// logging utility
  template <class Container>
  string ToString(const Container& v) {
    std::stringstream ss;
    ss << "{";
    size_t i = 0;
    for (const auto& e : v) {
      if (i != 0) {
        ss << ", ";
      }
      ss << e.ToString();
      ++i;
    }
    ss << "}";
    return ss.str();
  }

  void VerifyFDefNodes(const tensorflow::FunctionDef& fdef,