strongly_connected_components.h

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// Copyright 2010-2018 Google LLC
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

// This code computes the strongly connected components of a directed graph,
// and presents them sorted by reverse topological order.
//
// It implements an efficient version of Tarjan's strongly connected components
// algorithm published in: Tarjan, R. E. (1972), "Depth-first search and linear
// graph algorithms", SIAM Journal on Computing.
//
// A description can also be found here:
// http://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm
//
// SIMPLE EXAMPLE:
//
// Fill a std::vector<std::vector<int>> graph; representing your graph adjacency
// lists. That is, graph[i] contains the nodes adjacent to node #i. The nodes
// must be integers in [0, num_nodes). Then just do:
//
// std::vector<std::vector<int>> components;
// FindStronglyConnectedComponents(
//     static_cast<int>(graph.size()), graph, &components);
//
// The nodes of each strongly connected components will be listed in each
// subvector of components. The components appear in reverse topological order:
// outgoing arcs from a component will only be towards earlier components.
//
// IMPORTANT: num_nodes will be the number of nodes of the graph. Its type
// is the type used internally by the algorithm. It is why it is better to
// convert it to int or even int32 rather than using size_t which takes 64 bits.

#ifndef UTIL_GRAPH_STRONGLY_CONNECTED_COMPONENTS_H_
#define UTIL_GRAPH_STRONGLY_CONNECTED_COMPONENTS_H_

#include <limits>
#include <vector>

#include "ortools/base/logging.h"
#include "ortools/base/macros.h"

// Finds the strongly connected components of a directed graph. It is templated
// so it can be used in many contexts. See the simple example above for the
// easiest use case.
//
// The requirement of the different types are:
// - The type NodeIndex must be an integer type representing a node of the
//   graph. The nodes must be in [0, num_nodes). It can be unsigned.
// - The type Graph must provide a [] operator such that the following code
//   iterates over the adjacency list of the given node:
//     for (const NodeIndex head : graph[node]) {}
// - The type SccOutput must implement the function:
//     emplace_back(NodeIndex const* begin, NodeIndex const* end);
//   It will be called with the connected components of the given graph as they
//   are found (In the reverse topological order).
//
// More practical details on the algorithm:
// - It deals properly with self-loop and duplicate nodes.
// - It is really fast! and work in O(nodes + edges).
// - Its memory usage is also bounded by O(nodes + edges) but in practice it
//   uses less than the input graph.
template <typename NodeIndex, typename Graph, typename SccOutput>
void FindStronglyConnectedComponents(const NodeIndex num_nodes,
                                     const Graph& graph, SccOutput* components);

// A simple custom output class that just counts the number of SCC. Not
// allocating many vectors can save both space and speed if your graph is large.
//
// Note: If this matters, you probably don't want to use
// std::vector<std::vector<int>> as an input either. See StaticGraph in
// ortools/graph/graph.h for an efficient graph data structure compatible with
// this algorithm.
template <typename NodeIndex>
struct SccCounterOutput {
  int number_of_components = 0;
  void emplace_back(NodeIndex const* b, NodeIndex const* e) {
    ++number_of_components;
  }
  // This is just here so this class can transparently replace a code that
  // use std::vector<std::vector<int>> as an SccOutput, and get its size with
  // size().
  int size() const { return number_of_components; }
};

// This implementation is slightly different than a classical iterative version
// of Tarjan's strongly connected components algorithm. But basically it is
// still an iterative DFS.
//
// TODO(user): Possible optimizations:
// - Try to reserve the vectors which sizes are bounded by num_nodes.
// - Use an index rather than doing push_back(), pop_back() on them.
// - For a client needing many Scc computations one after another, it could be
//   better to wrap this in a class so we don't need to allocate the stacks at
//   each computation.
template <typename NodeIndex, typename Graph, typename SccOutput>
void FindStronglyConnectedComponents(const NodeIndex num_nodes,
                                     const Graph& graph,
                                     SccOutput* components) {
  // Each node expanded by the DFS will be pushed on this stack. A node is only
  // popped back when its strongly connected component has been explored and
  // outputted.
  std::vector<NodeIndex> scc_stack;

  // This is equivalent to the "low link" of a node in Tarjan's algorithm.
  // Basically, scc_start_index.back() represent the 1-based index in scc_stack
  // of the beginning of the current strongly connected component. All the
  // nodes after this index will be on the same component.
  std::vector<NodeIndex> scc_start_index;

  // Optimization. This will always be equal to scc_start_index.back() except
  // when scc_stack is empty, in which case its value does not matter.
  NodeIndex current_scc_start = 0;

  // Each node is assigned an index which changes 2 times in the algorithm:
  // - Everyone starts with an index of 0 which means unexplored.
  // - The first time they are explored by the DFS and pushed on scc_stack, they
  //   get their 1-based index on this stack.
  // - Once they have been processed and outputted to components, they are said
  //   to be settled, and their index become kSettledIndex.
  std::vector<NodeIndex> node_index(num_nodes, 0);
  constexpr NodeIndex kSettledIndex = std::numeric_limits<NodeIndex>::max();

  // This is a well known way to do an efficient iterative DFS. Each time a node
  // is explored, all its adjacent nodes are pushed on this stack. The iterative
  // dfs processes the nodes one by one by popping them back from here.
  std::vector<NodeIndex> node_to_process;

  // Loop over all the nodes not yet settled and start a DFS from each of them.
  for (NodeIndex base_node = 0; base_node < num_nodes; ++base_node) {
    if (node_index[base_node] != 0) continue;
    DCHECK_EQ(0, node_to_process.size());
    node_to_process.push_back(base_node);
    do {
      const NodeIndex node = node_to_process.back();
      const NodeIndex index = node_index[node];
      if (index == 0) {
        // We continue the dfs from this node and set its 1-based index.
        scc_stack.push_back(node);
        current_scc_start = scc_stack.size();
        node_index[node] = current_scc_start;
        scc_start_index.push_back(current_scc_start);

        // Enqueue all its adjacent nodes.
        NodeIndex min_head_index = kSettledIndex;
        for (const NodeIndex head : graph[node]) {
          const NodeIndex head_index = node_index[head];
          if (head_index == 0) {
            node_to_process.push_back(head);
          } else {
            // Note that if head_index == kSettledIndex, nothing happens.
            min_head_index = std::min(min_head_index, head_index);
          }
        }

        // Update the start of this strongly connected component.
        // Note that scc_start_index can never be empty since it first element
        // is 1 and by definition min_head_index is 1-based and can't be 0.
        while (current_scc_start > min_head_index) {
          scc_start_index.pop_back();
          current_scc_start = scc_start_index.back();
        }
      } else {
        node_to_process.pop_back();
        if (current_scc_start == index) {
          // We found a strongly connected component.
          components->emplace_back(&scc_stack[current_scc_start - 1],
                                   &scc_stack[0] + scc_stack.size());
          for (int i = current_scc_start - 1; i < scc_stack.size(); ++i) {
            node_index[scc_stack[i]] = kSettledIndex;
          }
          scc_stack.resize(current_scc_start - 1);
          scc_start_index.pop_back();
          current_scc_start =
              scc_start_index.empty() ? 0 : scc_start_index.back();
        }
      }
    } while (!node_to_process.empty());
  }
}

#endif  // UTIL_GRAPH_STRONGLY_CONNECTED_COMPONENTS_H_