Before revision:  However, the implementation of these algorithms often requires an enhanced indexing structure: to achieve optimality we either need to build completely new indexes  or we must populate the database with several different instantiations of common indexes such as B+-trees.
After revision:  However, the implementation of these algorithms often requires an enhanced indexing structure: to achieve optimality we either need to build completely new indexes or we must populate the database with several instantiations of indexes such as B+-trees.