Review of Last Class - PowerPoint PPT Presentation

CS293S Redundancy Removal Yufei Ding

Review of Last Class � Consideration of optimization � Sources of inefficiency � Components of optimization � Paradigms of optimization � Redundancy Elimination � Types of intermediate representations of a program � Removing redundant expressions 2

Topics of This Class � Removing redundant expressions � DAG: version tracking � Linear representation: value numbering � Scope of optimization � Basic block � Extended basic block � Region (dominator) 3

1. IR: Syntax Tree m <-- 2 x y x z n <-- 3 x y x z o <-- 2 x y - z 4

DAG: Eliminating Identical Subtrees m <-- 2 x y x z n <-- 3 x y x z o <-- 2 x y - z Using hash-based constructor: • key is the textual notion of operators and operands; • value is the node number or address. 5

Version Tracking m <-- 2 x y x z y <-- 3 x y x z o <-- 2 x y - z Associate a counter with each variable for version tracking m1 <-- 2 x y1 x z1 y2 <-- 3 x y1 x z1 o1 <-- 2 x y2 - z1 6

Missing Any Opportunities? Original Code a ¬ x + y z ¬ y d ¬ 17 c ¬ x + z � Is there any redundant expression? � Can the DAG-based approach recognize it? The hash-based constructor uses a textual notion of “equality”, - so y equals y, independent of its value. - z does not equal y, independent of its value. 7

Linear Representation: Linear IR Example: Three Address Code Statement form: � x ¬ y op z With 1 operator (op ) and, at most, 3 names (x, y, z) Example: t ¬ 2 * y becomes z ¬ x - 2 * y z ¬ x - t Statements are executed sequentially. 8

Local Value Numbering Basic Algorithm For each expression e (assuming in the form result e <-- o 1 op o 2 ) � Get value numbers for operands from hash lookup table, represented by VN( o 1 ), VN( o 2 ). If no such key exists, insert these keys to the hash table with new value numbers; � Search with hash key < op ,VN( o 1 ),VN( o 2 ) >, If no such key exists, 1. insert the key to the hash table with a new value number; 2. use the same value number for result e and insert them to hash table. else, get the value of this key, use the same value number for result e, and insert them to hash table. 9

Local Value Numbering to Find Redundancy An example Original Code With VNs Hash Table for VN a 3 ¬ x 1 + y 2 a ¬ x + y {<x,1>, <y,2>, <<+,1,2>,3>, <a,3>} z 2 ¬ y 2 {..., <z,2>} z ¬ y d 4 ¬ 17 {..., <z,2>, <17,4>, <d,4>} d ¬ 17 c 3 ¬ x 1 + z 2 c ¬ x + z {..., <z,2>, <17,4>, <d,4>, <c,3>} Compare Value Num with Version Num. (The way they get increase; why the latter is insufficient for the example. 10

Rewritten for Redundancy Elimination An example With VNs Hash Table for VN Original Code a 3 ¬ x 1 + y 2 {<x,1>, <y,2>, <<+,1,2>,3>, <a,3>} a ¬ x + y z 2 ¬ y 2 {..., <z,2>} z ¬ y d 4 ¬ 17 {..., <z,2>, <17,4>, <d,4>} d ¬ 17 c 3 ¬ x 1 + z 2 {..., <z,2>, <17,4>, <d,4>, <c,3>} c ¬ x + z Hash Table for Rewritten Rewritten {<1,x>, <2,y>, <3,a>} a ¬ x + y {<1,x>, <2,y>, <3,a>} z ¬ y {<1,x>, <2,y>, <3,a>, <4,17>} d ¬ 17 {<1,x>, <2,y>, <3,a>, <4,17>} c ¬ a

Resolving Overwritten Issue An example Original Code Rewritten With VNs a ¬ x + y a 3 ¬ x 1 + y 2 a ¬ x + y z ¬ y z ¬ y z 2 ¬ y 2 * b ¬ x + y * b 3 ¬ x 1 + y 2 * b ¬ a a ¬ 17 a 4 ¬ 17 a ¬ 17 * c ¬ x + y * c 3 ¬ x 1 + y 2 * c ¬ a Hash Table for Rewritten Two redundancies marked by * {<1,x>, <2,y>, <3,a>} {<1,x>, <2,y>, <3,a>} {<1,x>, <2,y>, <3,a>, <4,17>} Optional Solutions: {<1,x>, <2,y>, <3,a>, <4,17>} • Use c 3 ¬ b 3 • Save a 3 in t 3 • Rename around it (best) 12

Renaming: Renaming in Value Numbering • Give each value a unique name Example ( continued ) Original Code With VNs Rewritten a 03 ¬ x 01 + y 02 a 0 ¬ x 0 + y 0 a 0 ¬ x 0 + y 0 z 0 ¬ y 0 z 20 ¬ y 02 z 0 ¬ y 0 * b 03 ¬ x 01 + y 02 * b 0 ¬ x 0 + y 0 * b 0 ¬ a 0 a 14 ¬ 17 a 1 ¬ 17 a 1 ¬ 17 * c 03 ¬ x 01 + y 02 * c 0 ¬ x 0 + y 0 * c 0 ¬ a 0 Result: Hash Table for Rewritten • a 0 is available {<1,x 0 >, <2,y 0 >, <3,a 0 >} • Rewriting just {<1,x 0 >, <2,y 0 >, <3,a 0 >} works {<1,x 0 >, <2,y 0 >, <3,a 0 >, <4,17>} {<1,x 0 >, <2,y 0 >, <3,a 0 >, <4,17>} 13

Reordering based on associativity � The order in which expressions are written matters � Example: either 2 x y or y x z will be missed in left- or right-associative treatment. m <-- 2 x y x z n <-- 3 x y x z o <-- 2 x y - z � Example: 3 x a x 5 14

A Hard Problem: Pointer Assignments � *p = 17 could force every variable in a program to increase their version number. � A major motivation for pointer analysis Original Code a ¬ x + y b ¬ w + v *p ¬ 17 c ¬ x + y d ¬ w + v 15

So far… � Removing redundant expressions � DAG: version tracking � Linear representation: value numbering 16

Local Value Numbering <-> Linear IR Local Value Numbering A m ¬ a + b • 1 block at a time n ¬ a + b • Strong local results • No cross-block effects B C p ¬ c + d q ¬ a + b r ¬ c + d r ¬ c + d D E e ¬ b + 18 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 G y ¬ a + b Missed opportunities z ¬ c + d (need stronger methods) * 17

Basic blocks � A basic block is a maximal-length segment of straight-line, unpredicated code. In another word, it has one entry point (i.e., no code within it is the destination of a jump instruction), one exit point and no jump instructions contained within it. � Example m = 2; c = m + n; L2: if(c>0) goto L1; d = 4; goto L2; c = 5; L1: 18

CFG Control-flow graph (CFG) A m ¬ a + b n ¬ a + b • Nodes for basic blocks • Edges for branches B C p ¬ c + d q ¬ a + b • Basis for many program r ¬ c + d r ¬ c + d analysis & transformation D E e ¬ b + 18 e ¬ a + 17 s ¬ a + b t ¬ c + d u ¬ e + f u ¬ e + f F v ¬ a + b w ¬ c + d This CFG, G = (N,E) x ¬ e + f • N = {A,B,C,D,E,F,G} G • E = {(A,B),(A,C),(B,G),(C,D), y ¬ a + b z ¬ c + d (C,E),(D,F),(E,F),(F,E)} • |N| = 7, |E| = 8 19

Extended basic block (EBB) A m ¬ a + b n ¬ a + b B C p ¬ c + d q ¬ a + b r ¬ c + d r ¬ c + d D E e ¬ b + 18 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 G y ¬ a + b z ¬ c + d � An EBB is a set of blocks B1, B2, ..., Bn, where Bi, 2 <= i <= n has a unique predecessor, which is in the EBB. � May have multiple exits � A tree structure