android/guava/src/com/google/common/collect/Sets.java

/*
 * Copyright (C) 2007 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.collect;

import static com.google.common.base.Preconditions.checkArgument;
import static com.google.common.base.Preconditions.checkNotNull;
import static com.google.common.collect.CollectPreconditions.checkNonnegative;

import com.google.common.annotations.Beta;
import com.google.common.annotations.GwtCompatible;
import com.google.common.annotations.GwtIncompatible;
import com.google.common.base.Predicate;
import com.google.common.base.Predicates;
import com.google.common.collect.Collections2.FilteredCollection;
import com.google.common.math.IntMath;
import com.google.errorprone.annotations.CanIgnoreReturnValue;
import java.io.Serializable;
import java.util.AbstractSet;
import java.util.Arrays;
import java.util.BitSet;
import java.util.Collection;
import java.util.Collections;
import java.util.Comparator;
import java.util.EnumSet;
import java.util.HashSet;
import java.util.Iterator;
import java.util.LinkedHashSet;
import java.util.List;
import java.util.Map;
import java.util.NavigableSet;
import java.util.NoSuchElementException;
import java.util.Set;
import java.util.SortedSet;
import java.util.TreeSet;
import java.util.concurrent.ConcurrentHashMap;
import java.util.concurrent.CopyOnWriteArraySet;
import org.checkerframework.checker.nullness.compatqual.MonotonicNonNullDecl;
import org.checkerframework.checker.nullness.compatqual.NullableDecl;

/**
 * Static utility methods pertaining to {@link Set} instances. Also see this class's counterparts
 * {@link Lists}, {@link Maps} and {@link Queues}.
 *
 * <p>See the Guava User Guide article on <a href=
 * "https://github.com/google/guava/wiki/CollectionUtilitiesExplained#sets"> {@code Sets}</a>.
 *
 * @author Kevin Bourrillion
 * @author Jared Levy
 * @author Chris Povirk
 * @since 2.0
 */
@GwtCompatible(emulated = true)
public final class Sets {
  private Sets() {}

  /**
   * {@link AbstractSet} substitute without the potentially-quadratic {@code removeAll}
   * implementation.
   */
  abstract static class ImprovedAbstractSet<E> extends AbstractSet<E> {
    @Override
    public boolean removeAll(Collection<?> c) {
      return removeAllImpl(this, c);
    }

    @Override
    public boolean retainAll(Collection<?> c) {
      return super.retainAll(checkNotNull(c)); // GWT compatibility
    }
  }

  /**
   * Returns an immutable set instance containing the given enum elements. Internally, the returned
   * set will be backed by an {@link EnumSet}.
   *
   * <p>The iteration order of the returned set follows the enum's iteration order, not the order in
   * which the elements are provided to the method.
   *
   * @param anElement one of the elements the set should contain
   * @param otherElements the rest of the elements the set should contain
   * @return an immutable set containing those elements, minus duplicates
   */
  // http://code.google.com/p/google-web-toolkit/issues/detail?id=3028
  @GwtCompatible(serializable = true)
  public static <E extends Enum<E>> ImmutableSet<E> immutableEnumSet(
      E anElement, E... otherElements) {
    return ImmutableEnumSet.asImmutable(EnumSet.of(anElement, otherElements));
  }

  /**
   * Returns an immutable set instance containing the given enum elements. Internally, the returned
   * set will be backed by an {@link EnumSet}.
   *
   * <p>The iteration order of the returned set follows the enum's iteration order, not the order in
   * which the elements appear in the given collection.
   *
   * @param elements the elements, all of the same {@code enum} type, that the set should contain
   * @return an immutable set containing those elements, minus duplicates
   */
  // http://code.google.com/p/google-web-toolkit/issues/detail?id=3028
  @GwtCompatible(serializable = true)
  public static <E extends Enum<E>> ImmutableSet<E> immutableEnumSet(Iterable<E> elements) {
    if (elements instanceof ImmutableEnumSet) {
      return (ImmutableEnumSet<E>) elements;
    } else if (elements instanceof Collection) {
      Collection<E> collection = (Collection<E>) elements;
      if (collection.isEmpty()) {
        return ImmutableSet.of();
      } else {
        return ImmutableEnumSet.asImmutable(EnumSet.copyOf(collection));
      }
    } else {
      Iterator<E> itr = elements.iterator();
      if (itr.hasNext()) {
        EnumSet<E> enumSet = EnumSet.of(itr.next());
        Iterators.addAll(enumSet, itr);
        return ImmutableEnumSet.asImmutable(enumSet);
      } else {
        return ImmutableSet.of();
      }
    }
  }

  /**
   * Returns a new, <i>mutable</i> {@code EnumSet} instance containing the given elements in their
   * natural order. This method behaves identically to {@link EnumSet#copyOf(Collection)}, but also
   * accepts non-{@code Collection} iterables and empty iterables.
   */
  public static <E extends Enum<E>> EnumSet<E> newEnumSet(
      Iterable<E> iterable, Class<E> elementType) {
    EnumSet<E> set = EnumSet.noneOf(elementType);
    Iterables.addAll(set, iterable);
    return set;
  }

  // HashSet

  /**
   * Creates a <i>mutable</i>, initially empty {@code HashSet} instance.
   *
   * <p><b>Note:</b> if mutability is not required, use {@link ImmutableSet#of()} instead. If {@code
   * E} is an {@link Enum} type, use {@link EnumSet#noneOf} instead. Otherwise, strongly consider
   * using a {@code LinkedHashSet} instead, at the cost of increased memory footprint, to get
   * deterministic iteration behavior.
   *
   * <p><b>Note for Java 7 and later:</b> this method is now unnecessary and should be treated as
   * deprecated. Instead, use the {@code HashSet} constructor directly, taking advantage of the new
   * <a href="http://goo.gl/iz2Wi">"diamond" syntax</a>.
   */
  public static <E> HashSet<E> newHashSet() {
    return new HashSet<E>();
  }

  /**
   * Creates a <i>mutable</i> {@code HashSet} instance initially containing the given elements.
   *
   * <p><b>Note:</b> if elements are non-null and won't be added or removed after this point, use
   * {@link ImmutableSet#of()} or {@link ImmutableSet#copyOf(Object[])} instead. If {@code E} is an
   * {@link Enum} type, use {@link EnumSet#of(Enum, Enum[])} instead. Otherwise, strongly consider
   * using a {@code LinkedHashSet} instead, at the cost of increased memory footprint, to get
   * deterministic iteration behavior.
   *
   * <p>This method is just a small convenience, either for {@code newHashSet(}{@link Arrays#asList
   * asList}{@code (...))}, or for creating an empty set then calling {@link Collections#addAll}.
   * This method is not actually very useful and will likely be deprecated in the future.
   */
  public static <E> HashSet<E> newHashSet(E... elements) {
    HashSet<E> set = newHashSetWithExpectedSize(elements.length);
    Collections.addAll(set, elements);
    return set;
  }

  /**
   * Creates a <i>mutable</i> {@code HashSet} instance containing the given elements. A very thin
   * convenience for creating an empty set then calling {@link Collection#addAll} or {@link
   * Iterables#addAll}.
   *
   * <p><b>Note:</b> if mutability is not required and the elements are non-null, use {@link
   * ImmutableSet#copyOf(Iterable)} instead. (Or, change {@code elements} to be a {@link
   * FluentIterable} and call {@code elements.toSet()}.)
   *
   * <p><b>Note:</b> if {@code E} is an {@link Enum} type, use {@link #newEnumSet(Iterable, Class)}
   * instead.
   *
   * <p><b>Note for Java 7 and later:</b> if {@code elements} is a {@link Collection}, you don't
   * need this method. Instead, use the {@code HashSet} constructor directly, taking advantage of
   * the new <a href="http://goo.gl/iz2Wi">"diamond" syntax</a>.
   *
   * <p>Overall, this method is not very useful and will likely be deprecated in the future.
   */
  public static <E> HashSet<E> newHashSet(Iterable<? extends E> elements) {
    return (elements instanceof Collection)
        ? new HashSet<E>(Collections2.cast(elements))
        : newHashSet(elements.iterator());
  }

  /**
   * Creates a <i>mutable</i> {@code HashSet} instance containing the given elements. A very thin
   * convenience for creating an empty set and then calling {@link Iterators#addAll}.
   *
   * <p><b>Note:</b> if mutability is not required and the elements are non-null, use {@link
   * ImmutableSet#copyOf(Iterator)} instead.
   *
   * <p><b>Note:</b> if {@code E} is an {@link Enum} type, you should create an {@link EnumSet}
   * instead.
   *
   * <p>Overall, this method is not very useful and will likely be deprecated in the future.
   */
  public static <E> HashSet<E> newHashSet(Iterator<? extends E> elements) {
    HashSet<E> set = newHashSet();
    Iterators.addAll(set, elements);
    return set;
  }

  /**
   * Returns a new hash set using the smallest initial table size that can hold {@code expectedSize}
   * elements without resizing. Note that this is not what {@link HashSet#HashSet(int)} does, but it
   * is what most users want and expect it to do.
   *
   * <p>This behavior can't be broadly guaranteed, but has been tested with OpenJDK 1.7 and 1.8.
   *
   * @param expectedSize the number of elements you expect to add to the returned set
   * @return a new, empty hash set with enough capacity to hold {@code expectedSize} elements
   *     without resizing
   * @throws IllegalArgumentException if {@code expectedSize} is negative
   */
  public static <E> HashSet<E> newHashSetWithExpectedSize(int expectedSize) {
    return new HashSet<E>(Maps.capacity(expectedSize));
  }

  /**
   * Creates a thread-safe set backed by a hash map. The set is backed by a {@link
   * ConcurrentHashMap} instance, and thus carries the same concurrency guarantees.
   *
   * <p>Unlike {@code HashSet}, this class does NOT allow {@code null} to be used as an element. The
   * set is serializable.
   *
   * @return a new, empty thread-safe {@code Set}
   * @since 15.0
   */
  public static <E> Set<E> newConcurrentHashSet() {
    return Collections.newSetFromMap(new ConcurrentHashMap<E, Boolean>());
  }

  /**
   * Creates a thread-safe set backed by a hash map and containing the given elements. The set is
   * backed by a {@link ConcurrentHashMap} instance, and thus carries the same concurrency
   * guarantees.
   *
   * <p>Unlike {@code HashSet}, this class does NOT allow {@code null} to be used as an element. The
   * set is serializable.
   *
   * @param elements the elements that the set should contain
   * @return a new thread-safe set containing those elements (minus duplicates)
   * @throws NullPointerException if {@code elements} or any of its contents is null
   * @since 15.0
   */
  public static <E> Set<E> newConcurrentHashSet(Iterable<? extends E> elements) {
    Set<E> set = newConcurrentHashSet();
    Iterables.addAll(set, elements);
    return set;
  }

  // LinkedHashSet

  /**
   * Creates a <i>mutable</i>, empty {@code LinkedHashSet} instance.
   *
   * <p><b>Note:</b> if mutability is not required, use {@link ImmutableSet#of()} instead.
   *
   * <p><b>Note for Java 7 and later:</b> this method is now unnecessary and should be treated as
   * deprecated. Instead, use the {@code LinkedHashSet} constructor directly, taking advantage of
   * the new <a href="http://goo.gl/iz2Wi">"diamond" syntax</a>.
   *
   * @return a new, empty {@code LinkedHashSet}
   */
  public static <E> LinkedHashSet<E> newLinkedHashSet() {
    return new LinkedHashSet<E>();
  }

  /**
   * Creates a <i>mutable</i> {@code LinkedHashSet} instance containing the given elements in order.
   *
   * <p><b>Note:</b> if mutability is not required and the elements are non-null, use {@link
   * ImmutableSet#copyOf(Iterable)} instead.
   *
   * <p><b>Note for Java 7 and later:</b> if {@code elements} is a {@link Collection}, you don't
   * need this method. Instead, use the {@code LinkedHashSet} constructor directly, taking advantage
   * of the new <a href="http://goo.gl/iz2Wi">"diamond" syntax</a>.
   *
   * <p>Overall, this method is not very useful and will likely be deprecated in the future.
   *
   * @param elements the elements that the set should contain, in order
   * @return a new {@code LinkedHashSet} containing those elements (minus duplicates)
   */
  public static <E> LinkedHashSet<E> newLinkedHashSet(Iterable<? extends E> elements) {
    if (elements instanceof Collection) {
      return new LinkedHashSet<E>(Collections2.cast(elements));
    }
    LinkedHashSet<E> set = newLinkedHashSet();
    Iterables.addAll(set, elements);
    return set;
  }

  /**
   * Creates a {@code LinkedHashSet} instance, with a high enough "initial capacity" that it
   * <i>should</i> hold {@code expectedSize} elements without growth. This behavior cannot be
   * broadly guaranteed, but it is observed to be true for OpenJDK 1.7. It also can't be guaranteed
   * that the method isn't inadvertently <i>oversizing</i> the returned set.
   *
   * @param expectedSize the number of elements you expect to add to the returned set
   * @return a new, empty {@code LinkedHashSet} with enough capacity to hold {@code expectedSize}
   *     elements without resizing
   * @throws IllegalArgumentException if {@code expectedSize} is negative
   * @since 11.0
   */
  public static <E> LinkedHashSet<E> newLinkedHashSetWithExpectedSize(int expectedSize) {
    return new LinkedHashSet<E>(Maps.capacity(expectedSize));
  }

  // TreeSet

  /**
   * Creates a <i>mutable</i>, empty {@code TreeSet} instance sorted by the natural sort ordering of
   * its elements.
   *
   * <p><b>Note:</b> if mutability is not required, use {@link ImmutableSortedSet#of()} instead.
   *
   * <p><b>Note for Java 7 and later:</b> this method is now unnecessary and should be treated as
   * deprecated. Instead, use the {@code TreeSet} constructor directly, taking advantage of the new
   * <a href="http://goo.gl/iz2Wi">"diamond" syntax</a>.
   *
   * @return a new, empty {@code TreeSet}
   */
  public static <E extends Comparable> TreeSet<E> newTreeSet() {
    return new TreeSet<E>();
  }

  /**
   * Creates a <i>mutable</i> {@code TreeSet} instance containing the given elements sorted by their
   * natural ordering.
   *
   * <p><b>Note:</b> if mutability is not required, use {@link ImmutableSortedSet#copyOf(Iterable)}
   * instead.
   *
   * <p><b>Note:</b> If {@code elements} is a {@code SortedSet} with an explicit comparator, this
   * method has different behavior than {@link TreeSet#TreeSet(SortedSet)}, which returns a {@code
   * TreeSet} with that comparator.
   *
   * <p><b>Note for Java 7 and later:</b> this method is now unnecessary and should be treated as
   * deprecated. Instead, use the {@code TreeSet} constructor directly, taking advantage of the new
   * <a href="http://goo.gl/iz2Wi">"diamond" syntax</a>.
   *
   * <p>This method is just a small convenience for creating an empty set and then calling {@link
   * Iterables#addAll}. This method is not very useful and will likely be deprecated in the future.
   *
   * @param elements the elements that the set should contain
   * @return a new {@code TreeSet} containing those elements (minus duplicates)
   */
  public static <E extends Comparable> TreeSet<E> newTreeSet(Iterable<? extends E> elements) {
    TreeSet<E> set = newTreeSet();
    Iterables.addAll(set, elements);
    return set;
  }

  /**
   * Creates a <i>mutable</i>, empty {@code TreeSet} instance with the given comparator.
   *
   * <p><b>Note:</b> if mutability is not required, use {@code
   * ImmutableSortedSet.orderedBy(comparator).build()} instead.
   *
   * <p><b>Note for Java 7 and later:</b> this method is now unnecessary and should be treated as
   * deprecated. Instead, use the {@code TreeSet} constructor directly, taking advantage of the new
   * <a href="http://goo.gl/iz2Wi">"diamond" syntax</a>. One caveat to this is that the {@code
   * TreeSet} constructor uses a null {@code Comparator} to mean "natural ordering," whereas this
   * factory rejects null. Clean your code accordingly.
   *
   * @param comparator the comparator to use to sort the set
   * @return a new, empty {@code TreeSet}
   * @throws NullPointerException if {@code comparator} is null
   */
  public static <E> TreeSet<E> newTreeSet(Comparator<? super E> comparator) {
    return new TreeSet<E>(checkNotNull(comparator));
  }

  /**
   * Creates an empty {@code Set} that uses identity to determine equality. It compares object
   * references, instead of calling {@code equals}, to determine whether a provided object matches
   * an element in the set. For example, {@code contains} returns {@code false} when passed an
   * object that equals a set member, but isn't the same instance. This behavior is similar to the
   * way {@code IdentityHashMap} handles key lookups.
   *
   * @since 8.0
   */
  public static <E> Set<E> newIdentityHashSet() {
    return Collections.newSetFromMap(Maps.<E, Boolean>newIdentityHashMap());
  }

  /**
   * Creates an empty {@code CopyOnWriteArraySet} instance.
   *
   * <p><b>Note:</b> if you need an immutable empty {@link Set}, use {@link Collections#emptySet}
   * instead.
   *
   * @return a new, empty {@code CopyOnWriteArraySet}
   * @since 12.0
   */
  @GwtIncompatible // CopyOnWriteArraySet
  public static <E> CopyOnWriteArraySet<E> newCopyOnWriteArraySet() {
    return new CopyOnWriteArraySet<E>();
  }

  /**
   * Creates a {@code CopyOnWriteArraySet} instance containing the given elements.
   *
   * @param elements the elements that the set should contain, in order
   * @return a new {@code CopyOnWriteArraySet} containing those elements
   * @since 12.0
   */
  @GwtIncompatible // CopyOnWriteArraySet
  public static <E> CopyOnWriteArraySet<E> newCopyOnWriteArraySet(Iterable<? extends E> elements) {
    // We copy elements to an ArrayList first, rather than incurring the
    // quadratic cost of adding them to the COWAS directly.
    Collection<? extends E> elementsCollection =
        (elements instanceof Collection)
            ? Collections2.cast(elements)
            : Lists.newArrayList(elements);
    return new CopyOnWriteArraySet<E>(elementsCollection);
  }

  /**
   * Creates an {@code EnumSet} consisting of all enum values that are not in the specified
   * collection. If the collection is an {@link EnumSet}, this method has the same behavior as
   * {@link EnumSet#complementOf}. Otherwise, the specified collection must contain at least one
   * element, in order to determine the element type. If the collection could be empty, use {@link
   * #complementOf(Collection, Class)} instead of this method.
   *
   * @param collection the collection whose complement should be stored in the enum set
   * @return a new, modifiable {@code EnumSet} containing all values of the enum that aren't present
   *     in the given collection
   * @throws IllegalArgumentException if {@code collection} is not an {@code EnumSet} instance and
   *     contains no elements
   */
  public static <E extends Enum<E>> EnumSet<E> complementOf(Collection<E> collection) {
    if (collection instanceof EnumSet) {
      return EnumSet.complementOf((EnumSet<E>) collection);
    }
    checkArgument(
        !collection.isEmpty(), "collection is empty; use the other version of this method");
    Class<E> type = collection.iterator().next().getDeclaringClass();
    return makeComplementByHand(collection, type);
  }

  /**
   * Creates an {@code EnumSet} consisting of all enum values that are not in the specified
   * collection. This is equivalent to {@link EnumSet#complementOf}, but can act on any input
   * collection, as long as the elements are of enum type.
   *
   * @param collection the collection whose complement should be stored in the {@code EnumSet}
   * @param type the type of the elements in the set
   * @return a new, modifiable {@code EnumSet} initially containing all the values of the enum not
   *     present in the given collection
   */
  public static <E extends Enum<E>> EnumSet<E> complementOf(
      Collection<E> collection, Class<E> type) {
    checkNotNull(collection);
    return (collection instanceof EnumSet)
        ? EnumSet.complementOf((EnumSet<E>) collection)
        : makeComplementByHand(collection, type);
  }

  private static <E extends Enum<E>> EnumSet<E> makeComplementByHand(
      Collection<E> collection, Class<E> type) {
    EnumSet<E> result = EnumSet.allOf(type);
    result.removeAll(collection);
    return result;
  }

  /**
   * Returns a set backed by the specified map. The resulting set displays the same ordering,
   * concurrency, and performance characteristics as the backing map. In essence, this factory
   * method provides a {@link Set} implementation corresponding to any {@link Map} implementation.
   * There is no need to use this method on a {@link Map} implementation that already has a
   * corresponding {@link Set} implementation (such as {@link java.util.HashMap} or {@link
   * java.util.TreeMap}).
   *
   * <p>Each method invocation on the set returned by this method results in exactly one method
   * invocation on the backing map or its {@code keySet} view, with one exception. The {@code
   * addAll} method is implemented as a sequence of {@code put} invocations on the backing map.
   *
   * <p>The specified map must be empty at the time this method is invoked, and should not be
   * accessed directly after this method returns. These conditions are ensured if the map is created
   * empty, passed directly to this method, and no reference to the map is retained, as illustrated
   * in the following code fragment:
   *
   * <pre>{@code
   * Set<Object> identityHashSet = Sets.newSetFromMap(
   *     new IdentityHashMap<Object, Boolean>());
   * }</pre>
   *
   * <p>The returned set is serializable if the backing map is.
   *
   * @param map the backing map
   * @return the set backed by the map
   * @throws IllegalArgumentException if {@code map} is not empty
   * @deprecated Use {@link Collections#newSetFromMap} instead.
   */
  @Deprecated
  public static <E> Set<E> newSetFromMap(Map<E, Boolean> map) {
    return Collections.newSetFromMap(map);
  }

  /**
   * An unmodifiable view of a set which may be backed by other sets; this view will change as the
   * backing sets do. Contains methods to copy the data into a new set which will then remain
   * stable. There is usually no reason to retain a reference of type {@code SetView}; typically,
   * you either use it as a plain {@link Set}, or immediately invoke {@link #immutableCopy} or
   * {@link #copyInto} and forget the {@code SetView} itself.
   *
   * @since 2.0
   */
  public abstract static class SetView<E> extends AbstractSet<E> {
    private SetView() {} // no subclasses but our own

    /**
     * Returns an immutable copy of the current contents of this set view. Does not support null
     * elements.
     *
     * <p><b>Warning:</b> this may have unexpected results if a backing set of this view uses a
     * nonstandard notion of equivalence, for example if it is a {@link TreeSet} using a comparator
     * that is inconsistent with {@link Object#equals(Object)}.
     */
    public ImmutableSet<E> immutableCopy() {
      return ImmutableSet.copyOf(this);
    }

    /**
     * Copies the current contents of this set view into an existing set. This method has equivalent
     * behavior to {@code set.addAll(this)}, assuming that all the sets involved are based on the
     * same notion of equivalence.
     *
     * @return a reference to {@code set}, for convenience
     */
    // Note: S should logically extend Set<? super E> but can't due to either
    // some javac bug or some weirdness in the spec, not sure which.
    @CanIgnoreReturnValue
    public <S extends Set<E>> S copyInto(S set) {
      set.addAll(this);
      return set;
    }

    /**
     * Guaranteed to throw an exception and leave the collection unmodified.
     *
     * @throws UnsupportedOperationException always
     * @deprecated Unsupported operation.
     */
    @CanIgnoreReturnValue
    @Deprecated
    @Override
    public final boolean add(E e) {
      throw new UnsupportedOperationException();
    }

    /**
     * Guaranteed to throw an exception and leave the collection unmodified.
     *
     * @throws UnsupportedOperationException always
     * @deprecated Unsupported operation.
     */
    @CanIgnoreReturnValue
    @Deprecated
    @Override
    public final boolean remove(Object object) {
      throw new UnsupportedOperationException();
    }

    /**
     * Guaranteed to throw an exception and leave the collection unmodified.
     *
     * @throws UnsupportedOperationException always
     * @deprecated Unsupported operation.
     */
    @CanIgnoreReturnValue
    @Deprecated
    @Override
    public final boolean addAll(Collection<? extends E> newElements) {
      throw new UnsupportedOperationException();
    }

    /**
     * Guaranteed to throw an exception and leave the collection unmodified.
     *
     * @throws UnsupportedOperationException always
     * @deprecated Unsupported operation.
     */
    @CanIgnoreReturnValue
    @Deprecated
    @Override
    public final boolean removeAll(Collection<?> oldElements) {
      throw new UnsupportedOperationException();
    }

    /**
     * Guaranteed to throw an exception and leave the collection unmodified.
     *
     * @throws UnsupportedOperationException always
     * @deprecated Unsupported operation.
     */
    @CanIgnoreReturnValue
    @Deprecated
    @Override
    public final boolean retainAll(Collection<?> elementsToKeep) {
      throw new UnsupportedOperationException();
    }

    /**
     * Guaranteed to throw an exception and leave the collection unmodified.
     *
     * @throws UnsupportedOperationException always
     * @deprecated Unsupported operation.
     */
    @Deprecated
    @Override
    public final void clear() {
      throw new UnsupportedOperationException();
    }

    /**
     * Scope the return type to {@link UnmodifiableIterator} to ensure this is an unmodifiable view.
     *
     * @since 20.0 (present with return type {@link Iterator} since 2.0)
     */
    @Override
    public abstract UnmodifiableIterator<E> iterator();
  }

  /**
   * Returns an unmodifiable <b>view</b> of the union of two sets. The returned set contains all
   * elements that are contained in either backing set. Iterating over the returned set iterates
   * first over all the elements of {@code set1}, then over each element of {@code set2}, in order,
   * that is not contained in {@code set1}.
   *
   * <p>Results are undefined if {@code set1} and {@code set2} are sets based on different
   * equivalence relations (as {@link HashSet}, {@link TreeSet}, and the {@link Map#keySet} of an
   * {@code IdentityHashMap} all are).
   */
  public static <E> SetView<E> union(final Set<? extends E> set1, final Set<? extends E> set2) {
    checkNotNull(set1, "set1");
    checkNotNull(set2, "set2");

    return new SetView<E>() {
      @Override
      public int size() {
        int size = set1.size();
        for (E e : set2) {
          if (!set1.contains(e)) {
            size++;
          }
        }
        return size;
      }

      @Override
      public boolean isEmpty() {
        return set1.isEmpty() && set2.isEmpty();
      }

      @Override
      public UnmodifiableIterator<E> iterator() {
        return new AbstractIterator<E>() {
          final Iterator<? extends E> itr1 = set1.iterator();
          final Iterator<? extends E> itr2 = set2.iterator();

          @Override
          protected E computeNext() {
            if (itr1.hasNext()) {
              return itr1.next();
            }
            while (itr2.hasNext()) {
              E e = itr2.next();
              if (!set1.contains(e)) {
                return e;
              }
            }
            return endOfData();
          }
        };
      }

      @Override
      public boolean contains(Object object) {
        return set1.contains(object) || set2.contains(object);
      }

      @Override
      public <S extends Set<E>> S copyInto(S set) {
        set.addAll(set1);
        set.addAll(set2);
        return set;
      }

      @Override
      public ImmutableSet<E> immutableCopy() {
        return new ImmutableSet.Builder<E>().addAll(set1).addAll(set2).build();
      }
    };
  }

  /**
   * Returns an unmodifiable <b>view</b> of the intersection of two sets. The returned set contains
   * all elements that are contained by both backing sets. The iteration order of the returned set
   * matches that of {@code set1}.
   *
   * <p>Results are undefined if {@code set1} and {@code set2} are sets based on different
   * equivalence relations (as {@code HashSet}, {@code TreeSet}, and the keySet of an {@code
   * IdentityHashMap} all are).
   *
   * <p><b>Note:</b> The returned view performs slightly better when {@code set1} is the smaller of
   * the two sets. If you have reason to believe one of your sets will generally be smaller than the
   * other, pass it first. Unfortunately, since this method sets the generic type of the returned
   * set based on the type of the first set passed, this could in rare cases force you to make a
   * cast, for example:
   *
   * <pre>{@code
   * Set<Object> aFewBadObjects = ...
   * Set<String> manyBadStrings = ...
   *
   * // impossible for a non-String to be in the intersection
   * SuppressWarnings("unchecked")
   * Set<String> badStrings = (Set) Sets.intersection(
   *     aFewBadObjects, manyBadStrings);
   * }</pre>
   *
   * <p>This is unfortunate, but should come up only very rarely.
   */
  public static <E> SetView<E> intersection(final Set<E> set1, final Set<?> set2) {
    checkNotNull(set1, "set1");
    checkNotNull(set2, "set2");

    return new SetView<E>() {
      @Override
      public UnmodifiableIterator<E> iterator() {
        return new AbstractIterator<E>() {
          final Iterator<E> itr = set1.iterator();

          @Override
          protected E computeNext() {
            while (itr.hasNext()) {
              E e = itr.next();
              if (set2.contains(e)) {
                return e;
              }
            }
            return endOfData();
          }
        };
      }

      @Override
      public int size() {
        int size = 0;
        for (E e : set1) {
          if (set2.contains(e)) {
            size++;
          }
        }
        return size;
      }

      @Override
      public boolean isEmpty() {
        return Collections.disjoint(set2, set1);
      }

      @Override
      public boolean contains(Object object) {
        return set1.contains(object) && set2.contains(object);
      }

      @Override
      public boolean containsAll(Collection<?> collection) {
        return set1.containsAll(collection) && set2.containsAll(collection);
      }
    };
  }

  /**
   * Returns an unmodifiable <b>view</b> of the difference of two sets. The returned set contains
   * all elements that are contained by {@code set1} and not contained by {@code set2}. {@code set2}
   * may also contain elements not present in {@code set1}; these are simply ignored. The iteration
   * order of the returned set matches that of {@code set1}.
   *
   * <p>Results are undefined if {@code set1} and {@code set2} are sets based on different
   * equivalence relations (as {@code HashSet}, {@code TreeSet}, and the keySet of an {@code
   * IdentityHashMap} all are).
   */
  public static <E> SetView<E> difference(final Set<E> set1, final Set<?> set2) {
    checkNotNull(set1, "set1");
    checkNotNull(set2, "set2");

    return new SetView<E>() {
      @Override
      public UnmodifiableIterator<E> iterator() {
        return new AbstractIterator<E>() {
          final Iterator<E> itr = set1.iterator();

          @Override
          protected E computeNext() {
            while (itr.hasNext()) {
              E e = itr.next();
              if (!set2.contains(e)) {
                return e;
              }
            }
            return endOfData();
          }
        };
      }

      @Override
      public int size() {
        int size = 0;
        for (E e : set1) {
          if (!set2.contains(e)) {
            size++;
          }
        }
        return size;
      }

      @Override
      public boolean isEmpty() {
        return set2.containsAll(set1);
      }

      @Override
      public boolean contains(Object element) {
        return set1.contains(element) && !set2.contains(element);
      }
    };
  }

  /**
   * Returns an unmodifiable <b>view</b> of the symmetric difference of two sets. The returned set
   * contains all elements that are contained in either {@code set1} or {@code set2} but not in
   * both. The iteration order of the returned set is undefined.
   *
   * <p>Results are undefined if {@code set1} and {@code set2} are sets based on different
   * equivalence relations (as {@code HashSet}, {@code TreeSet}, and the keySet of an {@code
   * IdentityHashMap} all are).
   *
   * @since 3.0
   */
  public static <E> SetView<E> symmetricDifference(
      final Set<? extends E> set1, final Set<? extends E> set2) {
    checkNotNull(set1, "set1");
    checkNotNull(set2, "set2");

    return new SetView<E>() {
      @Override
      public UnmodifiableIterator<E> iterator() {
        final Iterator<? extends E> itr1 = set1.iterator();
        final Iterator<? extends E> itr2 = set2.iterator();
        return new AbstractIterator<E>() {
          @Override
          public E computeNext() {
            while (itr1.hasNext()) {
              E elem1 = itr1.next();
              if (!set2.contains(elem1)) {
                return elem1;
              }
            }
            while (itr2.hasNext()) {
              E elem2 = itr2.next();
              if (!set1.contains(elem2)) {
                return elem2;
              }
            }
            return endOfData();
          }
        };
      }

      @Override
      public int size() {
        int size = 0;
        for (E e : set1) {
          if (!set2.contains(e)) {
            size++;
          }
        }
        for (E e : set2) {
          if (!set1.contains(e)) {
            size++;
          }
        }
        return size;
      }

      @Override
      public boolean isEmpty() {
        return set1.equals(set2);
      }

      @Override
      public boolean contains(Object element) {
        return set1.contains(element) ^ set2.contains(element);
      }
    };
  }

  /**
   * Returns the elements of {@code unfiltered} that satisfy a predicate. The returned set is a live
   * view of {@code unfiltered}; changes to one affect the other.
   *
   * <p>The resulting set's iterator does not support {@code remove()}, but all other set methods
   * are supported. When given an element that doesn't satisfy the predicate, the set's {@code
   * add()} and {@code addAll()} methods throw an {@link IllegalArgumentException}. When methods
   * such as {@code removeAll()} and {@code clear()} are called on the filtered set, only elements
   * that satisfy the filter will be removed from the underlying set.
   *
   * <p>The returned set isn't threadsafe or serializable, even if {@code unfiltered} is.
   *
   * <p>Many of the filtered set's methods, such as {@code size()}, iterate across every element in
   * the underlying set and determine which elements satisfy the filter. When a live view is
   * <i>not</i> needed, it may be faster to copy {@code Iterables.filter(unfiltered, predicate)} and
   * use the copy.
   *
   * <p><b>Warning:</b> {@code predicate} must be <i>consistent with equals</i>, as documented at
   * {@link Predicate#apply}. Do not provide a predicate such as {@code
   * Predicates.instanceOf(ArrayList.class)}, which is inconsistent with equals. (See {@link
   * Iterables#filter(Iterable, Class)} for related functionality.)
   *
   * <p><b>Java 8 users:</b> many use cases for this method are better addressed by {@link
   * java.util.stream.Stream#filter}. This method is not being deprecated, but we gently encourage
   * you to migrate to streams.
   */
  // TODO(kevinb): how to omit that last sentence when building GWT javadoc?
  public static <E> Set<E> filter(Set<E> unfiltered, Predicate<? super E> predicate) {
    if (unfiltered instanceof SortedSet) {
      return filter((SortedSet<E>) unfiltered, predicate);
    }
    if (unfiltered instanceof FilteredSet) {
      // Support clear(), removeAll(), and retainAll() when filtering a filtered
      // collection.
      FilteredSet<E> filtered = (FilteredSet<E>) unfiltered;
      Predicate<E> combinedPredicate = Predicates.<E>and(filtered.predicate, predicate);
      return new FilteredSet<E>((Set<E>) filtered.unfiltered, combinedPredicate);
    }

    return new FilteredSet<E>(checkNotNull(unfiltered), checkNotNull(predicate));
  }

  /**
   * Returns the elements of a {@code SortedSet}, {@code unfiltered}, that satisfy a predicate. The
   * returned set is a live view of {@code unfiltered}; changes to one affect the other.
   *
   * <p>The resulting set's iterator does not support {@code remove()}, but all other set methods
   * are supported. When given an element that doesn't satisfy the predicate, the set's {@code
   * add()} and {@code addAll()} methods throw an {@link IllegalArgumentException}. When methods
   * such as {@code removeAll()} and {@code clear()} are called on the filtered set, only elements
   * that satisfy the filter will be removed from the underlying set.
   *
   * <p>The returned set isn't threadsafe or serializable, even if {@code unfiltered} is.
   *
   * <p>Many of the filtered set's methods, such as {@code size()}, iterate across every element in
   * the underlying set and determine which elements satisfy the filter. When a live view is
   * <i>not</i> needed, it may be faster to copy {@code Iterables.filter(unfiltered, predicate)} and
   * use the copy.
   *
   * <p><b>Warning:</b> {@code predicate} must be <i>consistent with equals</i>, as documented at
   * {@link Predicate#apply}. Do not provide a predicate such as {@code
   * Predicates.instanceOf(ArrayList.class)}, which is inconsistent with equals. (See {@link
   * Iterables#filter(Iterable, Class)} for related functionality.)
   *
   * @since 11.0
   */
  public static <E> SortedSet<E> filter(SortedSet<E> unfiltered, Predicate<? super E> predicate) {
    if (unfiltered instanceof FilteredSet) {
      // Support clear(), removeAll(), and retainAll() when filtering a filtered
      // collection.
      FilteredSet<E> filtered = (FilteredSet<E>) unfiltered;
      Predicate<E> combinedPredicate = Predicates.<E>and(filtered.predicate, predicate);
      return new FilteredSortedSet<E>((SortedSet<E>) filtered.unfiltered, combinedPredicate);
    }

    return new FilteredSortedSet<E>(checkNotNull(unfiltered), checkNotNull(predicate));
  }

  /**
   * Returns the elements of a {@code NavigableSet}, {@code unfiltered}, that satisfy a predicate.
   * The returned set is a live view of {@code unfiltered}; changes to one affect the other.
   *
   * <p>The resulting set's iterator does not support {@code remove()}, but all other set methods
   * are supported. When given an element that doesn't satisfy the predicate, the set's {@code
   * add()} and {@code addAll()} methods throw an {@link IllegalArgumentException}. When methods
   * such as {@code removeAll()} and {@code clear()} are called on the filtered set, only elements
   * that satisfy the filter will be removed from the underlying set.
   *
   * <p>The returned set isn't threadsafe or serializable, even if {@code unfiltered} is.
   *
   * <p>Many of the filtered set's methods, such as {@code size()}, iterate across every element in
   * the underlying set and determine which elements satisfy the filter. When a live view is
   * <i>not</i> needed, it may be faster to copy {@code Iterables.filter(unfiltered, predicate)} and
   * use the copy.
   *
   * <p><b>Warning:</b> {@code predicate} must be <i>consistent with equals</i>, as documented at
   * {@link Predicate#apply}. Do not provide a predicate such as {@code
   * Predicates.instanceOf(ArrayList.class)}, which is inconsistent with equals. (See {@link
   * Iterables#filter(Iterable, Class)} for related functionality.)
   *
   * @since 14.0
   */
  @GwtIncompatible // NavigableSet
  @SuppressWarnings("unchecked")
  public static <E> NavigableSet<E> filter(
      NavigableSet<E> unfiltered, Predicate<? super E> predicate) {
    if (unfiltered instanceof FilteredSet) {
      // Support clear(), removeAll(), and retainAll() when filtering a filtered
      // collection.
      FilteredSet<E> filtered = (FilteredSet<E>) unfiltered;
      Predicate<E> combinedPredicate = Predicates.<E>and(filtered.predicate, predicate);
      return new FilteredNavigableSet<E>((NavigableSet<E>) filtered.unfiltered, combinedPredicate);
    }

    return new FilteredNavigableSet<E>(checkNotNull(unfiltered), checkNotNull(predicate));
  }

  private static class FilteredSet<E> extends FilteredCollection<E> implements Set<E> {
    FilteredSet(Set<E> unfiltered, Predicate<? super E> predicate) {
      super(unfiltered, predicate);
    }

    @Override
    public boolean equals(@NullableDecl Object object) {
      return equalsImpl(this, object);
    }

    @Override
    public int hashCode() {
      return hashCodeImpl(this);
    }
  }

  private static class FilteredSortedSet<E> extends FilteredSet<E> implements SortedSet<E> {

    FilteredSortedSet(SortedSet<E> unfiltered, Predicate<? super E> predicate) {
      super(unfiltered, predicate);
    }

    @Override
    public Comparator<? super E> comparator() {
      return ((SortedSet<E>) unfiltered).comparator();
    }

    @Override
    public SortedSet<E> subSet(E fromElement, E toElement) {
      return new FilteredSortedSet<E>(
          ((SortedSet<E>) unfiltered).subSet(fromElement, toElement), predicate);
    }

    @Override
    public SortedSet<E> headSet(E toElement) {
      return new FilteredSortedSet<E>(((SortedSet<E>) unfiltered).headSet(toElement), predicate);
    }

    @Override
    public SortedSet<E> tailSet(E fromElement) {
      return new FilteredSortedSet<E>(((SortedSet<E>) unfiltered).tailSet(fromElement), predicate);
    }

    @Override
    public E first() {
      return Iterators.find(unfiltered.iterator(), predicate);
    }

    @Override
    public E last() {
      SortedSet<E> sortedUnfiltered = (SortedSet<E>) unfiltered;
      while (true) {
        E element = sortedUnfiltered.last();
        if (predicate.apply(element)) {
          return element;
        }
        sortedUnfiltered = sortedUnfiltered.headSet(element);
      }
    }
  }

  @GwtIncompatible // NavigableSet
  private static class FilteredNavigableSet<E> extends FilteredSortedSet<E>
      implements NavigableSet<E> {
    FilteredNavigableSet(NavigableSet<E> unfiltered, Predicate<? super E> predicate) {
      super(unfiltered, predicate);
    }

    NavigableSet<E> unfiltered() {
      return (NavigableSet<E>) unfiltered;
    }

    @Override
    @NullableDecl
    public E lower(E e) {
      return Iterators.find(unfiltered().headSet(e, false).descendingIterator(), predicate, null);
    }

    @Override
    @NullableDecl
    public E floor(E e) {
      return Iterators.find(unfiltered().headSet(e, true).descendingIterator(), predicate, null);
    }

    @Override
    public E ceiling(E e) {
      return Iterables.find(unfiltered().tailSet(e, true), predicate, null);
    }

    @Override
    public E higher(E e) {
      return Iterables.find(unfiltered().tailSet(e, false), predicate, null);
    }

    @Override
    public E pollFirst() {
      return Iterables.removeFirstMatching(unfiltered(), predicate);
    }

    @Override
    public E pollLast() {
      return Iterables.removeFirstMatching(unfiltered().descendingSet(), predicate);
    }

    @Override
    public NavigableSet<E> descendingSet() {
      return Sets.filter(unfiltered().descendingSet(), predicate);
    }

    @Override
    public Iterator<E> descendingIterator() {
      return Iterators.filter(unfiltered().descendingIterator(), predicate);
    }

    @Override
    public E last() {
      return Iterators.find(unfiltered().descendingIterator(), predicate);
    }

    @Override
    public NavigableSet<E> subSet(
        E fromElement, boolean fromInclusive, E toElement, boolean toInclusive) {
      return filter(
          unfiltered().subSet(fromElement, fromInclusive, toElement, toInclusive), predicate);
    }

    @Override
    public NavigableSet<E> headSet(E toElement, boolean inclusive) {
      return filter(unfiltered().headSet(toElement, inclusive), predicate);
    }

    @Override
    public NavigableSet<E> tailSet(E fromElement, boolean inclusive) {
      return filter(unfiltered().tailSet(fromElement, inclusive), predicate);
    }
  }

  /**
   * Returns every possible list that can be formed by choosing one element from each of the given
   * sets in order; the "n-ary <a href="http://en.wikipedia.org/wiki/Cartesian_product">Cartesian
   * product</a>" of the sets. For example:
   *
   * <pre>{@code
   * Sets.cartesianProduct(ImmutableList.of(
   *     ImmutableSet.of(1, 2),
   *     ImmutableSet.of("A", "B", "C")))
   * }</pre>
   *
   * <p>returns a set containing six lists:
   *
   * <ul>
   *   <li>{@code ImmutableList.of(1, "A")}
   *   <li>{@code ImmutableList.of(1, "B")}
   *   <li>{@code ImmutableList.of(1, "C")}
   *   <li>{@code ImmutableList.of(2, "A")}
   *   <li>{@code ImmutableList.of(2, "B")}
   *   <li>{@code ImmutableList.of(2, "C")}
   * </ul>
   *
   * <p>The result is guaranteed to be in the "traditional", lexicographical order for Cartesian
   * products that you would get from nesting for loops:
   *
   * <pre>{@code
   * for (B b0 : sets.get(0)) {
   *   for (B b1 : sets.get(1)) {
   *     ...
   *     ImmutableList<B> tuple = ImmutableList.of(b0, b1, ...);
   *     // operate on tuple
   *   }
   * }
   * }</pre>
   *
   * <p>Note that if any input set is empty, the Cartesian product will also be empty. If no sets at
   * all are provided (an empty list), the resulting Cartesian product has one element, an empty
   * list (counter-intuitive, but mathematically consistent).
   *
   * <p><i>Performance notes:</i> while the cartesian product of sets of size {@code m, n, p} is a
   * set of size {@code m x n x p}, its actual memory consumption is much smaller. When the
   * cartesian set is constructed, the input sets are merely copied. Only as the resulting set is
   * iterated are the individual lists created, and these are not retained after iteration.
   *
   * @param sets the sets to choose elements from, in the order that the elements chosen from those
   *     sets should appear in the resulting lists
   * @param <B> any common base class shared by all axes (often just {@link Object})
   * @return the Cartesian product, as an immutable set containing immutable lists
   * @throws NullPointerException if {@code sets}, any one of the {@code sets}, or any element of a
   *     provided set is null
   * @since 2.0
   */
  public static <B> Set<List<B>> cartesianProduct(List<? extends Set<? extends B>> sets) {
    return CartesianSet.create(sets);
  }

  /**
   * Returns every possible list that can be formed by choosing one element from each of the given
   * sets in order; the "n-ary <a href="http://en.wikipedia.org/wiki/Cartesian_product">Cartesian
   * product</a>" of the sets. For example:
   *
   * <pre>{@code
   * Sets.cartesianProduct(
   *     ImmutableSet.of(1, 2),
   *     ImmutableSet.of("A", "B", "C"))
   * }</pre>
   *
   * <p>returns a set containing six lists:
   *
   * <ul>
   *   <li>{@code ImmutableList.of(1, "A")}
   *   <li>{@code ImmutableList.of(1, "B")}
   *   <li>{@code ImmutableList.of(1, "C")}
   *   <li>{@code ImmutableList.of(2, "A")}
   *   <li>{@code ImmutableList.of(2, "B")}
   *   <li>{@code ImmutableList.of(2, "C")}
   * </ul>
   *
   * <p>The result is guaranteed to be in the "traditional", lexicographical order for Cartesian
   * products that you would get from nesting for loops:
   *
   * <pre>{@code
   * for (B b0 : sets.get(0)) {
   *   for (B b1 : sets.get(1)) {
   *     ...
   *     ImmutableList<B> tuple = ImmutableList.of(b0, b1, ...);
   *     // operate on tuple
   *   }
   * }
   * }</pre>
   *
   * <p>Note that if any input set is empty, the Cartesian product will also be empty. If no sets at
   * all are provided (an empty list), the resulting Cartesian product has one element, an empty
   * list (counter-intuitive, but mathematically consistent).
   *
   * <p><i>Performance notes:</i> while the cartesian product of sets of size {@code m, n, p} is a
   * set of size {@code m x n x p}, its actual memory consumption is much smaller. When the
   * cartesian set is constructed, the input sets are merely copied. Only as the resulting set is
   * iterated are the individual lists created, and these are not retained after iteration.
   *
   * @param sets the sets to choose elements from, in the order that the elements chosen from those
   *     sets should appear in the resulting lists
   * @param <B> any common base class shared by all axes (often just {@link Object})
   * @return the Cartesian product, as an immutable set containing immutable lists
   * @throws NullPointerException if {@code sets}, any one of the {@code sets}, or any element of a
   *     provided set is null
   * @since 2.0
   */
  @SafeVarargs
  public static <B> Set<List<B>> cartesianProduct(Set<? extends B>... sets) {
    return cartesianProduct(Arrays.asList(sets));
  }

  private static final class CartesianSet<E> extends ForwardingCollection<List<E>>
      implements Set<List<E>> {
    private final transient ImmutableList<ImmutableSet<E>> axes;
    private final transient CartesianList<E> delegate;

    static <E> Set<List<E>> create(List<? extends Set<? extends E>> sets) {
      ImmutableList.Builder<ImmutableSet<E>> axesBuilder = new ImmutableList.Builder<>(sets.size());
      for (Set<? extends E> set : sets) {
        ImmutableSet<E> copy = ImmutableSet.copyOf(set);
        if (copy.isEmpty()) {
          return ImmutableSet.of();
        }
        axesBuilder.add(copy);
      }
      final ImmutableList<ImmutableSet<E>> axes = axesBuilder.build();
      ImmutableList<List<E>> listAxes =
          new ImmutableList<List<E>>() {
            @Override
            public int size() {
              return axes.size();
            }

            @Override
            public List<E> get(int index) {
              return axes.get(index).asList();
            }

            @Override
            boolean isPartialView() {
              return true;
            }
          };
      return new CartesianSet<E>(axes, new CartesianList<E>(listAxes));
    }

    private CartesianSet(ImmutableList<ImmutableSet<E>> axes, CartesianList<E> delegate) {
      this.axes = axes;
      this.delegate = delegate;
    }

    @Override
    protected Collection<List<E>> delegate() {
      return delegate;
    }

    @Override
    public boolean equals(@NullableDecl Object object) {
      // Warning: this is broken if size() == 0, so it is critical that we
      // substitute an empty ImmutableSet to the user in place of this
      if (object instanceof CartesianSet) {
        CartesianSet<?> that = (CartesianSet<?>) object;
        return this.axes.equals(that.axes);
      }
      return super.equals(object);
    }

    @Override
    public int hashCode() {
      // Warning: this is broken if size() == 0, so it is critical that we
      // substitute an empty ImmutableSet to the user in place of this

      // It's a weird formula, but tests prove it works.
      int adjust = size() - 1;
      for (int i = 0; i < axes.size(); i++) {
        adjust *= 31;
        adjust = ~~adjust;
        // in GWT, we have to deal with integer overflow carefully
      }
      int hash = 1;
      for (Set<E> axis : axes) {
        hash = 31 * hash + (size() / axis.size() * axis.hashCode());

        hash = ~~hash;
      }
      hash += adjust;
      return ~~hash;
    }
  }

  /**
   * Returns the set of all possible subsets of {@code set}. For example, {@code
   * powerSet(ImmutableSet.of(1, 2))} returns the set {@code {{}, {1}, {2}, {1, 2}}}.
   *
   * <p>Elements appear in these subsets in the same iteration order as they appeared in the input
   * set. The order in which these subsets appear in the outer set is undefined. Note that the power
   * set of the empty set is not the empty set, but a one-element set containing the empty set.
   *
   * <p>The returned set and its constituent sets use {@code equals} to decide whether two elements
   * are identical, even if the input set uses a different concept of equivalence.
   *
   * <p><i>Performance notes:</i> while the power set of a set with size {@code n} is of size {@code
   * 2^n}, its memory usage is only {@code O(n)}. When the power set is constructed, the input set
   * is merely copied. Only as the power set is iterated are the individual subsets created, and
   * these subsets themselves occupy only a small constant amount of memory.
   *
   * @param set the set of elements to construct a power set from
   * @return the power set, as an immutable set of immutable sets
   * @throws IllegalArgumentException if {@code set} has more than 30 unique elements (causing the
   *     power set size to exceed the {@code int} range)
   * @throws NullPointerException if {@code set} is or contains {@code null}
   * @see <a href="http://en.wikipedia.org/wiki/Power_set">Power set article at Wikipedia</a>
   * @since 4.0
   */
  @GwtCompatible(serializable = false)
  public static <E> Set<Set<E>> powerSet(Set<E> set) {
    return new PowerSet<E>(set);
  }

  private static final class SubSet<E> extends AbstractSet<E> {
    private final ImmutableMap<E, Integer> inputSet;
    private final int mask;

    SubSet(ImmutableMap<E, Integer> inputSet, int mask) {
      this.inputSet = inputSet;
      this.mask = mask;
    }

    @Override
    public Iterator<E> iterator() {
      return new UnmodifiableIterator<E>() {
        final ImmutableList<E> elements = inputSet.keySet().asList();
        int remainingSetBits = mask;

        @Override
        public boolean hasNext() {
          return remainingSetBits != 0;
        }

        @Override
        public E next() {
          int index = Integer.numberOfTrailingZeros(remainingSetBits);
          if (index == 32) {
            throw new NoSuchElementException();
          }
          remainingSetBits &= ~(1 << index);
          return elements.get(index);
        }
      };
    }

    @Override
    public int size() {
      return Integer.bitCount(mask);
    }

    @Override
    public boolean contains(@NullableDecl Object o) {
      Integer index = inputSet.get(o);
      return index != null && (mask & (1 << index)) != 0;
    }
  }

  private static final class PowerSet<E> extends AbstractSet<Set<E>> {
    final ImmutableMap<E, Integer> inputSet;

    PowerSet(Set<E> input) {
      checkArgument(
          input.size() <= 30, "Too many elements to create power set: %s > 30", input.size());
      this.inputSet = Maps.indexMap(input);
    }

    @Override
    public int size() {
      return 1 << inputSet.size();
    }

    @Override
    public boolean isEmpty() {
      return false;
    }

    @Override
    public Iterator<Set<E>> iterator() {
      return new AbstractIndexedListIterator<Set<E>>(size()) {
        @Override
        protected Set<E> get(final int setBits) {
          return new SubSet<E>(inputSet, setBits);
        }
      };
    }

    @Override
    public boolean contains(@NullableDecl Object obj) {
      if (obj instanceof Set) {
        Set<?> set = (Set<?>) obj;
        return inputSet.keySet().containsAll(set);
      }
      return false;
    }

    @Override
    public boolean equals(@NullableDecl Object obj) {
      if (obj instanceof PowerSet) {
        PowerSet<?> that = (PowerSet<?>) obj;
        return inputSet.equals(that.inputSet);
      }
      return super.equals(obj);
    }

    @Override
    public int hashCode() {
      /*
       * The sum of the sums of the hash codes in each subset is just the sum of
       * each input element's hash code times the number of sets that element
       * appears in. Each element appears in exactly half of the 2^n sets, so:
       */
      return inputSet.keySet().hashCode() << (inputSet.size() - 1);
    }

    @Override
    public String toString() {
      return "powerSet(" + inputSet + ")";
    }
  }

  /**
   * Returns the set of all subsets of {@code set} of size {@code size}. For example, {@code
   * combinations(ImmutableSet.of(1, 2, 3), 2)} returns the set {@code {{1, 2}, {1, 3}, {2, 3}}}.
   *
   * <p>Elements appear in these subsets in the same iteration order as they appeared in the input
   * set. The order in which these subsets appear in the outer set is undefined.
   *
   * <p>The returned set and its constituent sets use {@code equals} to decide whether two elements
   * are identical, even if the input set uses a different concept of equivalence.
   *
   * <p><i>Performance notes:</i> the memory usage of the returned set is only {@code O(n)}. When
   * the result set is constructed, the input set is merely copied. Only as the result set is
   * iterated are the individual subsets created. Each of these subsets occupies an additional O(n)
   * memory but only for as long as the user retains a reference to it. That is, the set returned by
   * {@code combinations} does not retain the individual subsets.
   *
   * @param set the set of elements to take combinations of
   * @param size the number of elements per combination
   * @return the set of all combinations of {@code size} elements from {@code set}
   * @throws IllegalArgumentException if {@code size} is not between 0 and {@code set.size()}
   *     inclusive
   * @throws NullPointerException if {@code set} is or contains {@code null}
   * @since 23.0
   */
  @Beta
  public static <E> Set<Set<E>> combinations(Set<E> set, final int size) {
    final ImmutableMap<E, Integer> index = Maps.indexMap(set);
    checkNonnegative(size, "size");
    checkArgument(size <= index.size(), "size (%s) must be <= set.size() (%s)", size, index.size());
    if (size == 0) {
      return ImmutableSet.<Set<E>>of(ImmutableSet.<E>of());
    } else if (size == index.size()) {
      return ImmutableSet.<Set<E>>of(index.keySet());
    }
    return new AbstractSet<Set<E>>() {
      @Override
      public boolean contains(@NullableDecl Object o) {
        if (o instanceof Set) {
          Set<?> s = (Set<?>) o;
          return s.size() == size && index.keySet().containsAll(s);
        }
        return false;
      }

      @Override
      public Iterator<Set<E>> iterator() {
        return new AbstractIterator<Set<E>>() {
          final BitSet bits = new BitSet(index.size());

          @Override
          protected Set<E> computeNext() {
            if (bits.isEmpty()) {
              bits.set(0, size);
            } else {
              int firstSetBit = bits.nextSetBit(0);
              int bitToFlip = bits.nextClearBit(firstSetBit);

              if (bitToFlip == index.size()) {
                return endOfData();
              }
              /*
               * The current set in sorted order looks like
               * {firstSetBit, firstSetBit + 1, ..., bitToFlip - 1, ...}
               * where it does *not* contain bitToFlip.
               *
               * The next combination is
               *
               * {0, 1, ..., bitToFlip - firstSetBit - 2, bitToFlip, ...}
               *
               * This is lexicographically next if you look at the combinations in descending order
               * e.g. {2, 1, 0}, {3, 1, 0}, {3, 2, 0}, {3, 2, 1}, {4, 1, 0}...
               */

              bits.set(0, bitToFlip - firstSetBit - 1);
              bits.clear(bitToFlip - firstSetBit - 1, bitToFlip);
              bits.set(bitToFlip);
            }
            final BitSet copy = (BitSet) bits.clone();
            return new AbstractSet<E>() {
              @Override
              public boolean contains(@NullableDecl Object o) {
                Integer i = index.get(o);
                return i != null && copy.get(i);
              }

              @Override
              public Iterator<E> iterator() {
                return new AbstractIterator<E>() {
                  int i = -1;

                  @Override
                  protected E computeNext() {
                    i = copy.nextSetBit(i + 1);
                    if (i == -1) {
                      return endOfData();
                    }
                    return index.keySet().asList().get(i);
                  }
                };
              }

              @Override
              public int size() {
                return size;
              }
            };
          }
        };
      }

      @Override
      public int size() {
        return IntMath.binomial(index.size(), size);
      }

      @Override
      public String toString() {
        return "Sets.combinations(" + index.keySet() + ", " + size + ")";
      }
    };
  }

  /** An implementation for {@link Set#hashCode()}. */
  static int hashCodeImpl(Set<?> s) {
    int hashCode = 0;
    for (Object o : s) {
      hashCode += o != null ? o.hashCode() : 0;

      hashCode = ~~hashCode;
      // Needed to deal with unusual integer overflow in GWT.
    }
    return hashCode;
  }

  /** An implementation for {@link Set#equals(Object)}. */
  static boolean equalsImpl(Set<?> s, @NullableDecl Object object) {
    if (s == object) {
      return true;
    }
    if (object instanceof Set) {
      Set<?> o = (Set<?>) object;

      try {
        return s.size() == o.size() && s.containsAll(o);
      } catch (NullPointerException | ClassCastException ignored) {
        return false;
      }
    }
    return false;
  }

  /**
   * Returns an unmodifiable view of the specified navigable set. This method allows modules to
   * provide users with "read-only" access to internal navigable sets. Query operations on the
   * returned set "read through" to the specified set, and attempts to modify the returned set,
   * whether direct or via its collection views, result in an {@code UnsupportedOperationException}.
   *
   * <p>The returned navigable set will be serializable if the specified navigable set is
   * serializable.
   *
   * @param set the navigable set for which an unmodifiable view is to be returned
   * @return an unmodifiable view of the specified navigable set
   * @since 12.0
   */
  public static <E> NavigableSet<E> unmodifiableNavigableSet(NavigableSet<E> set) {
    if (set instanceof ImmutableCollection || set instanceof UnmodifiableNavigableSet) {
      return set;
    }
    return new UnmodifiableNavigableSet<E>(set);
  }

  static final class UnmodifiableNavigableSet<E> extends ForwardingSortedSet<E>
      implements NavigableSet<E>, Serializable {
    private final NavigableSet<E> delegate;
    private final SortedSet<E> unmodifiableDelegate;

    UnmodifiableNavigableSet(NavigableSet<E> delegate) {
      this.delegate = checkNotNull(delegate);
      this.unmodifiableDelegate = Collections.unmodifiableSortedSet(delegate);
    }

    @Override
    protected SortedSet<E> delegate() {
      return unmodifiableDelegate;
    }

    @Override
    public E lower(E e) {
      return delegate.lower(e);
    }

    @Override
    public E floor(E e) {
      return delegate.floor(e);
    }

    @Override
    public E ceiling(E e) {
      return delegate.ceiling(e);
    }

    @Override
    public E higher(E e) {
      return delegate.higher(e);
    }

    @Override
    public E pollFirst() {
      throw new UnsupportedOperationException();
    }

    @Override
    public E pollLast() {
      throw new UnsupportedOperationException();
    }

    @MonotonicNonNullDecl private transient UnmodifiableNavigableSet<E> descendingSet;

    @Override
    public NavigableSet<E> descendingSet() {
      UnmodifiableNavigableSet<E> result = descendingSet;
      if (result == null) {
        result = descendingSet = new UnmodifiableNavigableSet<E>(delegate.descendingSet());
        result.descendingSet = this;
      }
      return result;
    }

    @Override
    public Iterator<E> descendingIterator() {
      return Iterators.unmodifiableIterator(delegate.descendingIterator());
    }

    @Override
    public NavigableSet<E> subSet(
        E fromElement, boolean fromInclusive, E toElement, boolean toInclusive) {
      return unmodifiableNavigableSet(
          delegate.subSet(fromElement, fromInclusive, toElement, toInclusive));
    }

    @Override
    public NavigableSet<E> headSet(E toElement, boolean inclusive) {
      return unmodifiableNavigableSet(delegate.headSet(toElement, inclusive));
    }

    @Override
    public NavigableSet<E> tailSet(E fromElement, boolean inclusive) {
      return unmodifiableNavigableSet(delegate.tailSet(fromElement, inclusive));
    }

    private static final long serialVersionUID = 0;
  }

  /**
   * Returns a synchronized (thread-safe) navigable set backed by the specified navigable set. In
   * order to guarantee serial access, it is critical that <b>all</b> access to the backing
   * navigable set is accomplished through the returned navigable set (or its views).
   *
   * <p>It is imperative that the user manually synchronize on the returned sorted set when
   * iterating over it or any of its {@code descendingSet}, {@code subSet}, {@code headSet}, or
   * {@code tailSet} views.
   *
   * <pre>{@code
   * NavigableSet<E> set = synchronizedNavigableSet(new TreeSet<E>());
   *  ...
   * synchronized (set) {
   *   // Must be in the synchronized block
   *   Iterator<E> it = set.iterator();
   *   while (it.hasNext()) {
   *     foo(it.next());
   *   }
   * }
   * }</pre>
   *
   * <p>or:
   *
   * <pre>{@code
   * NavigableSet<E> set = synchronizedNavigableSet(new TreeSet<E>());
   * NavigableSet<E> set2 = set.descendingSet().headSet(foo);
   *  ...
   * synchronized (set) { // Note: set, not set2!!!
   *   // Must be in the synchronized block
   *   Iterator<E> it = set2.descendingIterator();
   *   while (it.hasNext())
   *     foo(it.next());
   *   }
   * }
   * }</pre>
   *
   * <p>Failure to follow this advice may result in non-deterministic behavior.
   *
   * <p>The returned navigable set will be serializable if the specified navigable set is
   * serializable.
   *
   * @param navigableSet the navigable set to be "wrapped" in a synchronized navigable set.
   * @return a synchronized view of the specified navigable set.
   * @since 13.0
   */
  @GwtIncompatible // NavigableSet
  public static <E> NavigableSet<E> synchronizedNavigableSet(NavigableSet<E> navigableSet) {
    return Synchronized.navigableSet(navigableSet);
  }

  /** Remove each element in an iterable from a set. */
  static boolean removeAllImpl(Set<?> set, Iterator<?> iterator) {
    boolean changed = false;
    while (iterator.hasNext()) {
      changed |= set.remove(iterator.next());
    }
    return changed;
  }

  static boolean removeAllImpl(Set<?> set, Collection<?> collection) {
    checkNotNull(collection); // for GWT
    if (collection instanceof Multiset) {
      collection = ((Multiset<?>) collection).elementSet();
    }
    /*
     * AbstractSet.removeAll(List) has quadratic behavior if the list size
     * is just more than the set's size.  We augment the test by
     * assuming that sets have fast contains() performance, and other
     * collections don't.  See
     * http://code.google.com/p/guava-libraries/issues/detail?id=1013
     */
    if (collection instanceof Set && collection.size() > set.size()) {
      return Iterators.removeAll(set.iterator(), collection);
    } else {
      return removeAllImpl(set, collection.iterator());
    }
  }

  @GwtIncompatible // NavigableSet
  static class DescendingSet<E> extends ForwardingNavigableSet<E> {
    private final NavigableSet<E> forward;

    DescendingSet(NavigableSet<E> forward) {
      this.forward = forward;
    }

    @Override
    protected NavigableSet<E> delegate() {
      return forward;
    }

    @Override
    public E lower(E e) {
      return forward.higher(e);
    }

    @Override
    public E floor(E e) {
      return forward.ceiling(e);
    }

    @Override
    public E ceiling(E e) {
      return forward.floor(e);
    }

    @Override
    public E higher(E e) {
      return forward.lower(e);
    }

    @Override
    public E pollFirst() {
      return forward.pollLast();
    }

    @Override
    public E pollLast() {
      return forward.pollFirst();
    }

    @Override
    public NavigableSet<E> descendingSet() {
      return forward;
    }

    @Override
    public Iterator<E> descendingIterator() {
      return forward.iterator();
    }

    @Override
    public NavigableSet<E> subSet(
        E fromElement, boolean fromInclusive, E toElement, boolean toInclusive) {
      return forward.subSet(toElement, toInclusive, fromElement, fromInclusive).descendingSet();
    }

    @Override
    public SortedSet<E> subSet(E fromElement, E toElement) {
      return standardSubSet(fromElement, toElement);
    }

    @Override
    public NavigableSet<E> headSet(E toElement, boolean inclusive) {
      return forward.tailSet(toElement, inclusive).descendingSet();
    }

    @Override
    public SortedSet<E> headSet(E toElement) {
      return standardHeadSet(toElement);
    }

    @Override
    public NavigableSet<E> tailSet(E fromElement, boolean inclusive) {
      return forward.headSet(fromElement, inclusive).descendingSet();
    }

    @Override
    public SortedSet<E> tailSet(E fromElement) {
      return standardTailSet(fromElement);
    }

    @SuppressWarnings("unchecked")
    @Override
    public Comparator<? super E> comparator() {
      Comparator<? super E> forwardComparator = forward.comparator();
      if (forwardComparator == null) {
        return (Comparator) Ordering.natural().reverse();
      } else {
        return reverse(forwardComparator);
      }
    }

    // If we inline this, we get a javac error.
    private static <T> Ordering<T> reverse(Comparator<T> forward) {
      return Ordering.from(forward).reverse();
    }

    @Override
    public E first() {
      return forward.last();
    }

    @Override
    public E last() {
      return forward.first();
    }

    @Override
    public Iterator<E> iterator() {
      return forward.descendingIterator();
    }

    @Override
    public Object[] toArray() {
      return standardToArray();
    }

    @Override
    public <T> T[] toArray(T[] array) {
      return standardToArray(array);
    }

    @Override
    public String toString() {
      return standardToString();
    }
  }

  /**
   * Returns a view of the portion of {@code set} whose elements are contained by {@code range}.
   *
   * <p>This method delegates to the appropriate methods of {@link NavigableSet} (namely {@link
   * NavigableSet#subSet(Object, boolean, Object, boolean) subSet()}, {@link
   * NavigableSet#tailSet(Object, boolean) tailSet()}, and {@link NavigableSet#headSet(Object,
   * boolean) headSet()}) to actually construct the view. Consult these methods for a full
   * description of the returned view's behavior.
   *
   * <p><b>Warning:</b> {@code Range}s always represent a range of values using the values' natural
   * ordering. {@code NavigableSet} on the other hand can specify a custom ordering via a {@link
   * Comparator}, which can violate the natural ordering. Using this method (or in general using
   * {@code Range}) with unnaturally-ordered sets can lead to unexpected and undefined behavior.
   *
   * @since 20.0
   */
  @Beta
  @GwtIncompatible // NavigableSet
  public static <K extends Comparable<? super K>> NavigableSet<K> subSet(
      NavigableSet<K> set, Range<K> range) {
    if (set.comparator() != null
        && set.comparator() != Ordering.natural()
        && range.hasLowerBound()
        && range.hasUpperBound()) {
      checkArgument(
          set.comparator().compare(range.lowerEndpoint(), range.upperEndpoint()) <= 0,
          "set is using a custom comparator which is inconsistent with the natural ordering.");
    }
    if (range.hasLowerBound() && range.hasUpperBound()) {
      return set.subSet(
          range.lowerEndpoint(),
          range.lowerBoundType() == BoundType.CLOSED,
          range.upperEndpoint(),
          range.upperBoundType() == BoundType.CLOSED);
    } else if (range.hasLowerBound()) {
      return set.tailSet(range.lowerEndpoint(), range.lowerBoundType() == BoundType.CLOSED);
    } else if (range.hasUpperBound()) {
      return set.headSet(range.upperEndpoint(), range.upperBoundType() == BoundType.CLOSED);
    }
    return checkNotNull(set);
  }
}