std::map::size_type for a map whose value type is its own size_type

I have a std::map<std::pair<std::string, std::string>, float> that is taking up too much memory, and in order to use less memory, I've decided to map the unique strings to integers (e.g., std::map<std::string, int>, where each new unique string is mapped to the current size() of the map), and use those integer value as pairwise keys to the map, (e.g., std::map<std::pair<int, int>, float>).

Instead of int, I want to use std::map::size_type:

using map_index = std::map::size_type;
std::pair<map_index, map_index> key;

Of course, this doesn't compile because I need to supply the argument list for the map:

vector.cc:14:19: error: invalid use of template-name `std::map' without an argument list
 using map_index = std::map::size_type;

And this (in theory) is what I'm trying to achieve:

using map_index = std::map<std::string, map_index>::size_type;

which gives the following (expected) compiler error:

vector.cc:15:41: error: `map_index' was not declared in this scope
 using map_index = std::map<std::string, map_index>::size_type;

What is the proper way to get the correct std::map::size_type for a map whose value type is its own size_type?