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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 ssa
// 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.
// http://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,
// http://jgaa.info/accepted/2006/GeorgiadisTarjanWerneck2006.10.1.pdf
// to avoid the need for buckets of size > 1.
import (
"bytes"
"fmt"
"math/big"
"os"
"sort"
)
// Idom returns the block that immediately dominates b:
// its parent in the dominator tree, if any.
// Neither the entry node (b.Index==0) nor recover node
// (b==b.Parent().Recover()) 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, 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
}
// ltState holds the working state for Lengauer-Tarjan algorithm
// (during which domInfo.pre is repurposed for CFG DFS preorder number).
type ltState struct {
// Each slice is indexed by b.Index.
sdom []*BasicBlock // b's semidominator
parent []*BasicBlock // b's parent in DFS traversal of CFG
ancestor []*BasicBlock // b's ancestor with least sdom
}
// dfs implements the depth-first search part of the LT algorithm.
func (lt *ltState) dfs(v *BasicBlock, i int32, preorder []*BasicBlock) int32 {
preorder[i] = v
v.dom.pre = i // For now: DFS preorder of spanning tree of CFG
i++
lt.sdom[v.Index] = v
lt.link(nil, v)
for _, w := range v.Succs {
if lt.sdom[w.Index] == nil {
lt.parent[w.Index] = v
i = lt.dfs(w, i, preorder)
}
}
return i
}
// eval implements the EVAL part of the LT algorithm.
func (lt *ltState) eval(v *BasicBlock) *BasicBlock {
// TODO(adonovan): opt: do path compression per simple LT.
u := v
for ; lt.ancestor[v.Index] != nil; v = lt.ancestor[v.Index] {
if lt.sdom[v.Index].dom.pre < lt.sdom[u.Index].dom.pre {
u = v
}
}
return u
}
// link implements the LINK part of the LT algorithm.
func (lt *ltState) link(v, w *BasicBlock) {
lt.ancestor[w.Index] = v
}
// buildDomTree computes the dominator tree of f using the LT algorithm.
// Precondition: all blocks are reachable (e.g. optimizeBlocks has been run).
//
func buildDomTree(f *Function) {
// The step numbers refer to the original LT paper; the
// reordering is due to Georgiadis.
// Clear any previous domInfo.
for _, b := range f.Blocks {
b.dom = domInfo{}
}
n := len(f.Blocks)
// Allocate space for 5 contiguous [n]*BasicBlock arrays:
// sdom, parent, ancestor, preorder, buckets.
space := make([]*BasicBlock, 5*n, 5*n)
lt := ltState{
sdom: space[0:n],
parent: space[n : 2*n],
ancestor: space[2*n : 3*n],
}
// Step 1. Number vertices by depth-first preorder.
preorder := space[3*n : 4*n]
root := f.Blocks[0]
prenum := lt.dfs(root, 0, preorder)
recover := f.Recover
if recover != nil {
lt.dfs(recover, prenum, preorder)
}
buckets := space[4*n : 5*n]
copy(buckets, preorder)
// In reverse preorder...
for i := int32(n) - 1; i > 0; i-- {
w := preorder[i]
// Step 3. Implicitly define the immediate dominator of each node.
for v := buckets[i]; v != w; v = buckets[v.dom.pre] {
u := lt.eval(v)
if lt.sdom[u.Index].dom.pre < i {
v.dom.idom = u
} else {
v.dom.idom = w
}
}
// Step 2. Compute the semidominators of all nodes.
lt.sdom[w.Index] = lt.parent[w.Index]
for _, v := range w.Preds {
u := lt.eval(v)
if lt.sdom[u.Index].dom.pre < lt.sdom[w.Index].dom.pre {
lt.sdom[w.Index] = lt.sdom[u.Index]
}
}
lt.link(lt.parent[w.Index], w)
if lt.parent[w.Index] == lt.sdom[w.Index] {
w.dom.idom = lt.parent[w.Index]
} else {
buckets[i] = buckets[lt.sdom[w.Index].dom.pre]
buckets[lt.sdom[w.Index].dom.pre] = w
}
}
// The final 'Step 3' is now outside the loop.
for v := buckets[0]; v != root; v = buckets[v.dom.pre] {
v.dom.idom = root
}
// Step 4. Explicitly define the immediate dominator of each
// node, in preorder.
for _, w := range preorder[1:] {
if w == root || w == recover {
w.dom.idom = nil
} else {
if w.dom.idom != lt.sdom[w.Index] {
w.dom.idom = w.dom.idom.dom.idom
}
// Calculate Children relation as inverse of Idom.
w.dom.idom.dom.children = append(w.dom.idom.dom.children, w)
}
}
pre, post := numberDomTree(root, 0, 0)
if recover != nil {
numberDomTree(recover, pre, post)
}
// printDomTreeDot(os.Stderr, f) // debugging
// printDomTreeText(os.Stderr, root, 0) // debugging
if f.Prog.mode&SanityCheckFunctions != 0 {
sanityCheckDomTree(f)
}
}
// 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
}
// 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, b := range f.Blocks {
if i == 0 || b == f.Recover {
// 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 || b == f.Recover {
continue
}
// Compute intersection across predecessors.
var x big.Int
x.Set(&all)
for _, pred := range b.Preds {
x.And(&x, &D[pred.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).
// The Recover block (if any) must be treated specially so we skip it.
ok := true
for i := 0; i < n; i++ {
for j := 0; j < n; j++ {
b, c := f.Blocks[i], f.Blocks[j]
if c == f.Recover {
continue
}
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.
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.
func printDomTreeDot(buf *bytes.Buffer, 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)
}
}
fmt.Fprintln(buf, "}")
}