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1 // Copyright (c) 2015, the Dart project authors. Please see the AUTHORS file | |
2 // for details. All rights reserved. Use of this source code is governed by a | |
3 // BSD-style license that can be found in the LICENSE file. | |
4 | |
5 /// A library to work with graphs. It contains a couple algorithms, including | |
6 /// Tarjan's algorithm to compute strongest connected components in a graph and | |
7 /// Cooper et al's dominator algorithm. | |
8 /// | |
9 /// Portions of the code in this library was adapted from | |
10 /// `package:analyzer/src/generated/collection_utilities.dart`. | |
11 // TODO(sigmund): move this into a shared place? | |
12 library compiler.tool.graph; | |
13 | |
14 import 'dart:math' as math; | |
15 | |
16 abstract class Graph<N> { | |
17 Iterable<N> get nodes; | |
18 bool get isEmpty; | |
19 int get nodeCount; | |
20 Iterable<N> targetsOf(N source); | |
21 Iterable<N> sourcesOf(N source); | |
22 | |
23 /// Run a topological sort of the graph. Since the graph may contain cycles, | |
24 /// this results in a list of strongly connected components rather than a list | |
25 /// of nodes. The nodes in each strongly connected components only have edges | |
26 /// that point to nodes in the same component or earlier components. | |
27 List<List<N>> computeTopologicalSort() { | |
28 _SccFinder<N> finder = new _SccFinder<N>(this); | |
29 return finder.computeTopologicalSort(); | |
30 } | |
31 | |
32 /// Whether [source] can transitively reach [target]. | |
33 bool containsPath(N source, N target) { | |
34 Set<N> seen = new Set<N>(); | |
35 bool helper(N node) { | |
36 if (identical(node, target)) return true; | |
37 if (!seen.add(node)) return false; | |
38 return targetsOf(node).any(helper); | |
39 } | |
40 return helper(source); | |
41 } | |
42 | |
43 /// Returns all nodes reachable from [root] in post order. | |
44 List<N> postOrder(N root) { | |
45 var seen = new Set<N>(); | |
46 var result = <N>[]; | |
47 helper(n) { | |
48 if (!seen.add(n)) return; | |
49 targetsOf(n).forEach(helper); | |
50 result.add(n); | |
51 } | |
52 helper(root); | |
53 return result; | |
54 } | |
55 | |
56 /// Return a list of nodes that form a cycle containing the given node. If the | |
57 /// node is not part of this graph, then a list containing only the node | |
58 /// itself will be returned. | |
59 List<N> findCycleContaining(N node) { | |
60 assert(node != null); | |
61 _SccFinder<N> finder = new _SccFinder<N>(this); | |
62 return finder._componentContaining(node); | |
63 } | |
64 | |
65 Graph<N> dominatorTree(N root) { | |
66 var iDom = (new _DominatorFinder(this)..run(root)).immediateDominators; | |
67 var graph = new EdgeListGraph<N>(); | |
68 for (N node in iDom.keys) { | |
69 if (node != root) graph.addEdge(iDom[node], node); | |
70 } | |
71 return graph; | |
72 } | |
73 } | |
74 | |
75 class EdgeListGraph<N> extends Graph<N> { | |
76 /// Edges in the graph. | |
77 Map<N, Set<N>> _edges = new Map<N, Set<N>>(); | |
78 | |
79 /// The reverse of _edges. | |
80 Map<N, Set<N>> _revEdges = new Map<N, Set<N>>(); | |
81 | |
82 Iterable<N> get nodes => _edges.keys; | |
83 bool get isEmpty => _edges.isEmpty; | |
84 int get nodeCount => _edges.length; | |
85 | |
86 final _empty = new Set<N>(); | |
87 | |
88 Iterable<N> targetsOf(N source) => _edges[source] ?? _empty; | |
89 Iterable<N> sourcesOf(N source) => _revEdges[source] ?? _empty; | |
90 | |
91 void addEdge(N source, N target) { | |
92 assert(source != null); | |
93 assert(target != null); | |
94 addNode(source); | |
95 addNode(target); | |
96 _edges[source].add(target); | |
97 _revEdges[target].add(source); | |
98 } | |
99 | |
100 void addNode(N node) { | |
101 assert(node != null); | |
102 _edges.putIfAbsent(node, () => new Set<N>()); | |
103 _revEdges.putIfAbsent(node, () => new Set<N>()); | |
104 } | |
105 | |
106 /// Remove the edge from the given [source] node to the given [target] node. | |
107 /// If there was no such edge then the graph will be unmodified: the number of | |
108 /// edges will be the same and the set of nodes will be the same (neither node | |
109 /// will either be added or removed). | |
110 void removeEdge(N source, N target) { | |
111 _edges[source]?.remove(target); | |
112 } | |
113 | |
114 /// Remove the given node from this graph. As a consequence, any edges for | |
115 /// which that node was either a head or a tail will also be removed. | |
116 void removeNode(N node) { | |
117 _edges.remove(node); | |
118 var sources = _revEdges[node]; | |
119 if (sources == null) return; | |
120 for (var source in sources) { | |
121 _edges[source].remove(node); | |
122 } | |
123 } | |
124 | |
125 /// Remove all of the given nodes from this graph. As a consequence, any edges | |
126 /// for which those nodes were either a head or a tail will also be removed. | |
127 void removeAllNodes(List<N> nodes) => nodes.forEach(removeNode); | |
128 } | |
129 | |
130 /// Used by the [SccFinder] to maintain information about the nodes that have | |
131 /// been examined. There is an instance of this class per node in the graph. | |
132 class _NodeInfo<N> { | |
133 /// Depth of the node corresponding to this info. | |
134 int index = 0; | |
135 | |
136 /// Depth of the first node in a cycle. | |
137 int lowlink = 0; | |
138 | |
139 /// Whether the corresponding node is on the stack. Used to remove the need | |
140 /// for searching a collection for the node each time the question needs to be | |
141 /// asked. | |
142 bool onStack = false; | |
143 | |
144 /// Component that contains the corresponding node. | |
145 List<N> component; | |
146 | |
147 _NodeInfo(int depth) | |
148 : index = depth, lowlink = depth, onStack = false; | |
149 } | |
150 | |
151 /// Implements Tarjan's Algorithm for finding the strongly connected components | |
152 /// in a graph. | |
153 class _SccFinder<N> { | |
154 /// The graph to process. | |
155 final Graph<N> _graph; | |
156 | |
157 /// The index used to uniquely identify the depth of nodes. | |
158 int _index = 0; | |
159 | |
160 /// Nodes that are being visited in order to identify components. | |
161 List<N> _stack = new List<N>(); | |
162 | |
163 /// Information associated with each node. | |
164 Map<N, _NodeInfo<N>> _info = <N, _NodeInfo<N>>{}; | |
165 | |
166 /// All strongly connected components found, in topological sort order (each | |
167 /// node in a strongly connected component only has edges that point to nodes | |
168 /// in the same component or earlier components). | |
169 List<List<N>> _allComponents = new List<List<N>>(); | |
170 | |
171 _SccFinder(this._graph) : super(); | |
172 | |
173 /// Return a list containing the nodes that are part of the strongly connected | |
174 /// component that contains the given node. | |
175 List<N> _componentContaining(N node) => _strongConnect(node).component; | |
176 | |
177 /// Run Tarjan's algorithm and return the resulting list of strongly connected | |
178 /// components. The list is in topological sort order (each node in a strongly | |
179 /// connected component only has edges that point to nodes in the same | |
180 /// component or earlier components). | |
181 List<List<N>> computeTopologicalSort() { | |
182 for (N node in _graph.nodes) { | |
183 var nodeInfo = _info[node]; | |
184 if (nodeInfo == null) _strongConnect(node); | |
185 } | |
186 return _allComponents; | |
187 } | |
188 | |
189 /// Remove and return the top-most element from the stack. | |
190 N _pop() { | |
191 N node = _stack.removeAt(_stack.length - 1); | |
192 _info[node].onStack = false; | |
193 return node; | |
194 } | |
195 | |
196 /// Add the given node to the stack. | |
197 void _push(N node) { | |
198 _info[node].onStack = true; | |
199 _stack.add(node); | |
200 } | |
201 | |
202 /// Compute the strongly connected component that contains the given node as | |
203 /// well as any components containing nodes that are reachable from the given | |
204 /// component. | |
205 _NodeInfo<N> _strongConnect(N v) { | |
206 // Set the depth index for v to the smallest unused index | |
207 var vInfo = new _NodeInfo<N>(_index++); | |
208 _info[v] = vInfo; | |
209 _push(v); | |
210 | |
211 for (N w in _graph.targetsOf(v)) { | |
212 var wInfo = _info[w]; | |
213 if (wInfo == null) { | |
214 // Successor w has not yet been visited; recurse on it | |
215 wInfo = _strongConnect(w); | |
216 vInfo.lowlink = math.min(vInfo.lowlink, wInfo.lowlink); | |
217 } else if (wInfo.onStack) { | |
218 // Successor w is in stack S and hence in the current SCC | |
219 vInfo.lowlink = math.min(vInfo.lowlink, wInfo.index); | |
220 } | |
221 } | |
222 | |
223 // If v is a root node, pop the stack and generate an SCC | |
224 if (vInfo.lowlink == vInfo.index) { | |
225 var component = new List<N>(); | |
226 N w; | |
227 do { | |
228 w = _pop(); | |
229 component.add(w); | |
230 _info[w].component = component; | |
231 } while (!identical(w, v)); | |
232 _allComponents.add(component); | |
233 } | |
234 return vInfo; | |
235 } | |
236 } | |
237 | |
238 /// Computes dominators using (Cooper, Harvey, and Kennedy's | |
239 /// algorithm)[http://www.cs.rice.edu/~keith/EMBED/dom.pdf]. | |
240 class _DominatorFinder<N> { | |
241 final Graph<N> _graph; | |
242 Map<N, N> immediateDominators = {}; | |
243 Map<N, int> postOrderId = {}; | |
244 _DominatorFinder(this._graph); | |
245 | |
246 run(N root) { | |
247 immediateDominators[root] = root; | |
248 bool changed = true; | |
249 int i = 0; | |
250 var nodesInPostOrder = _graph.postOrder(root); | |
251 for (var n in nodesInPostOrder) { | |
252 postOrderId[n] = i++; | |
253 } | |
254 var nodesInReversedPostOrder = nodesInPostOrder.reversed; | |
255 while (changed) { | |
256 changed = false; | |
257 for (var n in nodesInReversedPostOrder) { | |
258 if (n == root) continue; | |
259 bool first = true; | |
260 var idom = null; | |
261 for (var p in _graph.sourcesOf(n)) { | |
262 if (immediateDominators[p] != null) { | |
263 if (first) { | |
264 idom = p; | |
265 first = false; | |
266 } else { | |
267 idom = _intersect(p, idom); | |
268 } | |
269 } | |
270 } | |
271 if (immediateDominators[n] != idom) { | |
272 immediateDominators[n] = idom; | |
273 changed = true; | |
274 } | |
275 } | |
276 } | |
277 } | |
278 | |
279 N _intersect(N b1, N b2) { | |
280 var finger1 = b1; | |
281 var finger2 = b2; | |
282 while (finger1 != finger2) { | |
283 while (postOrderId[finger1] < postOrderId[finger2]) { | |
284 finger1 = immediateDominators[finger1]; | |
285 } | |
286 while (postOrderId[finger2] < postOrderId[finger1]) { | |
287 finger2 = immediateDominators[finger2]; | |
288 } | |
289 } | |
290 return finger1; | |
291 } | |
292 } | |
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