https://github.com/google/guava/guava: Guava is a suite of core and expanded libraries that include utility classes, google's collections, io classes, and much much more. 
/*
 * Copyright (C) 2014 The Guava Authors
 *
 * 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.
 */

package com.google.common.graph;

import static com.google.common.base.Preconditions.checkArgument;
import static com.google.common.graph.GraphConstants.NODE_NOT_IN_GRAPH;

import com.google.common.annotations.Beta;
import com.google.common.base.Objects;
import com.google.common.collect.Iterables;
import com.google.common.collect.Maps;
import com.google.errorprone.annotations.CanIgnoreReturnValue;
import java.util.ArrayDeque;
import java.util.Collection;
import java.util.Collections;
import java.util.HashSet;
import java.util.LinkedHashSet;
import java.util.Map;
import java.util.Optional;
import java.util.Queue;
import java.util.Set;
import org.checkerframework.checker.nullness.qual.Nullable;


Static utility methods for Graph, ValueGraph, and Network instances. 
Author:James Sexton, Joshua O'Madadhain
Since:20.0
/**
 * Static utility methods for {@link Graph}, {@link ValueGraph}, and {@link Network} instances.
 *
 * @author James Sexton
 * @author Joshua O'Madadhain
 * @since 20.0
 */
@Beta
public final class Graphs {

  private Graphs() {}

  // Graph query methods

  
Returns true if graph has at least one cycle. A cycle is defined as a non-empty subset of edges in a graph arranged to form a path (a sequence of adjacent outgoing edges) starting and ending with the same node. 
This method will detect any non-empty cycle, including self-loops (a cycle of length 1).

/**
   * Returns true if {@code graph} has at least one cycle. A cycle is defined as a non-empty subset
   * of edges in a graph arranged to form a path (a sequence of adjacent outgoing edges) starting
   * and ending with the same node.
   *
   * <p>This method will detect any non-empty cycle, including self-loops (a cycle of length 1).
   */
  public static <N> boolean hasCycle(Graph<N> graph) {
    int numEdges = graph.edges().size();
    if (numEdges == 0) {
      return false; // An edge-free graph is acyclic by definition.
    }
    if (!graph.isDirected() && numEdges >= graph.nodes().size()) {
      return true; // Optimization for the undirected case: at least one cycle must exist.
    }

    Map<Object, NodeVisitState> visitedNodes =
        Maps.newHashMapWithExpectedSize(graph.nodes().size());
    for (N node : graph.nodes()) {
      if (subgraphHasCycle(graph, visitedNodes, node, null)) {
        return true;
      }
    }
    return false;
  }

  
Returns true if network has at least one cycle. A cycle is defined as a non-empty subset of edges in a graph arranged to form a path (a sequence of adjacent outgoing edges) starting and ending with the same node. 
This method will detect any non-empty cycle, including self-loops (a cycle of length 1).

/**
   * Returns true if {@code network} has at least one cycle. A cycle is defined as a non-empty
   * subset of edges in a graph arranged to form a path (a sequence of adjacent outgoing edges)
   * starting and ending with the same node.
   *
   * <p>This method will detect any non-empty cycle, including self-loops (a cycle of length 1).
   */
  public static boolean hasCycle(Network<?, ?> network) {
    // In a directed graph, parallel edges cannot introduce a cycle in an acyclic graph.
    // However, in an undirected graph, any parallel edge induces a cycle in the graph.
    if (!network.isDirected()
        && network.allowsParallelEdges()
        && network.edges().size() > network.asGraph().edges().size()) {
      return true;
    }
    return hasCycle(network.asGraph());
  }

  
Performs a traversal of the nodes reachable from node. If we ever reach a node we've already visited (following only outgoing edges and without reusing edges), we know there's a cycle in the graph. 
/**
   * Performs a traversal of the nodes reachable from {@code node}. If we ever reach a node we've
   * already visited (following only outgoing edges and without reusing edges), we know there's a
   * cycle in the graph.
   */
  private static <N> boolean subgraphHasCycle(
      Graph<N> graph, Map<Object, NodeVisitState> visitedNodes, N node, @Nullable N previousNode) {
    NodeVisitState state = visitedNodes.get(node);
    if (state == NodeVisitState.COMPLETE) {
      return false;
    }
    if (state == NodeVisitState.PENDING) {
      return true;
    }

    visitedNodes.put(node, NodeVisitState.PENDING);
    for (N nextNode : graph.successors(node)) {
      if (canTraverseWithoutReusingEdge(graph, nextNode, previousNode)
          && subgraphHasCycle(graph, visitedNodes, nextNode, node)) {
        return true;
      }
    }
    visitedNodes.put(node, NodeVisitState.COMPLETE);
    return false;
  }

  
Determines whether an edge has already been used during traversal. In the directed case a cycle
is always detected before reusing an edge, so no special logic is required. In the undirected
case, we must take care not to "backtrack" over an edge (i.e. going from A to B and then going
from B to A).

/**
   * Determines whether an edge has already been used during traversal. In the directed case a cycle
   * is always detected before reusing an edge, so no special logic is required. In the undirected
   * case, we must take care not to "backtrack" over an edge (i.e. going from A to B and then going
   * from B to A).
   */
  private static boolean canTraverseWithoutReusingEdge(
      Graph<?> graph, Object nextNode, @Nullable Object previousNode) {
    if (graph.isDirected() || !Objects.equal(previousNode, nextNode)) {
      return true;
    }
    // This falls into the undirected A->B->A case. The Graph interface does not support parallel
    // edges, so this traversal would require reusing the undirected AB edge.
    return false;
  }

  
Returns the transitive closure of graph. The transitive closure of a graph is another graph with an edge connecting node A to node B if node B is reachable from node A. 
This is a "snapshot" based on the current topology of graph, rather than a live view of the transitive closure of graph. In other words, the returned Graph will not be updated after modifications to graph. 
/**
   * Returns the transitive closure of {@code graph}. The transitive closure of a graph is another
   * graph with an edge connecting node A to node B if node B is {@link #reachableNodes(Graph,
   * Object) reachable} from node A.
   *
   * <p>This is a "snapshot" based on the current topology of {@code graph}, rather than a live view
   * of the transitive closure of {@code graph}. In other words, the returned {@link Graph} will not
   * be updated after modifications to {@code graph}.
   */
  // TODO(b/31438252): Consider potential optimizations for this algorithm.
  public static <N> Graph<N> transitiveClosure(Graph<N> graph) {
    MutableGraph<N> transitiveClosure = GraphBuilder.from(graph).allowsSelfLoops(true).build();
    // Every node is, at a minimum, reachable from itself. Since the resulting transitive closure
    // will have no isolated nodes, we can skip adding nodes explicitly and let putEdge() do it.

    if (graph.isDirected()) {
      // Note: works for both directed and undirected graphs, but we only use in the directed case.
      for (N node : graph.nodes()) {
        for (N reachableNode : reachableNodes(graph, node)) {
          transitiveClosure.putEdge(node, reachableNode);
        }
      }
    } else {
      // An optimization for the undirected case: for every node B reachable from node A,
      // node A and node B have the same reachability set.
      Set<N> visitedNodes = new HashSet<N>();
      for (N node : graph.nodes()) {
        if (!visitedNodes.contains(node)) {
          Set<N> reachableNodes = reachableNodes(graph, node);
          visitedNodes.addAll(reachableNodes);
          int pairwiseMatch = 1; // start at 1 to include self-loops
          for (N nodeU : reachableNodes) {
            for (N nodeV : Iterables.limit(reachableNodes, pairwiseMatch++)) {
              transitiveClosure.putEdge(nodeU, nodeV);
            }
          }
        }
      }
    }

    return transitiveClosure;
  }

  
Returns the set of nodes that are reachable from node. Node B is defined as reachable from node A if there exists a path (a sequence of adjacent outgoing edges) starting at node A and ending at node B. Note that a node is always reachable from itself via a zero-length path. 
This is a "snapshot" based on the current topology of graph, rather than a live view of the set of nodes reachable from node. In other words, the returned Set will not be updated after modifications to graph. 
Throws:
/**
   * Returns the set of nodes that are reachable from {@code node}. Node B is defined as reachable
   * from node A if there exists a path (a sequence of adjacent outgoing edges) starting at node A
   * and ending at node B. Note that a node is always reachable from itself via a zero-length path.
   *
   * <p>This is a "snapshot" based on the current topology of {@code graph}, rather than a live view
   * of the set of nodes reachable from {@code node}. In other words, the returned {@link Set} will
   * not be updated after modifications to {@code graph}.
   *
   * @throws IllegalArgumentException if {@code node} is not present in {@code graph}
   */
  public static <N> Set<N> reachableNodes(Graph<N> graph, N node) {
    checkArgument(graph.nodes().contains(node), NODE_NOT_IN_GRAPH, node);
    Set<N> visitedNodes = new LinkedHashSet<N>();
    Queue<N> queuedNodes = new ArrayDeque<N>();
    visitedNodes.add(node);
    queuedNodes.add(node);
    // Perform a breadth-first traversal rooted at the input node.
    while (!queuedNodes.isEmpty()) {
      N currentNode = queuedNodes.remove();
      for (N successor : graph.successors(currentNode)) {
        if (visitedNodes.add(successor)) {
          queuedNodes.add(successor);
        }
      }
    }
    return Collections.unmodifiableSet(visitedNodes);
  }

  // Graph mutation methods

  // Graph view methods

  
Returns a view of graph with the direction (if any) of every edge reversed. All other properties remain intact, and further updates to graph will be reflected in the view. 
/**
   * Returns a view of {@code graph} with the direction (if any) of every edge reversed. All other
   * properties remain intact, and further updates to {@code graph} will be reflected in the view.
   */
  public static <N> Graph<N> transpose(Graph<N> graph) {
    if (!graph.isDirected()) {
      return graph; // the transpose of an undirected graph is an identical graph
    }

    if (graph instanceof TransposedGraph) {
      return ((TransposedGraph<N>) graph).graph;
    }

    return new TransposedGraph<N>(graph);
  }

  
Returns a view of graph with the direction (if any) of every edge reversed. All other properties remain intact, and further updates to graph will be reflected in the view. 
/**
   * Returns a view of {@code graph} with the direction (if any) of every edge reversed. All other
   * properties remain intact, and further updates to {@code graph} will be reflected in the view.
   */
  public static <N, V> ValueGraph<N, V> transpose(ValueGraph<N, V> graph) {
    if (!graph.isDirected()) {
      return graph; // the transpose of an undirected graph is an identical graph
    }

    if (graph instanceof TransposedValueGraph) {
      return ((TransposedValueGraph<N, V>) graph).graph;
    }

    return new TransposedValueGraph<>(graph);
  }

  
Returns a view of network with the direction (if any) of every edge reversed. All other properties remain intact, and further updates to network will be reflected in the view. 
/**
   * Returns a view of {@code network} with the direction (if any) of every edge reversed. All other
   * properties remain intact, and further updates to {@code network} will be reflected in the view.
   */
  public static <N, E> Network<N, E> transpose(Network<N, E> network) {
    if (!network.isDirected()) {
      return network; // the transpose of an undirected network is an identical network
    }

    if (network instanceof TransposedNetwork) {
      return ((TransposedNetwork<N, E>) network).network;
    }

    return new TransposedNetwork<>(network);
  }

  // NOTE: this should work as long as the delegate graph's implementation of edges() (like that of
  // AbstractGraph) derives its behavior from calling successors().
  private static class TransposedGraph<N> extends ForwardingGraph<N> {
    private final Graph<N> graph;

    TransposedGraph(Graph<N> graph) {
      this.graph = graph;
    }

    @Override
    protected Graph<N> delegate() {
      return graph;
    }

    @Override
    public Set<N> predecessors(N node) {
      return delegate().successors(node); // transpose
    }

    @Override
    public Set<N> successors(N node) {
      return delegate().predecessors(node); // transpose
    }

    @Override
    public int inDegree(N node) {
      return delegate().outDegree(node); // transpose
    }

    @Override
    public int outDegree(N node) {
      return delegate().inDegree(node); // transpose
    }

    @Override
    public boolean hasEdgeConnecting(N nodeU, N nodeV) {
      return delegate().hasEdgeConnecting(nodeV, nodeU); // transpose
    }
  }

  // NOTE: this should work as long as the delegate graph's implementation of edges() (like that of
  // AbstractValueGraph) derives its behavior from calling successors().
  private static class TransposedValueGraph<N, V> extends ForwardingValueGraph<N, V> {
    private final ValueGraph<N, V> graph;

    TransposedValueGraph(ValueGraph<N, V> graph) {
      this.graph = graph;
    }

    @Override
    protected ValueGraph<N, V> delegate() {
      return graph;
    }

    @Override
    public Set<N> predecessors(N node) {
      return delegate().successors(node); // transpose
    }

    @Override
    public Set<N> successors(N node) {
      return delegate().predecessors(node); // transpose
    }

    @Override
    public int inDegree(N node) {
      return delegate().outDegree(node); // transpose
    }

    @Override
    public int outDegree(N node) {
      return delegate().inDegree(node); // transpose
    }

    @Override
    public boolean hasEdgeConnecting(N nodeU, N nodeV) {
      return delegate().hasEdgeConnecting(nodeV, nodeU); // transpose
    }

    @Override
    public Optional<V> edgeValue(N nodeU, N nodeV) {
      return delegate().edgeValue(nodeV, nodeU); // transpose
    }

    @Override
    public @Nullable V edgeValueOrDefault(N nodeU, N nodeV, @Nullable V defaultValue) {
      return delegate().edgeValueOrDefault(nodeV, nodeU, defaultValue); // transpose
    }
  }

  private static class TransposedNetwork<N, E> extends ForwardingNetwork<N, E> {
    private final Network<N, E> network;

    TransposedNetwork(Network<N, E> network) {
      this.network = network;
    }

    @Override
    protected Network<N, E> delegate() {
      return network;
    }

    @Override
    public Set<N> predecessors(N node) {
      return delegate().successors(node); // transpose
    }

    @Override
    public Set<N> successors(N node) {
      return delegate().predecessors(node); // transpose
    }

    @Override
    public int inDegree(N node) {
      return delegate().outDegree(node); // transpose
    }

    @Override
    public int outDegree(N node) {
      return delegate().inDegree(node); // transpose
    }

    @Override
    public Set<E> inEdges(N node) {
      return delegate().outEdges(node); // transpose
    }

    @Override
    public Set<E> outEdges(N node) {
      return delegate().inEdges(node); // transpose
    }

    @Override
    public EndpointPair<N> incidentNodes(E edge) {
      EndpointPair<N> endpointPair = delegate().incidentNodes(edge);
      return EndpointPair.of(network, endpointPair.nodeV(), endpointPair.nodeU()); // transpose
    }

    @Override
    public Set<E> edgesConnecting(N nodeU, N nodeV) {
      return delegate().edgesConnecting(nodeV, nodeU); // transpose
    }

    @Override
    public Optional<E> edgeConnecting(N nodeU, N nodeV) {
      return delegate().edgeConnecting(nodeV, nodeU); // transpose
    }

    @Override
    public E edgeConnectingOrNull(N nodeU, N nodeV) {
      return delegate().edgeConnectingOrNull(nodeV, nodeU); // transpose
    }

    @Override
    public boolean hasEdgeConnecting(N nodeU, N nodeV) {
      return delegate().hasEdgeConnecting(nodeV, nodeU); // transpose
    }
  }

  // Graph copy methods

  
Returns the subgraph of graph induced by nodes. This subgraph is a new graph that contains all of the nodes in nodes, and all of the edges from graph for which both nodes are contained by nodes. 
Throws:
/**
   * Returns the subgraph of {@code graph} induced by {@code nodes}. This subgraph is a new graph
   * that contains all of the nodes in {@code nodes}, and all of the {@link Graph#edges() edges}
   * from {@code graph} for which both nodes are contained by {@code nodes}.
   *
   * @throws IllegalArgumentException if any element in {@code nodes} is not a node in the graph
   */
  public static <N> MutableGraph<N> inducedSubgraph(Graph<N> graph, Iterable<? extends N> nodes) {
    MutableGraph<N> subgraph =
        (nodes instanceof Collection)
            ? GraphBuilder.from(graph).expectedNodeCount(((Collection) nodes).size()).build()
            : GraphBuilder.from(graph).build();
    for (N node : nodes) {
      subgraph.addNode(node);
    }
    for (N node : subgraph.nodes()) {
      for (N successorNode : graph.successors(node)) {
        if (subgraph.nodes().contains(successorNode)) {
          subgraph.putEdge(node, successorNode);
        }
      }
    }
    return subgraph;
  }

  
Returns the subgraph of graph induced by nodes. This subgraph is a new graph that contains all of the nodes in nodes, and all of the edges (and associated edge values) from graph for which both nodes are contained by 
nodes. 
Throws:
/**
   * Returns the subgraph of {@code graph} induced by {@code nodes}. This subgraph is a new graph
   * that contains all of the nodes in {@code nodes}, and all of the {@link Graph#edges() edges}
   * (and associated edge values) from {@code graph} for which both nodes are contained by {@code
   * nodes}.
   *
   * @throws IllegalArgumentException if any element in {@code nodes} is not a node in the graph
   */
  public static <N, V> MutableValueGraph<N, V> inducedSubgraph(
      ValueGraph<N, V> graph, Iterable<? extends N> nodes) {
    MutableValueGraph<N, V> subgraph =
        (nodes instanceof Collection)
            ? ValueGraphBuilder.from(graph).expectedNodeCount(((Collection) nodes).size()).build()
            : ValueGraphBuilder.from(graph).build();
    for (N node : nodes) {
      subgraph.addNode(node);
    }
    for (N node : subgraph.nodes()) {
      for (N successorNode : graph.successors(node)) {
        if (subgraph.nodes().contains(successorNode)) {
          subgraph.putEdgeValue(
              node, successorNode, graph.edgeValueOrDefault(node, successorNode, null));
        }
      }
    }
    return subgraph;
  }

  
Returns the subgraph of network induced by nodes. This subgraph is a new graph that contains all of the nodes in nodes, and all of the edges from network for which the incident nodes are both contained by nodes. 
Throws:
/**
   * Returns the subgraph of {@code network} induced by {@code nodes}. This subgraph is a new graph
   * that contains all of the nodes in {@code nodes}, and all of the {@link Network#edges() edges}
   * from {@code network} for which the {@link Network#incidentNodes(Object) incident nodes} are
   * both contained by {@code nodes}.
   *
   * @throws IllegalArgumentException if any element in {@code nodes} is not a node in the graph
   */
  public static <N, E> MutableNetwork<N, E> inducedSubgraph(
      Network<N, E> network, Iterable<? extends N> nodes) {
    MutableNetwork<N, E> subgraph =
        (nodes instanceof Collection)
            ? NetworkBuilder.from(network).expectedNodeCount(((Collection) nodes).size()).build()
            : NetworkBuilder.from(network).build();
    for (N node : nodes) {
      subgraph.addNode(node);
    }
    for (N node : subgraph.nodes()) {
      for (E edge : network.outEdges(node)) {
        N successorNode = network.incidentNodes(edge).adjacentNode(node);
        if (subgraph.nodes().contains(successorNode)) {
          subgraph.addEdge(node, successorNode, edge);
        }
      }
    }
    return subgraph;
  }

  
Creates a mutable copy of graph with the same nodes and edges. 
/** Creates a mutable copy of {@code graph} with the same nodes and edges. */
  public static <N> MutableGraph<N> copyOf(Graph<N> graph) {
    MutableGraph<N> copy = GraphBuilder.from(graph).expectedNodeCount(graph.nodes().size()).build();
    for (N node : graph.nodes()) {
      copy.addNode(node);
    }
    for (EndpointPair<N> edge : graph.edges()) {
      copy.putEdge(edge.nodeU(), edge.nodeV());
    }
    return copy;
  }

  
Creates a mutable copy of graph with the same nodes, edges, and edge values. 
/** Creates a mutable copy of {@code graph} with the same nodes, edges, and edge values. */
  public static <N, V> MutableValueGraph<N, V> copyOf(ValueGraph<N, V> graph) {
    MutableValueGraph<N, V> copy =
        ValueGraphBuilder.from(graph).expectedNodeCount(graph.nodes().size()).build();
    for (N node : graph.nodes()) {
      copy.addNode(node);
    }
    for (EndpointPair<N> edge : graph.edges()) {
      copy.putEdgeValue(
          edge.nodeU(), edge.nodeV(), graph.edgeValueOrDefault(edge.nodeU(), edge.nodeV(), null));
    }
    return copy;
  }

  
Creates a mutable copy of network with the same nodes and edges. 
/** Creates a mutable copy of {@code network} with the same nodes and edges. */
  public static <N, E> MutableNetwork<N, E> copyOf(Network<N, E> network) {
    MutableNetwork<N, E> copy =
        NetworkBuilder.from(network)
            .expectedNodeCount(network.nodes().size())
            .expectedEdgeCount(network.edges().size())
            .build();
    for (N node : network.nodes()) {
      copy.addNode(node);
    }
    for (E edge : network.edges()) {
      EndpointPair<N> endpointPair = network.incidentNodes(edge);
      copy.addEdge(endpointPair.nodeU(), endpointPair.nodeV(), edge);
    }
    return copy;
  }

  @CanIgnoreReturnValue
  static int checkNonNegative(int value) {
    checkArgument(value >= 0, "Not true that %s is non-negative.", value);
    return value;
  }

  @CanIgnoreReturnValue
  static long checkNonNegative(long value) {
    checkArgument(value >= 0, "Not true that %s is non-negative.", value);
    return value;
  }

  @CanIgnoreReturnValue
  static int checkPositive(int value) {
    checkArgument(value > 0, "Not true that %s is positive.", value);
    return value;
  }

  @CanIgnoreReturnValue
  static long checkPositive(long value) {
    checkArgument(value > 0, "Not true that %s is positive.", value);
    return value;
  }

  
An enum representing the state of a node during DFS. PENDING means that the node is on the stack of the DFS, while COMPLETE means that the node and all its successors have been already explored. Any node that has not been explored will not have a state at all. 
/**
   * An enum representing the state of a node during DFS. {@code PENDING} means that the node is on
   * the stack of the DFS, while {@code COMPLETE} means that the node and all its successors have
   * been already explored. Any node that has not been explored will not have a state at all.
   */
  private enum NodeVisitState {
    PENDING,
    COMPLETE
  }
}