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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, "}")
+}