578 lines
23 KiB
Rust
578 lines
23 KiB
Rust
//! Routine to compute the strongly connected components (SCCs) of a graph.
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//!
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//! Also computes as the resulting DAG if each SCC is replaced with a
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//! node in the graph. This uses [Tarjan's algorithm](
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//! https://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm)
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//! that completes in *O*(*n*) time.
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use crate::fx::FxHashSet;
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use crate::graph::vec_graph::VecGraph;
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use crate::graph::{DirectedGraph, GraphSuccessors, WithNumEdges, WithSuccessors};
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use rustc_index::{Idx, IndexSlice, IndexVec};
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use std::ops::Range;
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#[cfg(test)]
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mod tests;
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/// Strongly connected components (SCC) of a graph. The type `N` is
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/// the index type for the graph nodes and `S` is the index type for
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/// the SCCs. We can map from each node to the SCC that it
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/// participates in, and we also have the successors of each SCC.
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pub struct Sccs<N: Idx, S: Idx> {
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/// For each node, what is the SCC index of the SCC to which it
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/// belongs.
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scc_indices: IndexVec<N, S>,
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/// Data about each SCC.
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scc_data: SccData<S>,
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}
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pub struct SccData<S: Idx> {
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/// For each SCC, the range of `all_successors` where its
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/// successors can be found.
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ranges: IndexVec<S, Range<usize>>,
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/// Contains the successors for all the Sccs, concatenated. The
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/// range of indices corresponding to a given SCC is found in its
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/// SccData.
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all_successors: Vec<S>,
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}
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impl<N: Idx, S: Idx + Ord> Sccs<N, S> {
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pub fn new(graph: &(impl DirectedGraph<Node = N> + WithSuccessors)) -> Self {
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SccsConstruction::construct(graph)
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}
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pub fn scc_indices(&self) -> &IndexSlice<N, S> {
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&self.scc_indices
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}
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pub fn scc_data(&self) -> &SccData<S> {
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&self.scc_data
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}
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/// Returns the number of SCCs in the graph.
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pub fn num_sccs(&self) -> usize {
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self.scc_data.len()
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}
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/// Returns an iterator over the SCCs in the graph.
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///
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/// The SCCs will be iterated in **dependency order** (or **post order**),
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/// meaning that if `S1 -> S2`, we will visit `S2` first and `S1` after.
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/// This is convenient when the edges represent dependencies: when you visit
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/// `S1`, the value for `S2` will already have been computed.
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pub fn all_sccs(&self) -> impl Iterator<Item = S> {
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(0..self.scc_data.len()).map(S::new)
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}
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/// Returns the SCC to which a node `r` belongs.
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pub fn scc(&self, r: N) -> S {
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self.scc_indices[r]
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}
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/// Returns the successors of the given SCC.
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pub fn successors(&self, scc: S) -> &[S] {
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self.scc_data.successors(scc)
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}
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/// Construct the reverse graph of the SCC graph.
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pub fn reverse(&self) -> VecGraph<S> {
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VecGraph::new(
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self.num_sccs(),
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self.all_sccs()
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.flat_map(|source| {
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self.successors(source).iter().map(move |&target| (target, source))
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})
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.collect(),
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)
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}
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}
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impl<N: Idx, S: Idx + Ord> DirectedGraph for Sccs<N, S> {
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type Node = S;
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fn num_nodes(&self) -> usize {
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self.num_sccs()
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}
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}
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impl<N: Idx, S: Idx + Ord> WithNumEdges for Sccs<N, S> {
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fn num_edges(&self) -> usize {
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self.scc_data.all_successors.len()
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}
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}
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impl<'graph, N: Idx, S: Idx> GraphSuccessors<'graph> for Sccs<N, S> {
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type Item = S;
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type Iter = std::iter::Cloned<std::slice::Iter<'graph, S>>;
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}
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impl<N: Idx, S: Idx + Ord> WithSuccessors for Sccs<N, S> {
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fn successors(&self, node: S) -> <Self as GraphSuccessors<'_>>::Iter {
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self.successors(node).iter().cloned()
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}
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}
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impl<S: Idx> SccData<S> {
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/// Number of SCCs,
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fn len(&self) -> usize {
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self.ranges.len()
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}
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pub fn ranges(&self) -> &IndexSlice<S, Range<usize>> {
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&self.ranges
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}
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pub fn all_successors(&self) -> &Vec<S> {
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&self.all_successors
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}
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/// Returns the successors of the given SCC.
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fn successors(&self, scc: S) -> &[S] {
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// Annoyingly, `range` does not implement `Copy`, so we have
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// to do `range.start..range.end`:
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let range = &self.ranges[scc];
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&self.all_successors[range.start..range.end]
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}
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/// Creates a new SCC with `successors` as its successors and
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/// returns the resulting index.
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fn create_scc(&mut self, successors: impl IntoIterator<Item = S>) -> S {
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// Store the successors on `scc_successors_vec`, remembering
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// the range of indices.
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let all_successors_start = self.all_successors.len();
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self.all_successors.extend(successors);
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let all_successors_end = self.all_successors.len();
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debug!(
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"create_scc({:?}) successors={:?}",
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self.ranges.len(),
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&self.all_successors[all_successors_start..all_successors_end],
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);
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self.ranges.push(all_successors_start..all_successors_end)
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}
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}
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struct SccsConstruction<'c, G: DirectedGraph + WithSuccessors, S: Idx> {
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graph: &'c G,
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/// The state of each node; used during walk to record the stack
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/// and after walk to record what cycle each node ended up being
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/// in.
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node_states: IndexVec<G::Node, NodeState<G::Node, S>>,
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/// The stack of nodes that we are visiting as part of the DFS.
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node_stack: Vec<G::Node>,
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/// The stack of successors: as we visit a node, we mark our
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/// position in this stack, and when we encounter a successor SCC,
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/// we push it on the stack. When we complete an SCC, we can pop
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/// everything off the stack that was found along the way.
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successors_stack: Vec<S>,
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/// A set used to strip duplicates. As we accumulate successors
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/// into the successors_stack, we sometimes get duplicate entries.
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/// We use this set to remove those -- we also keep its storage
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/// around between successors to amortize memory allocation costs.
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duplicate_set: FxHashSet<S>,
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scc_data: SccData<S>,
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}
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#[derive(Copy, Clone, Debug)]
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enum NodeState<N, S> {
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/// This node has not yet been visited as part of the DFS.
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///
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/// After SCC construction is complete, this state ought to be
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/// impossible.
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NotVisited,
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/// This node is currently being walk as part of our DFS. It is on
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/// the stack at the depth `depth`.
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///
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/// After SCC construction is complete, this state ought to be
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/// impossible.
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BeingVisited { depth: usize },
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/// Indicates that this node is a member of the given cycle.
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InCycle { scc_index: S },
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/// Indicates that this node is a member of whatever cycle
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/// `parent` is a member of. This state is transient: whenever we
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/// see it, we try to overwrite it with the current state of
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/// `parent` (this is the "path compression" step of a union-find
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/// algorithm).
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InCycleWith { parent: N },
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}
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#[derive(Copy, Clone, Debug)]
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enum WalkReturn<S> {
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Cycle { min_depth: usize },
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Complete { scc_index: S },
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}
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impl<'c, G, S> SccsConstruction<'c, G, S>
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where
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G: DirectedGraph + WithSuccessors,
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S: Idx,
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{
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/// Identifies SCCs in the graph `G` and computes the resulting
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/// DAG. This uses a variant of [Tarjan's
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/// algorithm][wikipedia]. The high-level summary of the algorithm
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/// is that we do a depth-first search. Along the way, we keep a
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/// stack of each node whose successors are being visited. We
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/// track the depth of each node on this stack (there is no depth
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/// if the node is not on the stack). When we find that some node
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/// N with depth D can reach some other node N' with lower depth
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/// D' (i.e., D' < D), we know that N, N', and all nodes in
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/// between them on the stack are part of an SCC.
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///
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/// [wikipedia]: https://bit.ly/2EZIx84
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fn construct(graph: &'c G) -> Sccs<G::Node, S> {
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let num_nodes = graph.num_nodes();
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let mut this = Self {
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graph,
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node_states: IndexVec::from_elem_n(NodeState::NotVisited, num_nodes),
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node_stack: Vec::with_capacity(num_nodes),
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successors_stack: Vec::new(),
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scc_data: SccData { ranges: IndexVec::new(), all_successors: Vec::new() },
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duplicate_set: FxHashSet::default(),
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};
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let scc_indices = (0..num_nodes)
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.map(G::Node::new)
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.map(|node| match this.start_walk_from(node) {
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WalkReturn::Complete { scc_index } => scc_index,
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WalkReturn::Cycle { min_depth } => {
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panic!("`start_walk_node({node:?})` returned cycle with depth {min_depth:?}")
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}
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})
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.collect();
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Sccs { scc_indices, scc_data: this.scc_data }
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}
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fn start_walk_from(&mut self, node: G::Node) -> WalkReturn<S> {
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if let Some(result) = self.inspect_node(node) {
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result
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} else {
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self.walk_unvisited_node(node)
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}
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}
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/// Inspect a node during the DFS. We first examine its current
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/// state -- if it is not yet visited (`NotVisited`), return `None` so
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/// that the caller might push it onto the stack and start walking its
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/// successors.
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///
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/// If it is already on the DFS stack it will be in the state
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/// `BeingVisited`. In that case, we have found a cycle and we
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/// return the depth from the stack.
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///
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/// Otherwise, we are looking at a node that has already been
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/// completely visited. We therefore return `WalkReturn::Complete`
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/// with its associated SCC index.
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fn inspect_node(&mut self, node: G::Node) -> Option<WalkReturn<S>> {
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Some(match self.find_state(node) {
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NodeState::InCycle { scc_index } => WalkReturn::Complete { scc_index },
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NodeState::BeingVisited { depth: min_depth } => WalkReturn::Cycle { min_depth },
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NodeState::NotVisited => return None,
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NodeState::InCycleWith { parent } => panic!(
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"`find_state` returned `InCycleWith({parent:?})`, which ought to be impossible"
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),
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})
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}
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/// Fetches the state of the node `r`. If `r` is recorded as being
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/// in a cycle with some other node `r2`, then fetches the state
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/// of `r2` (and updates `r` to reflect current result). This is
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/// basically the "find" part of a standard union-find algorithm
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/// (with path compression).
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fn find_state(&mut self, mut node: G::Node) -> NodeState<G::Node, S> {
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// To avoid recursion we temporarily reuse the `parent` of each
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// InCycleWith link to encode a downwards link while compressing
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// the path. After we have found the root or deepest node being
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// visited, we traverse the reverse links and correct the node
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// states on the way.
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//
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// **Note**: This mutation requires that this is a leaf function
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// or at least that none of the called functions inspects the
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// current node states. Luckily, we are a leaf.
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// Remember one previous link. The termination condition when
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// following links downwards is then simply as soon as we have
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// found the initial self-loop.
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let mut previous_node = node;
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// Ultimately assigned by the parent when following
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// `InCycleWith` upwards.
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let node_state = loop {
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debug!("find_state(r = {:?} in state {:?})", node, self.node_states[node]);
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match self.node_states[node] {
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NodeState::InCycle { scc_index } => break NodeState::InCycle { scc_index },
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NodeState::BeingVisited { depth } => break NodeState::BeingVisited { depth },
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NodeState::NotVisited => break NodeState::NotVisited,
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NodeState::InCycleWith { parent } => {
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// We test this, to be extremely sure that we never
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// ever break our termination condition for the
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// reverse iteration loop.
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assert!(node != parent, "Node can not be in cycle with itself");
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// Store the previous node as an inverted list link
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self.node_states[node] = NodeState::InCycleWith { parent: previous_node };
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// Update to parent node.
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previous_node = node;
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node = parent;
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}
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}
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};
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// The states form a graph where up to one outgoing link is stored at
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// each node. Initially in general,
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//
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// E
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// ^
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// |
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// InCycleWith/BeingVisited/NotVisited
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// |
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// A-InCycleWith->B-InCycleWith…>C-InCycleWith->D-+
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// |
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// = node, previous_node
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//
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// After the first loop, this will look like
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// E
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// ^
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// |
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// InCycleWith/BeingVisited/NotVisited
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// |
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// +>A<-InCycleWith-B<…InCycleWith-C<-InCycleWith-D-+
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// | | | |
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// | InCycleWith | = node
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// +-+ =previous_node
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//
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// Note in particular that A will be linked to itself in a self-cycle
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// and no other self-cycles occur due to how InCycleWith is assigned in
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// the find phase implemented by `walk_unvisited_node`.
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//
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// We now want to compress the path, that is assign the state of the
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// link D-E to all other links.
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//
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// We can then walk backwards, starting from `previous_node`, and assign
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// each node in the list with the updated state. The loop terminates
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// when we reach the self-cycle.
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// Move backwards until we found the node where we started. We
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// will know when we hit the state where previous_node == node.
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loop {
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// Back at the beginning, we can return.
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if previous_node == node {
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return node_state;
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}
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// Update to previous node in the link.
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match self.node_states[previous_node] {
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NodeState::InCycleWith { parent: previous } => {
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node = previous_node;
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previous_node = previous;
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}
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// Only InCycleWith nodes were added to the reverse linked list.
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other => panic!("Invalid previous link while compressing cycle: {other:?}"),
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}
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debug!("find_state: parent_state = {:?}", node_state);
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// Update the node state from the parent state. The assigned
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// state is actually a loop invariant but it will only be
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// evaluated if there is at least one backlink to follow.
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// Fully trusting llvm here to find this loop optimization.
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match node_state {
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// Path compression, make current node point to the same root.
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NodeState::InCycle { .. } => {
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self.node_states[node] = node_state;
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}
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// Still visiting nodes, compress to cycle to the node
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// at that depth.
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NodeState::BeingVisited { depth } => {
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self.node_states[node] =
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NodeState::InCycleWith { parent: self.node_stack[depth] };
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}
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// These are never allowed as parent nodes. InCycleWith
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// should have been followed to a real parent and
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// NotVisited can not be part of a cycle since it should
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// have instead gotten explored.
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NodeState::NotVisited | NodeState::InCycleWith { .. } => {
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panic!("invalid parent state: {node_state:?}")
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}
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}
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}
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}
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/// Walks a node that has never been visited before.
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///
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/// Call this method when `inspect_node` has returned `None`. Having the
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/// caller decide avoids mutual recursion between the two methods and allows
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/// us to maintain an allocated stack for nodes on the path between calls.
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#[instrument(skip(self, initial), level = "debug")]
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fn walk_unvisited_node(&mut self, initial: G::Node) -> WalkReturn<S> {
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struct VisitingNodeFrame<G: DirectedGraph, Successors> {
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node: G::Node,
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iter: Option<Successors>,
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depth: usize,
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min_depth: usize,
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successors_len: usize,
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min_cycle_root: G::Node,
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successor_node: G::Node,
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}
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// Move the stack to a local variable. We want to utilize the existing allocation and
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// mutably borrow it without borrowing self at the same time.
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let mut successors_stack = core::mem::take(&mut self.successors_stack);
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debug_assert_eq!(successors_stack.len(), 0);
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let mut stack: Vec<VisitingNodeFrame<G, _>> = vec![VisitingNodeFrame {
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node: initial,
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depth: 0,
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min_depth: 0,
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iter: None,
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successors_len: 0,
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min_cycle_root: initial,
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successor_node: initial,
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}];
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let mut return_value = None;
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'recurse: while let Some(frame) = stack.last_mut() {
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let VisitingNodeFrame {
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node,
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depth,
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iter,
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successors_len,
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min_depth,
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min_cycle_root,
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successor_node,
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} = frame;
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let node = *node;
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let depth = *depth;
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let successors = match iter {
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Some(iter) => iter,
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None => {
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// This None marks that we still have the initialize this node's frame.
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debug!(?depth, ?node);
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debug_assert!(matches!(self.node_states[node], NodeState::NotVisited));
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// Push `node` onto the stack.
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self.node_states[node] = NodeState::BeingVisited { depth };
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self.node_stack.push(node);
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// Walk each successor of the node, looking to see if any of
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// them can reach a node that is presently on the stack. If
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// so, that means they can also reach us.
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*successors_len = successors_stack.len();
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// Set and return a reference, this is currently empty.
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iter.get_or_insert(self.graph.successors(node))
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}
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};
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// Now that iter is initialized, this is a constant for this frame.
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let successors_len = *successors_len;
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// Construct iterators for the nodes and walk results. There are two cases:
|
|
// * The walk of a successor node returned.
|
|
// * The remaining successor nodes.
|
|
let returned_walk =
|
|
return_value.take().into_iter().map(|walk| (*successor_node, Some(walk)));
|
|
|
|
let successor_walk = successors.map(|successor_node| {
|
|
debug!(?node, ?successor_node);
|
|
(successor_node, self.inspect_node(successor_node))
|
|
});
|
|
|
|
for (successor_node, walk) in returned_walk.chain(successor_walk) {
|
|
match walk {
|
|
Some(WalkReturn::Cycle { min_depth: successor_min_depth }) => {
|
|
// Track the minimum depth we can reach.
|
|
assert!(successor_min_depth <= depth);
|
|
if successor_min_depth < *min_depth {
|
|
debug!(?node, ?successor_min_depth);
|
|
*min_depth = successor_min_depth;
|
|
*min_cycle_root = successor_node;
|
|
}
|
|
}
|
|
|
|
Some(WalkReturn::Complete { scc_index: successor_scc_index }) => {
|
|
// Push the completed SCC indices onto
|
|
// the `successors_stack` for later.
|
|
debug!(?node, ?successor_scc_index);
|
|
successors_stack.push(successor_scc_index);
|
|
}
|
|
|
|
None => {
|
|
let depth = depth + 1;
|
|
debug!(?depth, ?successor_node);
|
|
// Remember which node the return value will come from.
|
|
frame.successor_node = successor_node;
|
|
// Start a new stack frame the step into it.
|
|
stack.push(VisitingNodeFrame {
|
|
node: successor_node,
|
|
depth,
|
|
iter: None,
|
|
successors_len: 0,
|
|
min_depth: depth,
|
|
min_cycle_root: successor_node,
|
|
successor_node,
|
|
});
|
|
continue 'recurse;
|
|
}
|
|
}
|
|
}
|
|
|
|
// Completed walk, remove `node` from the stack.
|
|
let r = self.node_stack.pop();
|
|
debug_assert_eq!(r, Some(node));
|
|
|
|
// Remove the frame, it's done.
|
|
let frame = stack.pop().unwrap();
|
|
|
|
// If `min_depth == depth`, then we are the root of the
|
|
// cycle: we can't reach anyone further down the stack.
|
|
|
|
// Pass the 'return value' down the stack.
|
|
// We return one frame at a time so there can't be another return value.
|
|
debug_assert!(return_value.is_none());
|
|
return_value = Some(if frame.min_depth == depth {
|
|
// Note that successor stack may have duplicates, so we
|
|
// want to remove those:
|
|
let deduplicated_successors = {
|
|
let duplicate_set = &mut self.duplicate_set;
|
|
duplicate_set.clear();
|
|
successors_stack
|
|
.drain(successors_len..)
|
|
.filter(move |&i| duplicate_set.insert(i))
|
|
};
|
|
let scc_index = self.scc_data.create_scc(deduplicated_successors);
|
|
self.node_states[node] = NodeState::InCycle { scc_index };
|
|
WalkReturn::Complete { scc_index }
|
|
} else {
|
|
// We are not the head of the cycle. Return back to our
|
|
// caller. They will take ownership of the
|
|
// `self.successors` data that we pushed.
|
|
self.node_states[node] = NodeState::InCycleWith { parent: frame.min_cycle_root };
|
|
WalkReturn::Cycle { min_depth: frame.min_depth }
|
|
});
|
|
}
|
|
|
|
// Keep the allocation we used for successors_stack.
|
|
self.successors_stack = successors_stack;
|
|
debug_assert_eq!(self.successors_stack.len(), 0);
|
|
|
|
return_value.unwrap()
|
|
}
|
|
}
|