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//! Finding the dominators in a control-flow graph.
//!
//! Algorithm based on Loukas Georgiadis,
//! "Linear-Time Algorithms for Dominators and Related Problems",
//! <ftp://ftp.cs.princeton.edu/techreports/2005/737.pdf>
//!
//! Additionally useful is the original Lengauer-Tarjan paper on this subject,
//! "A Fast Algorithm for Finding Dominators in a Flowgraph"
//! Thomas Lengauer and Robert Endre Tarjan.
//! <https://www.cs.princeton.edu/courses/archive/spr03/cs423/download/dominators.pdf>
use super::ControlFlowGraph;
use rustc_index::{Idx, IndexSlice, IndexVec};
use std::cmp::Ordering;
#[cfg(test)]
mod tests;
struct PreOrderFrame<Iter> {
pre_order_idx: PreorderIndex,
iter: Iter,
}
rustc_index::newtype_index! {
struct PreorderIndex {}
}
pub fn dominators<G: ControlFlowGraph>(graph: &G) -> Dominators<G::Node> {
// compute the post order index (rank) for each node
let mut post_order_rank = IndexVec::from_elem_n(0, graph.num_nodes());
// We allocate capacity for the full set of nodes, because most of the time
// most of the nodes *are* reachable.
let mut parent: IndexVec<PreorderIndex, PreorderIndex> =
IndexVec::with_capacity(graph.num_nodes());
let mut stack = vec![PreOrderFrame {
pre_order_idx: PreorderIndex::new(0),
iter: graph.successors(graph.start_node()),
}];
let mut pre_order_to_real: IndexVec<PreorderIndex, G::Node> =
IndexVec::with_capacity(graph.num_nodes());
let mut real_to_pre_order: IndexVec<G::Node, Option<PreorderIndex>> =
IndexVec::from_elem_n(None, graph.num_nodes());
pre_order_to_real.push(graph.start_node());
parent.push(PreorderIndex::new(0)); // the parent of the root node is the root for now.
real_to_pre_order[graph.start_node()] = Some(PreorderIndex::new(0));
let mut post_order_idx = 0;
// Traverse the graph, collecting a number of things:
//
// * Preorder mapping (to it, and back to the actual ordering)
// * Postorder mapping (used exclusively for `cmp_in_dominator_order` on the final product)
// * Parents for each vertex in the preorder tree
//
// These are all done here rather than through one of the 'standard'
// graph traversals to help make this fast.
'recurse: while let Some(frame) = stack.last_mut() {
while let Some(successor) = frame.iter.next() {
if real_to_pre_order[successor].is_none() {
let pre_order_idx = pre_order_to_real.push(successor);
real_to_pre_order[successor] = Some(pre_order_idx);
parent.push(frame.pre_order_idx);
stack.push(PreOrderFrame { pre_order_idx, iter: graph.successors(successor) });
continue 'recurse;
}
}
post_order_rank[pre_order_to_real[frame.pre_order_idx]] = post_order_idx;
post_order_idx += 1;
stack.pop();
}
let reachable_vertices = pre_order_to_real.len();
let mut idom = IndexVec::from_elem_n(PreorderIndex::new(0), reachable_vertices);
let mut semi = IndexVec::from_fn_n(std::convert::identity, reachable_vertices);
let mut label = semi.clone();
let mut bucket = IndexVec::from_elem_n(vec![], reachable_vertices);
let mut lastlinked = None;
// We loop over vertices in reverse preorder. This implements the pseudocode
// of the simple Lengauer-Tarjan algorithm. A few key facts are noted here
// which are helpful for understanding the code (full proofs and such are
// found in various papers, including one cited at the top of this file).
//
// For each vertex w (which is not the root),
// * semi[w] is a proper ancestor of the vertex w (i.e., semi[w] != w)
// * idom[w] is an ancestor of semi[w] (i.e., idom[w] may equal semi[w])
//
// An immediate dominator of w (idom[w]) is a vertex v where v dominates w
// and every other dominator of w dominates v. (Every vertex except the root has
// a unique immediate dominator.)
//
// A semidominator for a given vertex w (semi[w]) is the vertex v with minimum
// preorder number such that there exists a path from v to w in which all elements (other than w) have
// preorder numbers greater than w (i.e., this path is not the tree path to
// w).
for w in (PreorderIndex::new(1)..PreorderIndex::new(reachable_vertices)).rev() {
// Optimization: process buckets just once, at the start of the
// iteration. Do not explicitly empty the bucket (even though it will
// not be used again), to save some instructions.
//
// The bucket here contains the vertices whose semidominator is the
// vertex w, which we are guaranteed to have found: all vertices who can
// be semidominated by w must have a preorder number exceeding w, so
// they have been placed in the bucket.
//
// We compute a partial set of immediate dominators here.
for &v in bucket[w].iter() {
// This uses the result of Lemma 5 from section 2 from the original
// 1979 paper, to compute either the immediate or relative dominator
// for a given vertex v.
//
// eval returns a vertex y, for which semi[y] is minimum among
// vertices semi[v] +> y *> v. Note that semi[v] = w as we're in the
// w bucket.
//
// Given such a vertex y, semi[y] <= semi[v] and idom[y] = idom[v].
// If semi[y] = semi[v], though, idom[v] = semi[v].
//
// Using this, we can either set idom[v] to be:
// * semi[v] (i.e. w), if semi[y] is w
// * idom[y], otherwise
//
// We don't directly set to idom[y] though as it's not necessarily
// known yet. The second preorder traversal will cleanup by updating
// the idom for any that were missed in this pass.
let y = eval(&mut parent, lastlinked, &semi, &mut label, v);
idom[v] = if semi[y] < w { y } else { w };
}
// This loop computes the semi[w] for w.
semi[w] = w;
for v in graph.predecessors(pre_order_to_real[w]) {
// TL;DR: Reachable vertices may have unreachable predecessors, so ignore any of them.
//
// Ignore blocks which are not connected to the entry block.
//
// The algorithm that was used to traverse the graph and build the
// `pre_order_to_real` and `real_to_pre_order` vectors does so by
// starting from the entry block and following the successors.
// Therefore, any blocks not reachable from the entry block will be
// set to `None` in the `pre_order_to_real` vector.
//
// For example, in this graph, A and B should be skipped:
//
// ┌─────┐
// │ │
// └──┬──┘
// │
// ┌──▼──┐ ┌─────┐
// │ │ │ A │
// └──┬──┘ └──┬──┘
// │ │
// ┌───────┴───────┐ │
// │ │ │
// ┌──▼──┐ ┌──▼──┐ ┌──▼──┐
// │ │ │ │ │ B │
// └──┬──┘ └──┬──┘ └──┬──┘
// │ └──────┬─────┘
// ┌──▼──┐ │
// │ │ │
// └──┬──┘ ┌──▼──┐
// │ │ │
// │ └─────┘
// ┌──▼──┐
// │ │
// └──┬──┘
// │
// ┌──▼──┐
// │ │
// └─────┘
//
// ...this may be the case if a MirPass modifies the CFG to remove
// or rearrange certain blocks/edges.
let Some(v) = real_to_pre_order[v] else { continue };
// eval returns a vertex x from which semi[x] is minimum among
// vertices semi[v] +> x *> v.
//
// From Lemma 4 from section 2, we know that the semidominator of a
// vertex w is the minimum (by preorder number) vertex of the
// following:
//
// * direct predecessors of w with preorder number less than w
// * semidominators of u such that u > w and there exists (v, w)
// such that u *> v
//
// This loop therefore identifies such a minima. Note that any
// semidominator path to w must have all but the first vertex go
// through vertices numbered greater than w, so the reverse preorder
// traversal we are using guarantees that all of the information we
// might need is available at this point.
//
// The eval call will give us semi[x], which is either:
//
// * v itself, if v has not yet been processed
// * A possible 'best' semidominator for w.
let x = eval(&mut parent, lastlinked, &semi, &mut label, v);
semi[w] = std::cmp::min(semi[w], semi[x]);
}
// semi[w] is now semidominator(w) and won't change any more.
// Optimization: Do not insert into buckets if parent[w] = semi[w], as
// we then immediately know the idom.
//
// If we don't yet know the idom directly, then push this vertex into
// our semidominator's bucket, where it will get processed at a later
// stage to compute its immediate dominator.
let z = parent[w];
if z != semi[w] {
bucket[semi[w]].push(w);
} else {
idom[w] = z;
}
// Optimization: We share the parent array between processed and not
// processed elements; lastlinked represents the divider.
lastlinked = Some(w);
}
// Finalize the idoms for any that were not fully settable during initial
// traversal.
//
// If idom[w] != semi[w] then we know that we've stored vertex y from above
// into idom[w]. It is known to be our 'relative dominator', which means
// that it's one of w's ancestors and has the same immediate dominator as w,
// so use that idom.
for w in PreorderIndex::new(1)..PreorderIndex::new(reachable_vertices) {
if idom[w] != semi[w] {
idom[w] = idom[idom[w]];
}
}
let mut immediate_dominators = IndexVec::from_elem_n(None, graph.num_nodes());
for (idx, node) in pre_order_to_real.iter_enumerated() {
immediate_dominators[*node] = Some(pre_order_to_real[idom[idx]]);
}
let start_node = graph.start_node();
immediate_dominators[start_node] = None;
let time = compute_access_time(start_node, &immediate_dominators);
Dominators { start_node, post_order_rank, immediate_dominators, time }
}
/// Evaluate the link-eval virtual forest, providing the currently minimum semi
/// value for the passed `node` (which may be itself).
///
/// This maintains that for every vertex v, `label[v]` is such that:
///
/// ```text
/// semi[eval(v)] = min { semi[label[u]] | root_in_forest(v) +> u *> v }
/// ```
///
/// where `+>` is a proper ancestor and `*>` is just an ancestor.
#[inline]
fn eval(
ancestor: &mut IndexSlice<PreorderIndex, PreorderIndex>,
lastlinked: Option<PreorderIndex>,
semi: &IndexSlice<PreorderIndex, PreorderIndex>,
label: &mut IndexSlice<PreorderIndex, PreorderIndex>,
node: PreorderIndex,
) -> PreorderIndex {
if is_processed(node, lastlinked) {
compress(ancestor, lastlinked, semi, label, node);
label[node]
} else {
node
}
}
#[inline]
fn is_processed(v: PreorderIndex, lastlinked: Option<PreorderIndex>) -> bool {
if let Some(ll) = lastlinked { v >= ll } else { false }
}
#[inline]
fn compress(
ancestor: &mut IndexSlice<PreorderIndex, PreorderIndex>,
lastlinked: Option<PreorderIndex>,
semi: &IndexSlice<PreorderIndex, PreorderIndex>,
label: &mut IndexSlice<PreorderIndex, PreorderIndex>,
v: PreorderIndex,
) {
assert!(is_processed(v, lastlinked));
// Compute the processed list of ancestors
//
// We use a heap stack here to avoid recursing too deeply, exhausting the
// stack space.
let mut stack: smallvec::SmallVec<[_; 8]> = smallvec::smallvec![v];
let mut u = ancestor[v];
while is_processed(u, lastlinked) {
stack.push(u);
u = ancestor[u];
}
// Then in reverse order, popping the stack
for &[v, u] in stack.array_windows().rev() {
if semi[label[u]] < semi[label[v]] {
label[v] = label[u];
}
ancestor[v] = ancestor[u];
}
}
/// Tracks the list of dominators for each node.
#[derive(Clone, Debug)]
pub struct Dominators<N: Idx> {
start_node: N,
post_order_rank: IndexVec<N, usize>,
// Even though we track only the immediate dominator of each node, it's
// possible to get its full list of dominators by looking up the dominator
// of each dominator. (See the `impl Iterator for Iter` definition).
immediate_dominators: IndexVec<N, Option<N>>,
time: IndexVec<N, Time>,
}
impl<Node: Idx> Dominators<Node> {
/// Returns true if node is reachable from the start node.
pub fn is_reachable(&self, node: Node) -> bool {
node == self.start_node || self.immediate_dominators[node].is_some()
}
/// Returns the immediate dominator of node, if any.
pub fn immediate_dominator(&self, node: Node) -> Option<Node> {
self.immediate_dominators[node]
}
/// Provides an iterator over each dominator up the CFG, for the given Node.
/// See the `impl Iterator for Iter` definition to understand how this works.
pub fn dominators(&self, node: Node) -> Iter<'_, Node> {
assert!(self.is_reachable(node), "node {node:?} is not reachable");
Iter { dom_tree: self, node: Some(node) }
}
/// Provide deterministic ordering of nodes such that, if any two nodes have a dominator
/// relationship, the dominator will always precede the dominated. (The relative ordering
/// of two unrelated nodes will also be consistent, but otherwise the order has no
/// meaning.) This method cannot be used to determine if either Node dominates the other.
pub fn cmp_in_dominator_order(&self, lhs: Node, rhs: Node) -> Ordering {
self.post_order_rank[rhs].cmp(&self.post_order_rank[lhs])
}
/// Returns true if `a` dominates `b`.
///
/// # Panics
///
/// Panics if `b` is unreachable.
pub fn dominates(&self, a: Node, b: Node) -> bool {
let a = self.time[a];
let b = self.time[b];
assert!(b.start != 0, "node {b:?} is not reachable");
a.start <= b.start && b.finish <= a.finish
}
}
pub struct Iter<'dom, Node: Idx> {
dom_tree: &'dom Dominators<Node>,
node: Option<Node>,
}
impl<'dom, Node: Idx> Iterator for Iter<'dom, Node> {
type Item = Node;
fn next(&mut self) -> Option<Self::Item> {
if let Some(node) = self.node {
self.node = self.dom_tree.immediate_dominator(node);
Some(node)
} else {
None
}
}
}
/// Describes the number of vertices discovered at the time when processing of a particular vertex
/// started and when it finished. Both values are zero for unreachable vertices.
#[derive(Copy, Clone, Default, Debug)]
struct Time {
start: u32,
finish: u32,
}
fn compute_access_time<N: Idx>(
start_node: N,
immediate_dominators: &IndexSlice<N, Option<N>>,
) -> IndexVec<N, Time> {
// Transpose the dominator tree edges, so that child nodes of vertex v are stored in
// node[edges[v].start..edges[v].end].
let mut edges: IndexVec<N, std::ops::Range<u32>> =
IndexVec::from_elem(0..0, immediate_dominators);
for &idom in immediate_dominators.iter() {
if let Some(idom) = idom {
edges[idom].end += 1;
}
}
let mut m = 0;
for e in edges.iter_mut() {
m += e.end;
e.start = m;
e.end = m;
}
let mut node = IndexVec::from_elem_n(Idx::new(0), m.try_into().unwrap());
for (i, &idom) in immediate_dominators.iter_enumerated() {
if let Some(idom) = idom {
edges[idom].start -= 1;
node[edges[idom].start] = i;
}
}
// Perform a depth-first search of the dominator tree. Record the number of vertices discovered
// when vertex v is discovered first as time[v].start, and when its processing is finished as
// time[v].finish.
let mut time: IndexVec<N, Time> = IndexVec::from_elem(Time::default(), immediate_dominators);
let mut stack = Vec::new();
let mut discovered = 1;
stack.push(start_node);
time[start_node].start = discovered;
while let Some(&i) = stack.last() {
let e = &mut edges[i];
if e.start == e.end {
// Finish processing vertex i.
time[i].finish = discovered;
stack.pop();
} else {
let j = node[e.start];
e.start += 1;
// Start processing vertex j.
discovered += 1;
time[j].start = discovered;
stack.push(j);
}
}
time
}