1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
use std::fmt::Debug;
use std::hash::Hash;
use std::mem;
use std::ops::Deref;

use rustc_index::bit_set::BitMatrix;

use crate::frozen::Frozen;
use crate::fx::{FxHashSet, FxIndexSet};

#[cfg(test)]
mod tests;

#[derive(Clone, Debug)]
pub struct TransitiveRelationBuilder<T> {
    // List of elements. This is used to map from a T to a usize.
    elements: FxIndexSet<T>,

    // List of base edges in the graph. Require to compute transitive
    // closure.
    edges: FxHashSet<Edge>,
}

#[derive(Debug)]
pub struct TransitiveRelation<T> {
    // Frozen transitive relation elements and edges.
    builder: Frozen<TransitiveRelationBuilder<T>>,

    // Cached transitive closure derived from the edges.
    closure: Frozen<BitMatrix<usize, usize>>,
}

impl<T> Deref for TransitiveRelation<T> {
    type Target = Frozen<TransitiveRelationBuilder<T>>;

    fn deref(&self) -> &Self::Target {
        &self.builder
    }
}

impl<T: Clone> Clone for TransitiveRelation<T> {
    fn clone(&self) -> Self {
        TransitiveRelation {
            builder: Frozen::freeze(self.builder.deref().clone()),
            closure: Frozen::freeze(self.closure.deref().clone()),
        }
    }
}

// HACK(eddyb) manual impl avoids `Default` bound on `T`.
impl<T: Eq + Hash> Default for TransitiveRelationBuilder<T> {
    fn default() -> Self {
        TransitiveRelationBuilder { elements: Default::default(), edges: Default::default() }
    }
}

#[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Debug, Hash)]
struct Index(usize);

#[derive(Clone, PartialEq, Eq, Debug, Hash)]
struct Edge {
    source: Index,
    target: Index,
}

impl<T: Eq + Hash + Copy> TransitiveRelationBuilder<T> {
    pub fn is_empty(&self) -> bool {
        self.edges.is_empty()
    }

    pub fn elements(&self) -> impl Iterator<Item = &T> {
        self.elements.iter()
    }

    fn index(&self, a: T) -> Option<Index> {
        self.elements.get_index_of(&a).map(Index)
    }

    fn add_index(&mut self, a: T) -> Index {
        let (index, _added) = self.elements.insert_full(a);
        Index(index)
    }

    /// Applies the (partial) function to each edge and returns a new
    /// relation builder. If `f` returns `None` for any end-point,
    /// returns `None`.
    pub fn maybe_map<F, U>(&self, mut f: F) -> Option<TransitiveRelationBuilder<U>>
    where
        F: FnMut(T) -> Option<U>,
        U: Clone + Debug + Eq + Hash + Copy,
    {
        let mut result = TransitiveRelationBuilder::default();
        for edge in &self.edges {
            result.add(f(self.elements[edge.source.0])?, f(self.elements[edge.target.0])?);
        }
        Some(result)
    }

    /// Indicate that `a < b` (where `<` is this relation)
    pub fn add(&mut self, a: T, b: T) {
        let a = self.add_index(a);
        let b = self.add_index(b);
        let edge = Edge { source: a, target: b };
        self.edges.insert(edge);
    }

    /// Compute the transitive closure derived from the edges, and converted to
    /// the final result. After this, all elements will be immutable to maintain
    /// the correctness of the result.
    pub fn freeze(self) -> TransitiveRelation<T> {
        let mut matrix = BitMatrix::new(self.elements.len(), self.elements.len());
        let mut changed = true;
        while changed {
            changed = false;
            for edge in &self.edges {
                // add an edge from S -> T
                changed |= matrix.insert(edge.source.0, edge.target.0);

                // add all outgoing edges from T into S
                changed |= matrix.union_rows(edge.target.0, edge.source.0);
            }
        }
        TransitiveRelation { builder: Frozen::freeze(self), closure: Frozen::freeze(matrix) }
    }
}

impl<T: Eq + Hash + Copy> TransitiveRelation<T> {
    /// Applies the (partial) function to each edge and returns a new
    /// relation including transitive closures.
    pub fn maybe_map<F, U>(&self, f: F) -> Option<TransitiveRelation<U>>
    where
        F: FnMut(T) -> Option<U>,
        U: Clone + Debug + Eq + Hash + Copy,
    {
        Some(self.builder.maybe_map(f)?.freeze())
    }

    /// Checks whether `a < target` (transitively)
    pub fn contains(&self, a: T, b: T) -> bool {
        match (self.index(a), self.index(b)) {
            (Some(a), Some(b)) => self.with_closure(|closure| closure.contains(a.0, b.0)),
            (None, _) | (_, None) => false,
        }
    }

    /// Thinking of `x R y` as an edge `x -> y` in a graph, this
    /// returns all things reachable from `a`.
    ///
    /// Really this probably ought to be `impl Iterator<Item = &T>`, but
    /// I'm too lazy to make that work, and -- given the caching
    /// strategy -- it'd be a touch tricky anyhow.
    pub fn reachable_from(&self, a: T) -> Vec<T> {
        match self.index(a) {
            Some(a) => {
                self.with_closure(|closure| closure.iter(a.0).map(|i| self.elements[i]).collect())
            }
            None => vec![],
        }
    }

    /// Picks what I am referring to as the "postdominating"
    /// upper-bound for `a` and `b`. This is usually the least upper
    /// bound, but in cases where there is no single least upper
    /// bound, it is the "mutual immediate postdominator", if you
    /// imagine a graph where `a < b` means `a -> b`.
    ///
    /// This function is needed because region inference currently
    /// requires that we produce a single "UB", and there is no best
    /// choice for the LUB. Rather than pick arbitrarily, I pick a
    /// less good, but predictable choice. This should help ensure
    /// that region inference yields predictable results (though it
    /// itself is not fully sufficient).
    ///
    /// Examples are probably clearer than any prose I could write
    /// (there are corresponding tests below, btw). In each case,
    /// the query is `postdom_upper_bound(a, b)`:
    ///
    /// ```text
    /// // Returns Some(x), which is also LUB.
    /// a -> a1 -> x
    ///            ^
    ///            |
    /// b -> b1 ---+
    ///
    /// // Returns `Some(x)`, which is not LUB (there is none)
    /// // diagonal edges run left-to-right.
    /// a -> a1 -> x
    ///   \/       ^
    ///   /\       |
    /// b -> b1 ---+
    ///
    /// // Returns `None`.
    /// a -> a1
    /// b -> b1
    /// ```
    pub fn postdom_upper_bound(&self, a: T, b: T) -> Option<T> {
        let mubs = self.minimal_upper_bounds(a, b);
        self.mutual_immediate_postdominator(mubs)
    }

    /// Viewing the relation as a graph, computes the "mutual
    /// immediate postdominator" of a set of points (if one
    /// exists). See `postdom_upper_bound` for details.
    pub fn mutual_immediate_postdominator(&self, mut mubs: Vec<T>) -> Option<T> {
        loop {
            match mubs[..] {
                [] => return None,
                [mub] => return Some(mub),
                _ => {
                    let m = mubs.pop().unwrap();
                    let n = mubs.pop().unwrap();
                    mubs.extend(self.minimal_upper_bounds(n, m));
                }
            }
        }
    }

    /// Returns the set of bounds `X` such that:
    ///
    /// - `a < X` and `b < X`
    /// - there is no `Y != X` such that `a < Y` and `Y < X`
    ///   - except for the case where `X < a` (i.e., a strongly connected
    ///     component in the graph). In that case, the smallest
    ///     representative of the SCC is returned (as determined by the
    ///     internal indices).
    ///
    /// Note that this set can, in principle, have any size.
    pub fn minimal_upper_bounds(&self, a: T, b: T) -> Vec<T> {
        let (Some(mut a), Some(mut b)) = (self.index(a), self.index(b)) else {
            return vec![];
        };

        // in some cases, there are some arbitrary choices to be made;
        // it doesn't really matter what we pick, as long as we pick
        // the same thing consistently when queried, so ensure that
        // (a, b) are in a consistent relative order
        if a > b {
            mem::swap(&mut a, &mut b);
        }

        let lub_indices = self.with_closure(|closure| {
            // Easy case is when either a < b or b < a:
            if closure.contains(a.0, b.0) {
                return vec![b.0];
            }
            if closure.contains(b.0, a.0) {
                return vec![a.0];
            }

            // Otherwise, the tricky part is that there may be some c
            // where a < c and b < c. In fact, there may be many such
            // values. So here is what we do:
            //
            // 1. Find the vector `[X | a < X && b < X]` of all values
            //    `X` where `a < X` and `b < X`. In terms of the
            //    graph, this means all values reachable from both `a`
            //    and `b`. Note that this vector is also a set, but we
            //    use the term vector because the order matters
            //    to the steps below.
            //    - This vector contains upper bounds, but they are
            //      not minimal upper bounds. So you may have e.g.
            //      `[x, y, tcx, z]` where `x < tcx` and `y < tcx` and
            //      `z < x` and `z < y`:
            //
            //           z --+---> x ----+----> tcx
            //               |           |
            //               |           |
            //               +---> y ----+
            //
            //      In this case, we really want to return just `[z]`.
            //      The following steps below achieve this by gradually
            //      reducing the list.
            // 2. Pare down the vector using `pare_down`. This will
            //    remove elements from the vector that can be reached
            //    by an earlier element.
            //    - In the example above, this would convert `[x, y,
            //      tcx, z]` to `[x, y, z]`. Note that `x` and `y` are
            //      still in the vector; this is because while `z < x`
            //      (and `z < y`) holds, `z` comes after them in the
            //      vector.
            // 3. Reverse the vector and repeat the pare down process.
            //    - In the example above, we would reverse to
            //      `[z, y, x]` and then pare down to `[z]`.
            // 4. Reverse once more just so that we yield a vector in
            //    increasing order of index. Not necessary, but why not.
            //
            // I believe this algorithm yields a minimal set. The
            // argument is that, after step 2, we know that no element
            // can reach its successors (in the vector, not the graph).
            // After step 3, we know that no element can reach any of
            // its predecessors (because of step 2) nor successors
            // (because we just called `pare_down`)
            //
            // This same algorithm is used in `parents` below.

            let mut candidates = closure.intersect_rows(a.0, b.0); // (1)
            pare_down(&mut candidates, closure); // (2)
            candidates.reverse(); // (3a)
            pare_down(&mut candidates, closure); // (3b)
            candidates
        });

        lub_indices
            .into_iter()
            .rev() // (4)
            .map(|i| self.elements[i])
            .collect()
    }

    /// Given an element A, returns the maximal set {B} of elements B
    /// such that
    ///
    /// - A != B
    /// - A R B is true
    /// - for each i, j: `B[i]` R `B[j]` does not hold
    ///
    /// The intuition is that this moves "one step up" through a lattice
    /// (where the relation is encoding the `<=` relation for the lattice).
    /// So e.g., if the relation is `->` and we have
    ///
    /// ```text
    /// a -> b -> d -> f
    /// |              ^
    /// +--> c -> e ---+
    /// ```
    ///
    /// then `parents(a)` returns `[b, c]`. The `postdom_parent` function
    /// would further reduce this to just `f`.
    pub fn parents(&self, a: T) -> Vec<T> {
        let Some(a) = self.index(a) else {
            return vec![];
        };

        // Steal the algorithm for `minimal_upper_bounds` above, but
        // with a slight tweak. In the case where `a R a`, we remove
        // that from the set of candidates.
        let ancestors = self.with_closure(|closure| {
            let mut ancestors = closure.intersect_rows(a.0, a.0);

            // Remove anything that can reach `a`. If this is a
            // reflexive relation, this will include `a` itself.
            ancestors.retain(|&e| !closure.contains(e, a.0));

            pare_down(&mut ancestors, closure); // (2)
            ancestors.reverse(); // (3a)
            pare_down(&mut ancestors, closure); // (3b)
            ancestors
        });

        ancestors
            .into_iter()
            .rev() // (4)
            .map(|i| self.elements[i])
            .collect()
    }

    fn with_closure<OP, R>(&self, op: OP) -> R
    where
        OP: FnOnce(&BitMatrix<usize, usize>) -> R,
    {
        op(&self.closure)
    }

    /// Lists all the base edges in the graph: the initial _non-transitive_ set of element
    /// relations, which will be later used as the basis for the transitive closure computation.
    pub fn base_edges(&self) -> impl Iterator<Item = (T, T)> + '_ {
        self.edges
            .iter()
            .map(move |edge| (self.elements[edge.source.0], self.elements[edge.target.0]))
    }
}

/// Pare down is used as a step in the LUB computation. It edits the
/// candidates array in place by removing any element j for which
/// there exists an earlier element i<j such that i -> j. That is,
/// after you run `pare_down`, you know that for all elements that
/// remain in candidates, they cannot reach any of the elements that
/// come after them.
///
/// Examples follow. Assume that a -> b -> c and x -> y -> z.
///
/// - Input: `[a, b, x]`. Output: `[a, x]`.
/// - Input: `[b, a, x]`. Output: `[b, a, x]`.
/// - Input: `[a, x, b, y]`. Output: `[a, x]`.
fn pare_down(candidates: &mut Vec<usize>, closure: &BitMatrix<usize, usize>) {
    let mut i = 0;
    while let Some(&candidate_i) = candidates.get(i) {
        i += 1;

        let mut j = i;
        let mut dead = 0;
        while let Some(&candidate_j) = candidates.get(j) {
            if closure.contains(candidate_i, candidate_j) {
                // If `i` can reach `j`, then we can remove `j`. So just
                // mark it as dead and move on; subsequent indices will be
                // shifted into its place.
                dead += 1;
            } else {
                candidates[j - dead] = candidate_j;
            }
            j += 1;
        }
        candidates.truncate(j - dead);
    }
}