* @serialData number of distinct keys, and then for each distinct key: the key, the number of
   *     values for that key, and the key's values
   */
  @GwtIncompatible // java.io.ObjectOutputStream
  @J2ktIncompatible
  private void writeObject(ObjectOutputStream stream) throws IOException {
    stream.defaultWriteObject();
    Serialization.writeMultimap(this, stream);
  }

  @GwtIncompatible // java.io.ObjectInputStream

   * @serialData number of distinct keys, and then for each distinct key: the key, the number of
   *     values for that key, and the key's values
   */
  @GwtIncompatible // java.io.ObjectOutputStream
  @J2ktIncompatible
  private void writeObject(ObjectOutputStream stream) throws IOException {
    stream.defaultWriteObject();
    stream.writeObject(valueComparator());
    Serialization.writeMultimap(this, stream);
  }

    // there's no way to generate the empty backing map.

    /**
     * @serialData the factory and the backing map
     */
    @GwtIncompatible // java.io.ObjectOutputStream
    @J2ktIncompatible
    private void writeObject(ObjectOutputStream stream) throws IOException {
      stream.defaultWriteObject();
      stream.writeObject(factory);
      stream.writeObject(backingMap());
    }

    Set<Integer> keySet = map.keySet();
    Collection<Integer> values = map.values();

    ByteArrayOutputStream bytes = new ByteArrayOutputStream();
    ObjectOutputStream oos = new ObjectOutputStream(bytes);
    oos.writeObject(map);
    oos.flush();

    int mapSize = bytes.size();
    oos.writeObject(keySet);
    oos.writeObject(values);
    oos.close();

    // This object retains references to collections returned by keySet() and value(). This saves
    // bytes when the both the map and its keySet or value collection are written to the same
    // instance of ObjectOutputStream.

    // TODO(b/160980469): remove support for the old serialization format after some time
    private static final boolean USE_LEGACY_SERIALIZATION = true;

    private final Object keys;