--- /dev/null
+// 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)
+ 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)
+ 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, "}")
+}
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