Optimizing Compilers CISC 673 Spring 2011 More Control Flow

UUNIVERSITYNIVERSITY OFOF D DELAWARE ELAWARE • • C COMPUTER & OMPUTER & IINFORMATION NFORMATION SSCIENCES CIENCES DDEPARTMENTEPARTMENT

Optimizing CompilersCISC 673

Spring 2011More Control Flow

John CavazosUniversity of Delaware

UUNIVERSITYNIVERSITY OFOF D DELAWARE ELAWARE • • C COMPUTER & OMPUTER & IINFORMATION NFORMATION SSCIENCES CIENCES DDEPARTMENTEPARTMENT 2

Overview

Introduced control-flow analysis Basic blocks Control-flow graphs

Discuss application of graph algorithms: loops Spanning trees, depth-first spanning trees Dominators Reducibility Dominator tree Strongly-connected components


Dominance

Node d dominates node i (“d dom i” )if every path from Entry to i includes d

Properties of Dominators Reflexive: a dom a Transitive: if a dom b and b dom c

then a dom c Antisymmetric: if a dom b and b

dom a then b=a


Immediate Dominance

a idom b iff a dom b there is no c such that a dom c, c dom b (c a, c b)

Idom’s: each node has unique idom relation forms a dominator tree


Natural loops

Single entry node (d) no jumps into middle of loop d dominates all nodes in loop

Requires back edge into loop header (n→

d) n is tail, d is head single entry point

Natural loop of back edge (n→ d) d + {all nodes that can reach n with

touching d}


Reducible Loops

Reducible: hierarchical, “well-structured” flowgraph reducible iff

all loops in it natural

A

B

C

D

E

F G

Entry

A

B

C

D

E

F G

Entry

reducible graph irreducible graph


Reducible Graph Test

Graph is reducible iff …all back edges are ones whose

head dominates its tail


Reducible Loops (more examples)

Is this a Natural Loop? Why or why not?

Back edge

Loop Header



Yes, Natural Loop with Multiple Branches

Back edge

Loop Header



Is this a Natural Loop? Why or why not?



Not a Natural Loop: 4→ 1 is back edge, but 1 (head) does not dominate 4 (tail)


Why is this not a Reducible Graph?


Nonreducible Graph

B (head) does not dominate C (tail)


Reducibility Example

Some languages only permit procedures with reducible flowgraphs (e.g., Java)

“GOTO Considered Harmful”: can introduce irreducibility FORTRAN C C++


Dominance Tree

A

B

C

D

E

F G

Entry

Immediate and other dominators:(excluding Entry) a idom b; a dom a, b, c,

d, e, f, g b idom c; b dom b, c,

d, e, f, g c idom d; c dom c, d, e,

f, g d idom e; d dom d, e, f,

g e idom f, e idom g; e

dom e, f, g

A

B

C

D

E

F G

Entry

control-flow graph dominator tree


Dominator Tree?


Dominator Tree


Reducible Graph?

Construct Spanning Tree to identify back edges. Now check natural back edges property, i.e., head must dominate tail


Natural Loops

Now we can find natural loops Given back edge m → n, natural

loop is n (loop header) and nodes that can reach m without passing through n


Find Natural Loops?


Natural Loops

Back Edge Natural Loop J → G {G,H,J} G → D {D,E,F,G,H,J} D → C {C,D,E,F,G,H,J} H → C {C,D,E,F,G,H,J} I → A {A,B,C,D,E,F,G,H,I,J}


Strongly-Connected Components

What about irreducible flowgraphs? Most general loop form = strongly-

connected component (SCC): subgraph S such that every node in S

reachable from every other node by path including only edges in S

Maximal SCC: S is maximal SCC if it is the largest SCC

that contains S.


SCC Example

Entry

B1

B2

B3

Strongly-connected components (SCC)(B1, B2), (B2, B3),

(B1, B2, B3)

Maximal SCC(B1, B2, B3)


Computing Maximal SCCs

Tarjan’s algorithm: Computes all maximal SCCs Linear-time (in number of nodes and

edges) CLR algorithm:

Also linear-time Simpler:

Two depth-first searches and one “transpose”:reverse all graph edge


Next Time

Dataflow analysis Read Marlowe and Ryder paper

Optimizing Compilers CISC 673 Spring 2011 More Control Flow

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