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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) 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, "}") }