344 lines
9.4 KiB
Go
344 lines
9.4 KiB
Go
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// Copyright 2013 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package ssa
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// This file defines algorithms related to dominance.
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// Dominator tree construction ----------------------------------------
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//
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// We use the algorithm described in Lengauer & Tarjan. 1979. A fast
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// algorithm for finding dominators in a flowgraph.
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// http://doi.acm.org/10.1145/357062.357071
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//
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// We also apply the optimizations to SLT described in Georgiadis et
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// al, Finding Dominators in Practice, JGAA 2006,
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// http://jgaa.info/accepted/2006/GeorgiadisTarjanWerneck2006.10.1.pdf
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// to avoid the need for buckets of size > 1.
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import (
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"bytes"
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"fmt"
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"math/big"
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"os"
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"sort"
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)
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// Idom returns the block that immediately dominates b:
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// its parent in the dominator tree, if any.
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// Neither the entry node (b.Index==0) nor recover node
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// (b==b.Parent().Recover()) have a parent.
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//
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func (b *BasicBlock) Idom() *BasicBlock { return b.dom.idom }
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// Dominees returns the list of blocks that b immediately dominates:
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// its children in the dominator tree.
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//
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func (b *BasicBlock) Dominees() []*BasicBlock { return b.dom.children }
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// Dominates reports whether b dominates c.
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func (b *BasicBlock) Dominates(c *BasicBlock) bool {
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return b.dom.pre <= c.dom.pre && c.dom.post <= b.dom.post
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}
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type byDomPreorder []*BasicBlock
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func (a byDomPreorder) Len() int { return len(a) }
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func (a byDomPreorder) Swap(i, j int) { a[i], a[j] = a[j], a[i] }
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func (a byDomPreorder) Less(i, j int) bool { return a[i].dom.pre < a[j].dom.pre }
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// DomPreorder returns a new slice containing the blocks of f in
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// dominator tree preorder.
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//
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func (f *Function) DomPreorder() []*BasicBlock {
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n := len(f.Blocks)
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order := make(byDomPreorder, n)
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copy(order, f.Blocks)
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sort.Sort(order)
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return order
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}
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// domInfo contains a BasicBlock's dominance information.
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type domInfo struct {
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idom *BasicBlock // immediate dominator (parent in domtree)
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children []*BasicBlock // nodes immediately dominated by this one
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pre, post int32 // pre- and post-order numbering within domtree
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}
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// ltState holds the working state for Lengauer-Tarjan algorithm
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// (during which domInfo.pre is repurposed for CFG DFS preorder number).
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type ltState struct {
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// Each slice is indexed by b.Index.
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sdom []*BasicBlock // b's semidominator
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parent []*BasicBlock // b's parent in DFS traversal of CFG
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ancestor []*BasicBlock // b's ancestor with least sdom
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}
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// dfs implements the depth-first search part of the LT algorithm.
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func (lt *ltState) dfs(v *BasicBlock, i int32, preorder []*BasicBlock) int32 {
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preorder[i] = v
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v.dom.pre = i // For now: DFS preorder of spanning tree of CFG
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i++
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lt.sdom[v.Index] = v
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lt.link(nil, v)
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for _, w := range v.Succs {
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if lt.sdom[w.Index] == nil {
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lt.parent[w.Index] = v
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i = lt.dfs(w, i, preorder)
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}
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}
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return i
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}
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// eval implements the EVAL part of the LT algorithm.
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func (lt *ltState) eval(v *BasicBlock) *BasicBlock {
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// TODO(adonovan): opt: do path compression per simple LT.
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u := v
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for ; lt.ancestor[v.Index] != nil; v = lt.ancestor[v.Index] {
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if lt.sdom[v.Index].dom.pre < lt.sdom[u.Index].dom.pre {
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u = v
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}
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}
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return u
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}
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// link implements the LINK part of the LT algorithm.
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func (lt *ltState) link(v, w *BasicBlock) {
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lt.ancestor[w.Index] = v
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}
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// buildDomTree computes the dominator tree of f using the LT algorithm.
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// Precondition: all blocks are reachable (e.g. optimizeBlocks has been run).
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//
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func buildDomTree(f *Function) {
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// The step numbers refer to the original LT paper; the
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// reordering is due to Georgiadis.
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// Clear any previous domInfo.
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for _, b := range f.Blocks {
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b.dom = domInfo{}
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}
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n := len(f.Blocks)
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// Allocate space for 5 contiguous [n]*BasicBlock arrays:
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// sdom, parent, ancestor, preorder, buckets.
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space := make([]*BasicBlock, 5*n)
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lt := ltState{
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sdom: space[0:n],
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parent: space[n : 2*n],
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ancestor: space[2*n : 3*n],
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}
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// Step 1. Number vertices by depth-first preorder.
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preorder := space[3*n : 4*n]
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root := f.Blocks[0]
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prenum := lt.dfs(root, 0, preorder)
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recover := f.Recover
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if recover != nil {
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lt.dfs(recover, prenum, preorder)
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}
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buckets := space[4*n : 5*n]
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copy(buckets, preorder)
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// In reverse preorder...
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for i := int32(n) - 1; i > 0; i-- {
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w := preorder[i]
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// Step 3. Implicitly define the immediate dominator of each node.
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for v := buckets[i]; v != w; v = buckets[v.dom.pre] {
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u := lt.eval(v)
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if lt.sdom[u.Index].dom.pre < i {
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v.dom.idom = u
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} else {
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v.dom.idom = w
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}
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}
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// Step 2. Compute the semidominators of all nodes.
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lt.sdom[w.Index] = lt.parent[w.Index]
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for _, v := range w.Preds {
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u := lt.eval(v)
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if lt.sdom[u.Index].dom.pre < lt.sdom[w.Index].dom.pre {
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lt.sdom[w.Index] = lt.sdom[u.Index]
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}
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}
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lt.link(lt.parent[w.Index], w)
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if lt.parent[w.Index] == lt.sdom[w.Index] {
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w.dom.idom = lt.parent[w.Index]
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} else {
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buckets[i] = buckets[lt.sdom[w.Index].dom.pre]
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buckets[lt.sdom[w.Index].dom.pre] = w
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}
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}
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// The final 'Step 3' is now outside the loop.
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for v := buckets[0]; v != root; v = buckets[v.dom.pre] {
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v.dom.idom = root
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}
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// Step 4. Explicitly define the immediate dominator of each
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// node, in preorder.
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for _, w := range preorder[1:] {
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if w == root || w == recover {
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w.dom.idom = nil
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} else {
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if w.dom.idom != lt.sdom[w.Index] {
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w.dom.idom = w.dom.idom.dom.idom
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}
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// Calculate Children relation as inverse of Idom.
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w.dom.idom.dom.children = append(w.dom.idom.dom.children, w)
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}
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}
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pre, post := numberDomTree(root, 0, 0)
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if recover != nil {
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numberDomTree(recover, pre, post)
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}
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// printDomTreeDot(os.Stderr, f) // debugging
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// printDomTreeText(os.Stderr, root, 0) // debugging
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if f.Prog.mode&SanityCheckFunctions != 0 {
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sanityCheckDomTree(f)
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}
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}
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// numberDomTree sets the pre- and post-order numbers of a depth-first
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// traversal of the dominator tree rooted at v. These are used to
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// answer dominance queries in constant time.
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//
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func numberDomTree(v *BasicBlock, pre, post int32) (int32, int32) {
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v.dom.pre = pre
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pre++
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for _, child := range v.dom.children {
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pre, post = numberDomTree(child, pre, post)
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}
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v.dom.post = post
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post++
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return pre, post
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}
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// Testing utilities ----------------------------------------
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// sanityCheckDomTree checks the correctness of the dominator tree
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// computed by the LT algorithm by comparing against the dominance
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// relation computed by a naive Kildall-style forward dataflow
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// analysis (Algorithm 10.16 from the "Dragon" book).
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//
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func sanityCheckDomTree(f *Function) {
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n := len(f.Blocks)
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// D[i] is the set of blocks that dominate f.Blocks[i],
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// represented as a bit-set of block indices.
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D := make([]big.Int, n)
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one := big.NewInt(1)
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// all is the set of all blocks; constant.
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var all big.Int
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all.Set(one).Lsh(&all, uint(n)).Sub(&all, one)
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// Initialization.
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for i, b := range f.Blocks {
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if i == 0 || b == f.Recover {
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// A root is dominated only by itself.
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D[i].SetBit(&D[0], 0, 1)
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} else {
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// All other blocks are (initially) dominated
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// by every block.
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D[i].Set(&all)
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}
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}
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// Iteration until fixed point.
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for changed := true; changed; {
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changed = false
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for i, b := range f.Blocks {
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if i == 0 || b == f.Recover {
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continue
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}
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// Compute intersection across predecessors.
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var x big.Int
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x.Set(&all)
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for _, pred := range b.Preds {
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x.And(&x, &D[pred.Index])
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}
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x.SetBit(&x, i, 1) // a block always dominates itself.
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if D[i].Cmp(&x) != 0 {
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D[i].Set(&x)
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changed = true
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}
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}
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}
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// Check the entire relation. O(n^2).
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// The Recover block (if any) must be treated specially so we skip it.
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ok := true
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for i := 0; i < n; i++ {
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for j := 0; j < n; j++ {
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b, c := f.Blocks[i], f.Blocks[j]
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if c == f.Recover {
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continue
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}
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actual := b.Dominates(c)
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expected := D[j].Bit(i) == 1
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if actual != expected {
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fmt.Fprintf(os.Stderr, "dominates(%s, %s)==%t, want %t\n", b, c, actual, expected)
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ok = false
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}
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}
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}
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preorder := f.DomPreorder()
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for _, b := range f.Blocks {
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if got := preorder[b.dom.pre]; got != b {
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fmt.Fprintf(os.Stderr, "preorder[%d]==%s, want %s\n", b.dom.pre, got, b)
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ok = false
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}
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}
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if !ok {
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panic("sanityCheckDomTree failed for " + f.String())
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}
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}
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// Printing functions ----------------------------------------
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// printDomTree prints the dominator tree as text, using indentation.
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//lint:ignore U1000 used during debugging
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func printDomTreeText(buf *bytes.Buffer, v *BasicBlock, indent int) {
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fmt.Fprintf(buf, "%*s%s\n", 4*indent, "", v)
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for _, child := range v.dom.children {
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printDomTreeText(buf, child, indent+1)
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}
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}
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// printDomTreeDot prints the dominator tree of f in AT&T GraphViz
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// (.dot) format.
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//lint:ignore U1000 used during debugging
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func printDomTreeDot(buf *bytes.Buffer, f *Function) {
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fmt.Fprintln(buf, "//", f)
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fmt.Fprintln(buf, "digraph domtree {")
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for i, b := range f.Blocks {
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v := b.dom
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fmt.Fprintf(buf, "\tn%d [label=\"%s (%d, %d)\",shape=\"rectangle\"];\n", v.pre, b, v.pre, v.post)
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// TODO(adonovan): improve appearance of edges
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// belonging to both dominator tree and CFG.
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// Dominator tree edge.
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if i != 0 {
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fmt.Fprintf(buf, "\tn%d -> n%d [style=\"solid\",weight=100];\n", v.idom.dom.pre, v.pre)
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}
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// CFG edges.
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for _, pred := range b.Preds {
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fmt.Fprintf(buf, "\tn%d -> n%d [style=\"dotted\",weight=0];\n", pred.dom.pre, v.pre)
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}
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}
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fmt.Fprintln(buf, "}")
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}
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