1 // Copyright 2013 The Go Authors. All rights reserved.
2 // Use of this source code is governed by a BSD-style
3 // license that can be found in the LICENSE file.
4 5 package ssa
6 7 // This file defines algorithms related to dominance.
8 9 // Dominator tree construction ----------------------------------------
10 //
11 // We use the algorithm described in Lengauer & Tarjan. 1979. A fast
12 // algorithm for finding dominators in a flowgraph.
13 // http://doi.acm.org/10.1145/357062.357071
14 //
15 // We also apply the optimizations to SLT described in Georgiadis et
16 // al, Finding Dominators in Practice, JGAA 2006,
17 // http://jgaa.info/accepted/2006/GeorgiadisTarjanWerneck2006.10.1.pdf
18 // to avoid the need for buckets of size > 1.
19 20 import (
21 "bytes"
22 "fmt"
23 "math/big"
24 "os"
25 "slices"
26 "sort"
27 )
28 29 // Idom returns the block that immediately dominates b:
30 // its parent in the dominator tree, if any.
31 // Neither the entry node (b.Index==0) nor recover node
32 // (b==b.Parent().Recover()) have a parent.
33 func (b *BasicBlock) Idom() *BasicBlock { return b.dom.idom }
34 35 // Dominees returns the list of blocks that b immediately dominates:
36 // its children in the dominator tree.
37 func (b *BasicBlock) Dominees() []*BasicBlock { return b.dom.children }
38 39 // Dominates reports whether b dominates c.
40 func (b *BasicBlock) Dominates(c *BasicBlock) bool {
41 return b.dom.pre <= c.dom.pre && c.dom.post <= b.dom.post
42 }
43 44 // DomPreorder returns a new slice containing the blocks of f
45 // in a preorder traversal of the dominator tree.
46 func (f *Function) DomPreorder() []*BasicBlock {
47 slice := slices.Clone(f.Blocks)
48 sort.Slice(slice, func(i, j int) bool {
49 return slice[i].dom.pre < slice[j].dom.pre
50 })
51 return slice
52 }
53 54 // DomPostorder returns a new slice containing the blocks of f
55 // in a postorder traversal of the dominator tree.
56 // (This is not the same as a postdominance order.)
57 func (f *Function) DomPostorder() []*BasicBlock {
58 slice := slices.Clone(f.Blocks)
59 sort.Slice(slice, func(i, j int) bool {
60 return slice[i].dom.post < slice[j].dom.post
61 })
62 return slice
63 }
64 65 // domInfo contains a BasicBlock's dominance information.
66 type domInfo struct {
67 idom *BasicBlock // immediate dominator (parent in domtree)
68 children []*BasicBlock // nodes immediately dominated by this one
69 pre, post int32 // pre- and post-order numbering within domtree
70 }
71 72 // ltState holds the working state for Lengauer-Tarjan algorithm
73 // (during which domInfo.pre is repurposed for CFG DFS preorder number).
74 type ltState struct {
75 // Each slice is indexed by b.Index.
76 sdom []*BasicBlock // b's semidominator
77 parent []*BasicBlock // b's parent in DFS traversal of CFG
78 ancestor []*BasicBlock // b's ancestor with least sdom
79 }
80 81 // dfs implements the depth-first search part of the LT algorithm.
82 func (lt *ltState) dfs(v *BasicBlock, i int32, preorder []*BasicBlock) int32 {
83 preorder[i] = v
84 v.dom.pre = i // For now: DFS preorder of spanning tree of CFG
85 i++
86 lt.sdom[v.Index] = v
87 lt.link(nil, v)
88 for _, w := range v.Succs {
89 if lt.sdom[w.Index] == nil {
90 lt.parent[w.Index] = v
91 i = lt.dfs(w, i, preorder)
92 }
93 }
94 return i
95 }
96 97 // eval implements the EVAL part of the LT algorithm.
98 func (lt *ltState) eval(v *BasicBlock) *BasicBlock {
99 // TODO(adonovan): opt: do path compression per simple LT.
100 u := v
101 for ; lt.ancestor[v.Index] != nil; v = lt.ancestor[v.Index] {
102 if lt.sdom[v.Index].dom.pre < lt.sdom[u.Index].dom.pre {
103 u = v
104 }
105 }
106 return u
107 }
108 109 // link implements the LINK part of the LT algorithm.
110 func (lt *ltState) link(v, w *BasicBlock) {
111 lt.ancestor[w.Index] = v
112 }
113 114 // buildDomTree computes the dominator tree of f using the LT algorithm.
115 // Precondition: all blocks are reachable (e.g. optimizeBlocks has been run).
116 func buildDomTree(f *Function) {
117 // The step numbers refer to the original LT paper; the
118 // reordering is due to Georgiadis.
119 120 // Clear any previous domInfo.
121 for _, b := range f.Blocks {
122 b.dom = domInfo{}
123 }
124 125 n := len(f.Blocks)
126 // Allocate space for 5 contiguous [n]*BasicBlock arrays:
127 // sdom, parent, ancestor, preorder, buckets.
128 space := make([]*BasicBlock, 5*n)
129 lt := ltState{
130 sdom: space[0:n],
131 parent: space[n : 2*n],
132 ancestor: space[2*n : 3*n],
133 }
134 135 // Step 1. Number vertices by depth-first preorder.
136 preorder := space[3*n : 4*n]
137 root := f.Blocks[0]
138 prenum := lt.dfs(root, 0, preorder)
139 recover := f.Recover
140 if recover != nil {
141 lt.dfs(recover, prenum, preorder)
142 }
143 144 buckets := space[4*n : 5*n]
145 copy(buckets, preorder)
146 147 // In reverse preorder...
148 for i := int32(n) - 1; i > 0; i-- {
149 w := preorder[i]
150 151 // Step 3. Implicitly define the immediate dominator of each node.
152 for v := buckets[i]; v != w; v = buckets[v.dom.pre] {
153 u := lt.eval(v)
154 if lt.sdom[u.Index].dom.pre < i {
155 v.dom.idom = u
156 } else {
157 v.dom.idom = w
158 }
159 }
160 161 // Step 2. Compute the semidominators of all nodes.
162 lt.sdom[w.Index] = lt.parent[w.Index]
163 for _, v := range w.Preds {
164 u := lt.eval(v)
165 if lt.sdom[u.Index].dom.pre < lt.sdom[w.Index].dom.pre {
166 lt.sdom[w.Index] = lt.sdom[u.Index]
167 }
168 }
169 170 lt.link(lt.parent[w.Index], w)
171 172 if lt.parent[w.Index] == lt.sdom[w.Index] {
173 w.dom.idom = lt.parent[w.Index]
174 } else {
175 buckets[i] = buckets[lt.sdom[w.Index].dom.pre]
176 buckets[lt.sdom[w.Index].dom.pre] = w
177 }
178 }
179 180 // The final 'Step 3' is now outside the loop.
181 for v := buckets[0]; v != root; v = buckets[v.dom.pre] {
182 v.dom.idom = root
183 }
184 185 // Step 4. Explicitly define the immediate dominator of each
186 // node, in preorder.
187 for _, w := range preorder[1:] {
188 if w == root || w == recover {
189 w.dom.idom = nil
190 } else {
191 if w.dom.idom != lt.sdom[w.Index] {
192 w.dom.idom = w.dom.idom.dom.idom
193 }
194 // Calculate Children relation as inverse of Idom.
195 w.dom.idom.dom.children = append(w.dom.idom.dom.children, w)
196 }
197 }
198 199 pre, post := numberDomTree(root, 0, 0)
200 if recover != nil {
201 numberDomTree(recover, pre, post)
202 }
203 204 // printDomTreeDot(os.Stderr, f) // debugging
205 // printDomTreeText(os.Stderr, root, 0) // debugging
206 207 if f.Prog.mode&SanityCheckFunctions != 0 {
208 sanityCheckDomTree(f)
209 }
210 }
211 212 // numberDomTree sets the pre- and post-order numbers of a depth-first
213 // traversal of the dominator tree rooted at v. These are used to
214 // answer dominance queries in constant time.
215 func numberDomTree(v *BasicBlock, pre, post int32) (int32, int32) {
216 v.dom.pre = pre
217 pre++
218 for _, child := range v.dom.children {
219 pre, post = numberDomTree(child, pre, post)
220 }
221 v.dom.post = post
222 post++
223 return pre, post
224 }
225 226 // Testing utilities ----------------------------------------
227 228 // sanityCheckDomTree checks the correctness of the dominator tree
229 // computed by the LT algorithm by comparing against the dominance
230 // relation computed by a naive Kildall-style forward dataflow
231 // analysis (Algorithm 10.16 from the "Dragon" book).
232 func sanityCheckDomTree(f *Function) {
233 n := len(f.Blocks)
234 235 // D[i] is the set of blocks that dominate f.Blocks[i],
236 // represented as a bit-set of block indices.
237 D := make([]big.Int, n)
238 239 one := big.NewInt(1)
240 241 // all is the set of all blocks; constant.
242 var all big.Int
243 all.Set(one).Lsh(&all, uint(n)).Sub(&all, one)
244 245 // Initialization.
246 for i, b := range f.Blocks {
247 if i == 0 || b == f.Recover {
248 // A root is dominated only by itself.
249 D[i].SetBit(&D[0], 0, 1)
250 } else {
251 // All other blocks are (initially) dominated
252 // by every block.
253 D[i].Set(&all)
254 }
255 }
256 257 // Iteration until fixed point.
258 for changed := true; changed; {
259 changed = false
260 for i, b := range f.Blocks {
261 if i == 0 || b == f.Recover {
262 continue
263 }
264 // Compute intersection across predecessors.
265 var x big.Int
266 x.Set(&all)
267 for _, pred := range b.Preds {
268 x.And(&x, &D[pred.Index])
269 }
270 x.SetBit(&x, i, 1) // a block always dominates itself.
271 if D[i].Cmp(&x) != 0 {
272 D[i].Set(&x)
273 changed = true
274 }
275 }
276 }
277 278 // Check the entire relation. O(n^2).
279 // The Recover block (if any) must be treated specially so we skip it.
280 ok := true
281 for i := range n {
282 for j := range n {
283 b, c := f.Blocks[i], f.Blocks[j]
284 if c == f.Recover {
285 continue
286 }
287 actual := b.Dominates(c)
288 expected := D[j].Bit(i) == 1
289 if actual != expected {
290 fmt.Fprintf(os.Stderr, "dominates(%s, %s)==%t, want %t\n", b, c, actual, expected)
291 ok = false
292 }
293 }
294 }
295 296 preorder := f.DomPreorder()
297 for _, b := range f.Blocks {
298 if got := preorder[b.dom.pre]; got != b {
299 fmt.Fprintf(os.Stderr, "preorder[%d]==%s, want %s\n", b.dom.pre, got, b)
300 ok = false
301 }
302 }
303 304 if !ok {
305 panic("sanityCheckDomTree failed for " + f.String())
306 }
307 308 }
309 310 // Printing functions ----------------------------------------
311 312 // printDomTreeText prints the dominator tree as text, using indentation.
313 func printDomTreeText(buf *bytes.Buffer, v *BasicBlock, indent int) {
314 fmt.Fprintf(buf, "%*s%s\n", 4*indent, "", v)
315 for _, child := range v.dom.children {
316 printDomTreeText(buf, child, indent+1)
317 }
318 }
319 320 // printDomTreeDot prints the dominator tree of f in AT&T GraphViz
321 // (.dot) format.
322 // (unused; retained for debugging)
323 func printDomTreeDot(buf *bytes.Buffer, f *Function) {
324 fmt.Fprintln(buf, "//", f)
325 fmt.Fprintln(buf, "digraph domtree {")
326 for i, b := range f.Blocks {
327 v := b.dom
328 fmt.Fprintf(buf, "\tn%d [label=\"%s (%d, %d)\",shape=\"rectangle\"];\n", v.pre, b, v.pre, v.post)
329 // TODO(adonovan): improve appearance of edges
330 // belonging to both dominator tree and CFG.
331 332 // Dominator tree edge.
333 if i != 0 {
334 fmt.Fprintf(buf, "\tn%d -> n%d [style=\"solid\",weight=100];\n", v.idom.dom.pre, v.pre)
335 }
336 // CFG edges.
337 for _, pred := range b.Preds {
338 fmt.Fprintf(buf, "\tn%d -> n%d [style=\"dotted\",weight=0];\n", pred.dom.pre, v.pre)
339 }
340 }
341 fmt.Fprintln(buf, "}")
342 }
343