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// Copyright 2020 Ferdinand Bachmann // // Licensed under the Apache License, Version 2.0, <LICENSE-APACHE or // http://apache.org/licenses/LICENSE-2.0> or the MIT license <LICENSE-MIT or // http://opensource.org/licenses/MIT>, at your option. This file may not be // copied, modified, or distributed except according to those terms. //! An implementation of //! [Kahn's algorithm](https://en.wikipedia.org/wiki/Topological_sorting) //! for topological sorting and //! [Kosaraju's algorithm](https://en.wikipedia.org/wiki/Kosaraju%27s_algorithm) //! for strongly connected components. //! //! This crate provides: //! //! - an adjacency-list based graph data structure (`IndexGraph`) //! - an implementation of a topological sorting algorithm that runs in //! `O(V + E)` time and `O(V)` additional space (Kahn's algorithm) //! - an implementation of an algorithm that finds the strongly connected //! components of a graph in `O(V + E)` time and `O(V)` additional space //! (Kosaraju's algorithm) //! - both algorithms are available either as single methods (`.toposort()` and //! `.scc()`) or as a combined method (`.toposort_or_scc()`) on `IndexGraph` //! //! The `id-arena` feature adds an additional wrapper type (`ArenaGraph`) that //! allows topological sorting and finding of strongly connected components on //! arbitrary graph structures built with the `id-arena` crate by creating a //! proxy graph that is sorted and returning a list of indices into the original //! graph. //! //! # Example //! //! This example creates an `IndexGraph` of the example graph from the //! Wikipedia page for //! [Topological sorting](https://en.wikipedia.org/wiki/Topological_sorting). //! //! A copy of the graph with cycles in it is created to demonstrate finding //! of strongly connected components. //! //! ```rust //! use toposort_scc::IndexGraph; //! //! let g = IndexGraph::from_adjacency_list(&vec![ //! vec![3], //! vec![3, 4], //! vec![4, 7], //! vec![5, 6, 7], //! vec![6], //! vec![], //! vec![], //! vec![] //! ]); //! //! let mut g2 = g.clone(); //! g2.add_edge(0, 0); // trivial cycle [0] //! g2.add_edge(6, 2); // cycle [2, 4, 6] //! //! assert_eq!(g.toposort_or_scc(), Ok(vec![0, 1, 2, 3, 4, 5, 7, 6])); //! assert_eq!(g2.toposort_or_scc(), Err(vec![vec![0], vec![4, 2, 6]])); //! ``` use std::collections::VecDeque as Queue; use std::vec::IntoIter as VecIntoIter; use std::slice::Iter as SliceIter; use std::ops::Index; use std::mem; #[cfg(feature = "id-arena")] mod arena_graph; #[cfg(feature = "id-arena")] pub use arena_graph::*; /// An adjacency-list-based graph data structure /// /// Stores graph vertices as lists of incoming and outgoing edges by their /// index in the graph. No additional data is stored per vertex. #[derive(Debug, Clone)] pub struct IndexGraph { vertices: Vec<Vertex>, } /// A vertex in an `IndexGraph` /// /// Every vertex stores the vertices it is connected to via edges in both /// directions. #[derive(Debug, Clone, Default)] pub struct Vertex { in_degree: usize, out_degree: usize, pub in_edges: Vec<usize>, pub out_edges: Vec<usize>, } /// A builder object that allows to easily add edges to a graph /// /// It stores a vertex index, so that edges can be added specifying only the /// target edge or source edge. /// /// See `IndexGraph::from_graph()` for usage examples #[derive(Debug)] pub struct IndexGraphBuilder<'g> { graph: &'g mut IndexGraph, index: usize } impl IndexGraphBuilder<'_> { /// Returns a reference to the stored graph pub fn as_graph(&self) -> &IndexGraph { self.graph } /// Returns a mutable reference to the stored graph pub fn as_mut_graph(&mut self) -> &mut IndexGraph { self.graph } /// Returns the stored index pub fn index(&self) -> usize { self.index } /// Add an edge from the stored index to the passed index /// /// This method does not check for duplicate edges. pub fn add_out_edge(&mut self, index: usize) { self.graph.add_edge(self.index, index) } /// Add an edge from the passed index to the stored index /// /// This method does not check for duplicate edges. pub fn add_in_edge(&mut self, index: usize) { self.graph.add_edge(index, self.index) } } impl IndexGraph { /// Create a new graph with `len` vertices and no edges /// /// Edges can then be added with the `.add_edge()` method. /// /// # Example /// /// This example creates a graph with three vertices connected together in a /// cycle. /// /// ```rust /// use toposort_scc::IndexGraph; /// /// let mut g = IndexGraph::with_vertices(3); /// g.add_edge(0, 1); /// g.add_edge(1, 2); /// g.add_edge(2, 0); /// ``` pub fn with_vertices(len: usize) -> Self { let mut vertices = Vec::with_capacity(len); vertices.resize_with(len, Default::default); IndexGraph { vertices } } /// Create a new graph from a list of adjacent vertices /// /// The graph will contain outgoing edges from each vertex to the vertices /// in its adjacency list. /// /// # Example /// /// This example creates an `IndexGraph` of the example graph from the /// Wikipedia page for /// [Topological sorting](https://en.wikipedia.org/wiki/Topological_sorting). /// /// ```rust /// use toposort_scc::IndexGraph; /// /// let g = IndexGraph::from_adjacency_list(&vec![ /// vec![3], /// vec![3, 4], /// vec![4, 7], /// vec![5, 6, 7], /// vec![6], /// vec![], /// vec![], /// vec![] /// ]); /// ``` pub fn from_adjacency_list<S>(g: &[S]) -> Self where S: AsRef<[usize]> { IndexGraph::from_graph(g, |mut builder, edges| for &edge in edges.as_ref() { builder.add_out_edge(edge) }) } /// Create a new graph from an existing graph-like data structure /// /// The given closure will be called once for every element of `g`, with an /// `IndexGraphBuilder` instance so that edges can be easily added. /// /// This method is useful for creating `IndexGraphs` from existing /// structures. /// /// # Example /// /// This example creates a graph of dependencies in a hypothetical compiler /// or build tool, with edges from a dependency to the targets that use /// them. /// /// ```rust /// use toposort_scc::IndexGraph; /// /// // a target during compilation, having a name and dependencies /// struct Target { name: &'static str, deps: Vec<usize> } /// impl Target { /// fn new(name: &'static str, deps: Vec<usize>) -> Self { /// Target { name, deps } /// } /// } /// /// let targets = vec![ /// Target::new("program", vec![1, 2, 4]), /// Target::new("main.c", vec![3]), /// Target::new("util.c", vec![3]), /// Target::new("util.h", vec![]), /// Target::new("libfoo.so", vec![]) /// ]; /// /// let g = IndexGraph::from_graph(&targets, |mut builder, target| { /// for &dep in &target.deps { /// builder.add_in_edge(dep); /// } /// }); /// ``` /// /// To get a graph with edges in the other direction, use `.add_out_edge()` /// or the `.transpose()` method of the graph. /// /// More complicated graph structures or structures that don't store edges /// as indices will need to first create a `Map` to look up indices in. /// Alternatively, the `ArenaGraph` type from the `id-arena` feature can be /// used. pub fn from_graph<T, F>(g: &[T], mut f: F) -> Self where F: FnMut(IndexGraphBuilder<'_>, &T) { let mut graph = Self::with_vertices(g.len()); for (idx, element) in g.iter().enumerate() { f(IndexGraphBuilder { graph: &mut graph, index: idx }, element) } graph } /// Returns an iterator over the contained vertices pub fn iter(&self) -> SliceIter<'_, Vertex> { self.vertices.iter() } /// Add a new edge to the graph /// /// This method does not check for duplicate edges. pub fn add_edge(&mut self, from: usize, to: usize) { self.vertices[from].out_degree += 1; self.vertices[to].in_degree += 1; self.vertices[from].out_edges.push(to); self.vertices[to].in_edges.push(from); } /// Transpose the graph /// /// Inverts the direction of all edges in the graph pub fn transpose(&mut self) { for vertex in &mut self.vertices { mem::swap(&mut vertex.in_degree, &mut vertex.out_degree); mem::swap(&mut vertex.in_edges, &mut vertex.out_edges); } } /// Internal method that attempts to perform topological sort /// /// If the graph contains no cycles, finds the topological ordering of this /// graph using Kahn's algorithm and returns it as `Ok(sorted)`. /// /// If the graph contains cycles, returns the graph as Err(self). /// /// This method is not public because it breaks the invariants of /// `Vertex.in_degree` and `Vertex.out_degree` /// /// This method sets `Vertex.out_degree` to zero for every vertex so that /// the precondition of `IndexGraph::scc_internal()` is fulfilled fn try_toposort_internal(mut self) -> Result<Vec<usize>, IndexGraph> { let mut queue = Queue::new(); let mut sorted = Vec::new(); // Kahn's algorithm for toposort // enqueue vertices with in-degree zero for (idx, vertex) in self.vertices.iter_mut().enumerate() { // out_degree is unused in this algorithm // set out_degree to zero to be used as a 'visited' flag by // Kosaraju's algorithm later vertex.out_degree = 0; if vertex.in_degree == 0 { queue.push_back(idx); } } // add vertices from queue to sorted list // decrement in-degree of neighboring edges // add to queue if in-degree zero while let Some(idx) = queue.pop_front() { sorted.push(idx); for edge_idx in 0..self.vertices[idx].out_edges.len() { let next_idx = self.vertices[idx].out_edges[edge_idx]; self.vertices[next_idx].in_degree -= 1; if self.vertices[next_idx].in_degree == 0 { queue.push_back(next_idx); } } } // if every vertex appears in sorted list, sort is successful if sorted.len() == self.vertices.len() { Ok(sorted) } else { Err(self) } } /// Try to perform topological sort on the graph /// /// If the graph contains no cycles, finds the topological ordering of this /// graph using Kahn's algorithm and returns it as `Ok(sorted)`. /// /// If the graph contains cycles, returns the graph as `Err(self)`. /// /// For examples, see `IndexGraph::toposort()` pub fn try_toposort(self) -> Result<Vec<usize>, IndexGraph> { self.try_toposort_internal() .map_err(|mut graph| { for vertex in graph.vertices.iter_mut() { vertex.in_degree = vertex.in_edges.len(); vertex.out_degree = vertex.out_edges.len(); } graph }) } /// Perform topological sort on the graph /// /// If the graph contains no cycles, finds the topological ordering of this /// graph using Kahn's algorithm and returns it as `Some(sorted)`. /// /// If the graph contains cycles, returns `None`. /// /// # Example /// /// This example creates an `IndexGraph` of the example graph from the /// Wikipedia page for /// [Topological sorting](https://en.wikipedia.org/wiki/Topological_sorting) /// and performs a topological sort. /// /// A copy of the graph with cycles in it is created to demonstrate failure. /// /// ```rust /// use toposort_scc::IndexGraph; /// /// let g = IndexGraph::from_adjacency_list(&vec![ /// vec![3], /// vec![3, 4], /// vec![4, 7], /// vec![5, 6, 7], /// vec![6], /// vec![], /// vec![], /// vec![] /// ]); /// /// let mut g2 = g.clone(); /// g2.add_edge(0, 0); // trivial cycle [0] /// g2.add_edge(6, 2); // cycle [2, 4, 6] /// /// assert_eq!(g.toposort(), Some(vec![0, 1, 2, 3, 4, 5, 7, 6])); /// assert_eq!(g2.toposort(), None); /// ``` pub fn toposort(self) -> Option<Vec<usize>> { self.try_toposort_internal().ok() } /// Internal method that finds strongly connected components /// /// Finds the strongly connected components of this graph using Kosaraju's /// algorithm and returns them. /// /// This method is not public because it assumes `Vertex.out_degree` is /// zero for every vertex. fn scc_internal(mut self) -> Vec<Vec<usize>> { // assumes out_degree is zero everywhere, to be used as a 'visited' flag // empty graphs are always cycle-free if self.vertices.is_empty() { return Vec::new() } // Kosaraju's algorithm for strongly connected components // start depth-first search with first vertex let mut queue = Queue::new(); let mut dfs_stack = vec![(0, 0)]; self.vertices[0].out_degree = 1; // add vertices to queue in post-order while let Some((idx, edge_idx)) = dfs_stack.pop() { if edge_idx < self.vertices[idx].out_edges.len() { dfs_stack.push((idx, edge_idx + 1)); let next_idx = self.vertices[idx].out_edges[edge_idx]; if self.vertices[next_idx].out_degree == 0 { self.vertices[next_idx].out_degree = 1; dfs_stack.push((next_idx, 0)); } } else { queue.push_back(idx); } } // collect cycles by depth-first search in opposite edge direction // from each vertex in queue let mut cycles = Vec::new(); while let Some(root_idx) = queue.pop_back() { if self.vertices[root_idx].out_degree == 2 { continue } let mut cur_cycle = Vec::new(); dfs_stack.push((root_idx, 0)); while let Some((idx, edge_idx)) = dfs_stack.pop() { if edge_idx < self.vertices[idx].in_edges.len() { dfs_stack.push((idx, edge_idx + 1)); let next_idx = self.vertices[idx].in_edges[edge_idx]; if self.vertices[next_idx].out_degree == 1 { self.vertices[next_idx].out_degree = 2; dfs_stack.push((self.vertices[idx].in_edges[edge_idx], 0)); cur_cycle.push(next_idx); } } } if self.vertices[root_idx].out_degree == 2 { cycles.push(cur_cycle); } else { self.vertices[root_idx].out_degree = 2; } } // return collected cycles cycles } /// Find strongly connected components /// /// Finds the strongly connected components of this graph using Kosaraju's /// algorithm and returns them. /// /// # Example /// /// This examples creates an `IndexGraph` of the example graph from the /// Wikipedia page for /// [Strongly connected compoents](https://en.wikipedia.org/wiki/Strongly_connected_component) /// and finds the strongly connected components. /// /// ```rust /// use toposort_scc::IndexGraph; /// /// let g = IndexGraph::from_adjacency_list(&vec![ /// vec![1], /// vec![2, 4, 5], /// vec![3, 6], /// vec![2, 7], /// vec![0, 5], /// vec![6], /// vec![5], /// vec![3, 6] /// ]); /// /// assert_eq!(g.scc(), vec![vec![4, 1, 0], vec![3, 2, 7], vec![5, 6]]); /// ``` pub fn scc(mut self) -> Vec<Vec<usize>> { for vertex in self.vertices.iter_mut() { vertex.out_degree = 0; } self.scc_internal() } /// Perform topological sort or find strongly connected components /// /// If the graph contains no cycles, finds the topological ordering of this /// graph using Kahn's algorithm and returns it as `Ok(sorted)`. /// /// If the graph contains cycles, finds the strongly connected components of /// this graph using Kosaraju's algorithm and returns them as `Err(cycles)`. /// /// # Example /// /// This example creates an `IndexGraph` of the example graph from the /// Wikipedia page for /// [Topological sorting](https://en.wikipedia.org/wiki/Topological_sorting) /// and performs a topological sort. /// /// A copy of the graph with cycles in it is created to demonstrate finding /// of strongly connected components. /// /// ```rust /// use toposort_scc::IndexGraph; /// /// let g = IndexGraph::from_adjacency_list(&vec![ /// vec![3], /// vec![3, 4], /// vec![4, 7], /// vec![5, 6, 7], /// vec![6], /// vec![], /// vec![], /// vec![] /// ]); /// /// let mut g2 = g.clone(); /// g2.add_edge(0, 0); // trivial cycle [0] /// g2.add_edge(6, 2); // cycle [2, 4, 6] /// /// assert_eq!(g.toposort_or_scc(), Ok(vec![0, 1, 2, 3, 4, 5, 7, 6])); /// assert_eq!(g2.toposort_or_scc(), Err(vec![vec![0], vec![4, 2, 6]])); /// ``` pub fn toposort_or_scc(self) -> Result<Vec<usize>, Vec<Vec<usize>>> { match self.try_toposort_internal() { Ok(sorted) => Ok(sorted), Err(graph) => Err(graph.scc_internal()) } } } impl Index<usize> for IndexGraph { type Output = Vertex; fn index(&self, index: usize) -> &Vertex { &self.vertices[index] } } impl<'g> IntoIterator for &'g IndexGraph { type Item = &'g Vertex; type IntoIter = SliceIter<'g, Vertex>; fn into_iter(self) -> Self::IntoIter { self.vertices.iter() } } impl IntoIterator for IndexGraph { type Item = Vertex; type IntoIter = VecIntoIter<Vertex>; fn into_iter(self) -> Self::IntoIter { self.vertices.into_iter() } }