// 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 ( "fmt" "io" "math/big" "os" ) // domNode represents a node in the dominator tree. // // TODO(adonovan): export this, when ready. type domNode struct { Block *BasicBlock // the basic block; n.Block.dom == n Idom *domNode // immediate dominator (parent in dominator tree) Children []*domNode // nodes dominated by this one Level int // level number of node within tree; zero for root Pre, Post int // pre- and post-order numbering within dominator tree // Working state for Lengauer-Tarjan algorithm // (during which Pre is repurposed for CFG DFS preorder number). // TODO(adonovan): opt: measure allocating these as temps. semi *domNode // semidominator parent *domNode // parent in DFS traversal of CFG ancestor *domNode // ancestor with least sdom } // ltDfs implements the depth-first search part of the LT algorithm. func ltDfs(v *domNode, i int, preorder []*domNode) int { preorder[i] = v v.Pre = i // For now: DFS preorder of spanning tree of CFG i++ v.semi = v v.ancestor = nil for _, succ := range v.Block.Succs { if w := succ.dom; w.semi == nil { w.parent = v i = ltDfs(w, i, preorder) } } return i } // ltEval implements the EVAL part of the LT algorithm. func ltEval(v *domNode) *domNode { // TODO(adonovan): opt: do path compression per simple LT. u := v for ; v.ancestor != nil; v = v.ancestor { if v.semi.Pre < u.semi.Pre { u = v } } return u } // ltLink implements the LINK part of the LT algorithm. func ltLink(v, w *domNode) { w.ancestor = 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 // reodering is due to Georgiadis. // Initialize domNode nodes. for _, b := range f.Blocks { dom := b.dom if dom == nil { dom = &domNode{Block: b} b.dom = dom } else { dom.Block = b // reuse } } // Step 1. Number vertices by depth-first preorder. n := len(f.Blocks) preorder := make([]*domNode, n) root := f.Blocks[0].dom prenum := ltDfs(root, 0, preorder) var recover *domNode if f.Recover != nil { recover = f.Recover.dom ltDfs(recover, prenum, preorder) } buckets := make([]*domNode, n) copy(buckets, preorder) // In reverse preorder... for i := 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.Pre] { u := ltEval(v) if u.semi.Pre < i { v.Idom = u } else { v.Idom = w } } // Step 2. Compute the semidominators of all nodes. w.semi = w.parent for _, pred := range w.Block.Preds { v := pred.dom u := ltEval(v) if u.semi.Pre < w.semi.Pre { w.semi = u.semi } } ltLink(w.parent, w) if w.parent == w.semi { w.Idom = w.parent } else { buckets[i] = buckets[w.semi.Pre] buckets[w.semi.Pre] = w } } // The final 'Step 3' is now outside the loop. for v := buckets[0]; v != root; v = buckets[v.Pre] { v.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.Idom = nil } else { if w.Idom != w.semi { w.Idom = w.Idom.Idom } // Calculate Children relation as inverse of Idom. w.Idom.Children = append(w.Idom.Children, w) } // Clear working state. w.semi = nil w.parent = nil w.ancestor = nil } numberDomTree(root, 0, 0, 0) // 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. Also, it sets the level // numbers (zero for the root) used for frontier computation. // func numberDomTree(v *domNode, pre, post, level int) (int, int) { v.Level = level level++ v.Pre = pre pre++ for _, child := range v.Children { pre, post = numberDomTree(child, pre, post, level) } v.Post = post post++ return pre, post } // dominates returns true if b dominates c. // Requires that dominance information is up-to-date. // func dominates(b, c *BasicBlock) bool { return b.dom.Pre <= c.dom.Pre && c.dom.Post <= b.dom.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 { // The 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 := dominates(b, 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 } } } if !ok { panic("sanityCheckDomTree failed for " + f.String()) } } // Printing functions ---------------------------------------- // printDomTree prints the dominator tree as text, using indentation. func printDomTreeText(w io.Writer, v *domNode, indent int) { fmt.Fprintf(w, "%*s%s\n", 4*indent, "", v.Block) for _, child := range v.Children { printDomTreeText(w, child, indent+1) } } // printDomTreeDot prints the dominator tree of f in AT&T GraphViz // (.dot) format. func printDomTreeDot(w io.Writer, f *Function) { fmt.Fprintln(w, "//", f) fmt.Fprintln(w, "digraph domtree {") for i, b := range f.Blocks { v := b.dom fmt.Fprintf(w, "\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(w, "\tn%d -> n%d [style=\"solid\",weight=100];\n", v.Idom.Pre, v.Pre) } // CFG edges. for _, pred := range b.Preds { fmt.Fprintf(w, "\tn%d -> n%d [style=\"dotted\",weight=0];\n", pred.dom.Pre, v.Pre) } } fmt.Fprintln(w, "}") }