C++ :: Searching A MultiMap With Regex

I have a multimap with over 300k entries defined like so: std::multimap<std::string, std::string> filedata;

Using the following code and std::multimap::equal_range, I am able to successfully search for a word in the multimap and get needed data:

// Data with strings.
data = std::vector<std::string>();
// Get iterators to matched pairs.
std::pair <std::multimap<std::string, std::string>::iterator, std::multimap<std::string, std::string>::iterator> dat = filedata.equal_range(word);
// Go through each matched pair and get needed info.
for (std::multimap<std::string, std::string>::iterator iter = dat.first; iter != dat.second; iter++) {
data.push_back(iter->second);
}

Now, I would like to search the multimap using regular expressions (EX: std::regex("[a-z][a-e]h")). What is the fastest way to do this? Example code may look like:

std::pair <std::multimap<std::string, std::string>::iterator, std::multimap<std::string, std::string>::iterator> dat = filedata.equal_range_with_regex(std::regex("" + word + ""));. Pseudo-code / algorithms will be enough.