Control Flow II: Dominators and Loop Detection - Lecture 20, University of Michigan, Exams of Electrical and Electronics Engineering

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Control Flow II: Dominators, Loop Detection

EECS 483 – Lecture 20 University of Michigan Wednesday, November 15, 2006

• 1 -

Announcements and Reading

Simon’s office hours on Thurs (11/16) » Moved to 10am-12pm, 4817 CSE » Exam 1 problem 11 was incorrectly graded

y

Follow set for B should contain z

» See Simon to get your points back

Project 3/4 – postponed until Monday » Group formation for Project 3/

Today’s class material: » 10.4, end of 10.9 (dominator algorithm)

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Control Flow Analysis

™ Determining properties of the program branchstructure

» Static properties

Æ

Not executing the code

» Properties that exist regardless of the run-time branch

directions

» Use CFG » Optimize efficiency of control flow structure

™ Determine instruction execution properties

» Global optimization of computation operations » Discuss this later

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Dominator

™ Defn: Dominator

• Given a CFG(V, E, Entry, Exit), a node x dominates a node y, if every pathfrom the Entry block to y contains x ™ 3 properties of dominators

» Each BB dominates itself » If x dominates y, and y dominates z, then x dominates z » If x dominates z and y dominates z, then either x

dominates y or y dominates x

™ Intuition

» Given some BB, which blocks are guaranteed to have

executed prior to executing the BB

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Dominator Analysis

Compute dom(BBi) = set of BBsthat dominate BBi

Initialization

Dom(entry) = entry

Dom(everything else) = all nodes

Iterative computation

while change, do

y change = false y for each BB (except the entry BB) ‹ tmp(BB) = BB + {intersect of Domof all predecessor BB’s} ‹ if (tmp(BB) != dom(BB)) È dom(BB) = tmp(BB) È change = true

BB

BB

BB

BB

BB

BB

BB

Entry Exit

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Immediate Dominator

™ Defn: Immediate dominator (idom)– Each node n has a uniqueimmediate dominator mthat is the last dominatorof n on any path from theinitial node to n

» Closest node that

dominates

BB

BB

BB

BB

BB

BB

BB

Entry Exit

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Post Dominator

Reverse of dominator

Defn: Post Dominator

• Given a CFG(V, E, Entry, Exit), a node x post dominates anode y, if every path from y to the Exitcontains x

Intuition » Given some BB, which blocks are guaranteed to have executed after executing the BB

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Post Dominator Examples

BB

BB

BB

BB

BB

BB

BB

Entry Exit

BB2 BB

BB

BB

Entry BB1 BB6^ Exit

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Immediate Post Dominator

Defn: Immediate post dominator

(ipdom) –

Each node n has a uniqueimmediate postdominator m that is thefirst post dominator of non any path from n to theExit

Closest node that postdominates

First breadth-firstsuccessor that postdominates a node

BB

BB

BB

BB

BB

BB

BB

Entry Exit

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Class Problem

BB

BB2 BB

BB3 BB

Entry^ BB6^ BB7^ Exit

Calculate the PDOM set for each BB

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Natural Loops

Cycle suitable for optimization » Discuss opti later

2 properties: » Single entry point called the header

y

Header dominates

all blocks in the loop

» Must be one way to iterate the loop (ie at least 1 path back to the header from within theloop) called a backedge

Backedge detection » Edge, x Æ y where the target (y) dominates the source (x)

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Backedge Example

BB2 BB

BB

BB

Entry BB1 BB6^ Exit

dom(1) = E,1dom(2) = E,1,

dom(4) = E,1,2,3,

dom(3) = E,1,2,

dom(5) = E,1,2,3,

dom(6) = E,1,2,

BE = target dominatessourceE

Æ

1 : No

Æ

2 : No

Æ

3 : No

Æ

6 : No

Æ

4 : No

Æ

5 : No

Æ

3 : Yes

Æ

5 : No

Æ

3 : Yes

Æ

6 : No

Æ

2 : Yes

Æ

X : No

In this example, BE = edge from higher BB to lower BB, not always this easy!

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Loop Detection Example

BB2 BB

BB

BB

Entry BB1 BB6^ Exit

dom(1) = E,1 dom(2) = E,1,

dom(4) = E,1,2,3,

dom(3) = E,1,2,

dom(5) = E,1,2,3,

dom(6) = E,1,2,

Loop detection: 3 steps: •

Identify backedges

Compute LoopBB

Merge loops with the same header

Loop1: defined by 6

Æ

LoopBB = 2,3,4,5,

Loop2: defined by 4

Æ

LoopBB = 3,

Loop3: defined by 5

Æ

LoopBB = 3,4,

Merge loops 2,

LoopBB = 3,4,5 Backedges = 4

Æ

Æ

• 19 -

Class Problem

BB

BB

BB

BB

BB

BB

BB

Entry Exit

Find the loops What are the header(s)? What are the backedge(s)?