Is it permissible to use hotel and large restaurant restrooms to relieve oneself without staying there?

If shop owners do not permit the use of their facilities except for their guests, or if that is the common practice, then it is not permissible to use them. However, if they allow the general public, or if that is the common practice, then it is permissible to use them. This applies in situations of choice and ample opportunity. As for the one in dire necessity, it is permissible for him to use them, due to the general rulings concerning necessity.