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// Copyright 2013 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package ir
// This file defines algorithms related to dominance.
// Dominator tree construction ----------------------------------------
//
// We use the algorithm described in Lengauer & Tarjan. 1979. A fast
// algorithm for finding dominators in a flowgraph.
// https://doi.acm.org/10.1145/357062.357071
//
// We also apply the optimizations to SLT described in Georgiadis et
// al, Finding Dominators in Practice, JGAA 2006,
// https://jgaa.info/accepted/2006/GeorgiadisTarjanWerneck2006.10.1.pdf
// to avoid the need for buckets of size > 1.
import (
"bytes"
"fmt"
"io"
"math/big"
"os"
"sort"
)
// Idom returns the block that immediately dominates b:
// its parent in the dominator tree, if any.
// The entry node (b.Index==0) does not have a parent.
func (b *BasicBlock) Idom() *BasicBlock { return b.dom.idom }
// Dominees returns the list of blocks that b immediately dominates:
// its children in the dominator tree.
func (b *BasicBlock) Dominees() []*BasicBlock { return b.dom.children }
// Dominates reports whether b dominates c.
func (b *BasicBlock) Dominates(c *BasicBlock) bool {
return b.dom.pre <= c.dom.pre && c.dom.post <= b.dom.post
}
type byDomPreorder []*BasicBlock
func (a byDomPreorder) Len() int { return len(a) }
func (a byDomPreorder) Swap(i, j int) { a[i], a[j] = a[j], a[i] }
func (a byDomPreorder) Less(i, j int) bool { return a[i].dom.pre < a[j].dom.pre }
// DomPreorder returns a new slice containing the blocks of f in
// dominator tree preorder.
func (f *Function) DomPreorder() []*BasicBlock {
n := len(f.Blocks)
order := make(byDomPreorder, n)
copy(order, f.Blocks)
sort.Sort(order)
return order
}
// domInfo contains a BasicBlock's dominance information.
type domInfo struct {
idom *BasicBlock // immediate dominator (parent in domtree)
children []*BasicBlock // nodes immediately dominated by this one
pre, post int32 // pre- and post-order numbering within domtree
}
// buildDomTree computes the dominator tree of f using the LT algorithm.
// Precondition: all blocks are reachable (e.g. optimizeBlocks has been run).
func buildDomTree(fn *Function) {
// The step numbers refer to the original LT paper; the
// reordering is due to Georgiadis.
// Clear any previous domInfo.
for _, b := range fn.Blocks {
b.dom = domInfo{}
}
idoms := make([]*BasicBlock, len(fn.Blocks))
order := make([]*BasicBlock, 0, len(fn.Blocks))
seen := fn.blockset(0)
var dfs func(b *BasicBlock)
dfs = func(b *BasicBlock) {
if !seen.Add(b) {
return
}
for _, succ := range b.Succs {
dfs(succ)
}
if fn.fakeExits.Has(b) {
dfs(fn.Exit)
}
order = append(order, b)
b.post = len(order) - 1
}
dfs(fn.Blocks[0])
for i := 0; i < len(order)/2; i++ {
o := len(order) - i - 1
order[i], order[o] = order[o], order[i]
}
idoms[fn.Blocks[0].Index] = fn.Blocks[0]
changed := true
for changed {
changed = false
// iterate over all nodes in reverse postorder, except for the
// entry node
for _, b := range order[1:] {
var newIdom *BasicBlock
do := func(p *BasicBlock) {
if idoms[p.Index] == nil {
return
}
if newIdom == nil {
newIdom = p
} else {
finger1 := p
finger2 := newIdom
for finger1 != finger2 {
for finger1.post < finger2.post {
finger1 = idoms[finger1.Index]
}
for finger2.post < finger1.post {
finger2 = idoms[finger2.Index]
}
}
newIdom = finger1
}
}
for _, p := range b.Preds {
do(p)
}
if b == fn.Exit {
for _, p := range fn.Blocks {
if fn.fakeExits.Has(p) {
do(p)
}
}
}
if idoms[b.Index] != newIdom {
idoms[b.Index] = newIdom
changed = true
}
}
}
for i, b := range idoms {
fn.Blocks[i].dom.idom = b
if b == nil {
// malformed CFG
continue
}
if i == b.Index {
continue
}
b.dom.children = append(b.dom.children, fn.Blocks[i])
}
numberDomTree(fn.Blocks[0], 0, 0)
// printDomTreeDot(os.Stderr, fn) // debugging
// printDomTreeText(os.Stderr, root, 0) // debugging
if fn.Prog.mode&SanityCheckFunctions != 0 {
sanityCheckDomTree(fn)
}
}
// buildPostDomTree is like buildDomTree, but builds the post-dominator tree instead.
func buildPostDomTree(fn *Function) {
// The step numbers refer to the original LT paper; the
// reordering is due to Georgiadis.
// Clear any previous domInfo.
for _, b := range fn.Blocks {
b.pdom = domInfo{}
}
idoms := make([]*BasicBlock, len(fn.Blocks))
order := make([]*BasicBlock, 0, len(fn.Blocks))
seen := fn.blockset(0)
var dfs func(b *BasicBlock)
dfs = func(b *BasicBlock) {
if !seen.Add(b) {
return
}
for _, pred := range b.Preds {
dfs(pred)
}
if b == fn.Exit {
for _, p := range fn.Blocks {
if fn.fakeExits.Has(p) {
dfs(p)
}
}
}
order = append(order, b)
b.post = len(order) - 1
}
dfs(fn.Exit)
for i := 0; i < len(order)/2; i++ {
o := len(order) - i - 1
order[i], order[o] = order[o], order[i]
}
idoms[fn.Exit.Index] = fn.Exit
changed := true
for changed {
changed = false
// iterate over all nodes in reverse postorder, except for the
// exit node
for _, b := range order[1:] {
var newIdom *BasicBlock
do := func(p *BasicBlock) {
if idoms[p.Index] == nil {
return
}
if newIdom == nil {
newIdom = p
} else {
finger1 := p
finger2 := newIdom
for finger1 != finger2 {
for finger1.post < finger2.post {
finger1 = idoms[finger1.Index]
}
for finger2.post < finger1.post {
finger2 = idoms[finger2.Index]
}
}
newIdom = finger1
}
}
for _, p := range b.Succs {
do(p)
}
if fn.fakeExits.Has(b) {
do(fn.Exit)
}
if idoms[b.Index] != newIdom {
idoms[b.Index] = newIdom
changed = true
}
}
}
for i, b := range idoms {
fn.Blocks[i].pdom.idom = b
if b == nil {
// malformed CFG
continue
}
if i == b.Index {
continue
}
b.pdom.children = append(b.pdom.children, fn.Blocks[i])
}
numberPostDomTree(fn.Exit, 0, 0)
// printPostDomTreeDot(os.Stderr, fn) // debugging
// printPostDomTreeText(os.Stderr, fn.Exit, 0) // debugging
if fn.Prog.mode&SanityCheckFunctions != 0 { // XXX
sanityCheckDomTree(fn) // XXX
}
}
// numberDomTree sets the pre- and post-order numbers of a depth-first
// traversal of the dominator tree rooted at v. These are used to
// answer dominance queries in constant time.
func numberDomTree(v *BasicBlock, pre, post int32) (int32, int32) {
v.dom.pre = pre
pre++
for _, child := range v.dom.children {
pre, post = numberDomTree(child, pre, post)
}
v.dom.post = post
post++
return pre, post
}
// numberPostDomTree sets the pre- and post-order numbers of a depth-first
// traversal of the post-dominator tree rooted at v. These are used to
// answer post-dominance queries in constant time.
func numberPostDomTree(v *BasicBlock, pre, post int32) (int32, int32) {
8000
v.pdom.pre = pre
pre++
for _, child := range v.pdom.children {
pre, post = numberPostDomTree(child, pre, post)
}
v.pdom.post = post
post++
return pre, post
}
// Testing utilities ----------------------------------------
// sanityCheckDomTree checks the correctness of the dominator tree
// computed by the LT algorithm by comparing against the dominance
// relation computed by a naive Kildall-style forward dataflow
// analysis (Algorithm 10.16 from the "Dragon" book).
func sanityCheckDomTree(f *Function) {
n := len(f.Blocks)
// D[i] is the set of blocks that dominate f.Blocks[i],
// represented as a bit-set of block indices.
D := make([]big.Int, n)
one := big.NewInt(1)
// all is the set of all blocks; constant.
var all big.Int
all.Set(one).Lsh(&all, uint(n)).Sub(&all, one)
// Initialization.
for i := range f.Blocks {
if i == 0 {
// A root is dominated only by itself.
D[i].SetBit(&D[0], 0, 1)
} else {
// All other blocks are (initially) dominated
// by every block.
D[i].Set(&all)
}
}
// Iteration until fixed point.
for changed := true; changed; {
changed = false
for i, b := range f.Blocks {
if i == 0 {
continue
}
// Compute intersection across predecessors.
var x big.Int
x.Set(&all)
for _, pred := range b.Preds {
x.And(&x, &D[pred.Index])
}
if b == f.Exit {
for _, p := range f.Blocks {
if f.fakeExits.Has(p) {
x.And(&x, &D[p.Index])
}
}
}
x.SetBit(&x, i, 1) // a block always dominates itself.
if D[i].Cmp(&x) != 0 {
D[i].Set(&x)
changed = true
}
}
}
// Check the entire relation. O(n^2).
ok := true
for i := range n {
for j := range n {
b, c := f.Blocks[i], f.Blocks[j]
actual := b.Dominates(c)
expected := D[j].Bit(i) == 1
if actual != expected {
fmt.Fprintf(os.Stderr, "dominates(%s, %s)==%t, want %t\n", b, c, actual, expected)
ok = false
}
}
}
preorder := f.DomPreorder()
for _, b := range f.Blocks {
if got := preorder[b.dom.pre]; got != b {
fmt.Fprintf(os.Stderr, "preorder[%d]==%s, want %s\n", b.dom.pre, got, b)
ok = false
}
}
if !ok {
panic("sanityCheckDomTree failed for " + f.String())
}
}
// Printing functions ----------------------------------------
// printDomTree prints the dominator tree as text, using indentation.
//
//lint:ignore U1000 used during debugging
func printDomTreeText(buf *bytes.Buffer, v *BasicBlock, indent int) {
fmt.Fprintf(buf, "%*s%s\n", 4*indent, "", v)
for _, child := range v.dom.children {
printDomTreeText(buf, child, indent+1)
}
}
// printDomTreeDot prints the dominator tree of f in AT&T GraphViz
// (.dot) format.
//
//lint:ignore U1000 used during debugging
func printDomTreeDot(buf io.Writer, f *Function) {
fmt.Fprintln(buf, "//", f)
fmt.Fprintln(buf, "digraph domtree {")
for i, b := range f.Blocks {
v := b.dom
fmt.Fprintf(buf, "\tn%d [label=\"%s (%d, %d)\",shape=\"rectangle\"];\n", v.pre, b, v.pre, v.post)
// TODO(adonovan): improve appearance of edges
// belonging to both dominator tree and CFG.
// Dominator tree edge.
if i != 0 {
fmt.Fprintf(buf, "\tn%d -> n%d [style=\"solid\",weight=100];\n", v.idom.dom.pre, v.pre)
}
// CFG edges.
for _, pred := range b.Preds {
fmt.Fprintf(buf, "\tn%d -> n%d [style=\"dotted\",weight=0];\n", pred.dom.pre, v.pre)
}
if f.fakeExits.Has(b) {
fmt.Fprintf(buf, "\tn%d -> n%d [style=\"dotted\",weight=0,color=red];\n", b.dom.pre, f.Exit.dom.pre)
}
}
fmt.Fprintln(buf, "}")
}
// printDomTree prints the dominator tree as text, using indentation.
//
//lint:ignore U1000 used during debugging
func printPostDomTreeText(buf io.Writer, v *BasicBlock, indent int) {
fmt.Fprintf(buf, "%*s%s\n", 4*indent, "", v)
for _, child := range v.pdom.children {
printPostDomTreeText(buf, child, indent+1)
}
}
// printDomTreeDot prints the dominator tree of f in AT&T GraphViz
// (.dot) format.
//
//lint:ignore U1000 used during debugging
func printPostDomTreeDot(buf io.Writer, f *Function) {
fmt.Fprintln(buf, "//", f)
fmt.Fprintln(buf, "digraph pdomtree {")
for _, b := range f.Blocks {
v := b.pdom
fmt.Fprintf(buf, "\tn%d [label=\"%s (%d, %d)\",shape=\"rectangle\"];\n", v.pre, b, v.pre, v.post)
// TODO(adonovan): improve appearance of edges
// belonging to both dominator tree and CFG.
// Dominator tree edge.
if b != f.Exit {
fmt.Fprintf(buf, "\tn%d -> n%d [style=\"solid\",weight=100];\n", v.idom.pdom.pre, v.pre)
}
// CFG edges.
for _, pred := range b.Preds {
fmt.Fprintf(buf, "\tn%d -> n%d [style=\"dotted\",weight=0];\n", pred.pdom.pre, v.pre)
}
if f.fakeExits.Has(b) {
fmt.Fprintf(buf, "\tn%d -> n%d [style=\"dotted\",weight=0,color=red];\n", b.dom.pre, f.Exit.dom.pre)
}
}
fmt.Fprintln(buf, "}")
}