How can I create my own comparator for a map?

std::map takes up to four template type arguments, the third one being a comparator. E.g.:

struct cmpByStringLength {
    bool operator()(const std::string& a, const std::string& b) const {
        return a.length() < b.length();
    }
};

// ...
std::map<std::string, std::string, cmpByStringLength> myMap;

Alternatively you could also pass a comparator to maps constructor.

Note however that when comparing by length you can only have one string of each length in the map as a key.

Since C++11, you can also use a lambda expression instead of defining a comparator struct:

auto comp = [](const string& a, const string& b) { return a.length() < b.length(); };
map<string, string, decltype(comp)> my_map(comp);

my_map["1"]      = "a";
my_map["three"]  = "b";
my_map["two"]    = "c";
my_map["fouuur"] = "d";

for(auto const &kv : my_map)
    cout << kv.first << endl;

Output:

1
two
three
fouuur

I'd like to repeat the final note of Georg's answer: When comparing by length you can only have one string of each length in the map as a key.

Code on Ideone

Yes, the 3rd template parameter on map specifies the comparator, which is a binary predicate. Example:

struct ByLength : public std::binary_function<string, string, bool>
{
    bool operator()(const string& lhs, const string& rhs) const
    {
        return lhs.length() < rhs.length();
    }
};

int main()
{
    typedef map<string, string, ByLength> lenmap;
    lenmap mymap;

    mymap["one"] = "one";
    mymap["a"] = "a";
    mymap["fewbahr"] = "foobar";

    for( lenmap::const_iterator it = mymap.begin(), end = mymap.end(); it != end; ++it )
        cout << it->first << "\n";
}

Specify the type of the pointer to your comparison function as the 3rd type into the map, and provide the function pointer to the map constructor:
map<keyType, valueType, typeOfPointerToFunction> mapName(pointerToComparisonFunction);

Take a look at the example below for providing a comparison function to a map, with vector iterator as key and int as value.

#include "headers.h"

bool int_vector_iter_comp(const vector<int>::iterator iter1, const vector<int>::iterator iter2) {
    return *iter1 < *iter2;
}

int main() {
    // Without providing custom comparison function
    map<vector<int>::iterator, int> default_comparison;

    // Providing custom comparison function
    // Basic version
    map<vector<int>::iterator, int,
        bool (*)(const vector<int>::iterator iter1, const vector<int>::iterator iter2)>
        basic(int_vector_iter_comp);

    // use decltype
    map<vector<int>::iterator, int, decltype(int_vector_iter_comp)*> with_decltype(&int_vector_iter_comp);

    // Use type alias or using
    typedef bool my_predicate(const vector<int>::iterator iter1, const vector<int>::iterator iter2);
    map<vector<int>::iterator, int, my_predicate*> with_typedef(&int_vector_iter_comp);

    using my_predicate_pointer_type = bool (*)(const vector<int>::iterator iter1, const vector<int>::iterator iter2);
    map<vector<int>::iterator, int, my_predicate_pointer_type> with_using(&int_vector_iter_comp);


    // Testing 
    vector<int> v = {1, 2, 3};

    default_comparison.insert(pair<vector<int>::iterator, int>({v.end(), 0}));
    default_comparison.insert(pair<vector<int>::iterator, int>({v.begin(), 0}));
    default_comparison.insert(pair<vector<int>::iterator, int>({v.begin(), 1}));
    default_comparison.insert(pair<vector<int>::iterator, int>({v.begin() + 1, 1}));

    cout << "size: " << default_comparison.size() << endl;
    for (auto& p : default_comparison) {
        cout << *(p.first) << ": " << p.second << endl;
    }

    basic.insert(pair<vector<int>::iterator, int>({v.end(), 0}));
    basic.insert(pair<vector<int>::iterator, int>({v.begin(), 0}));
    basic.insert(pair<vector<int>::iterator, int>({v.begin(), 1}));
    basic.insert(pair<vector<int>::iterator, int>({v.begin() + 1, 1}));

    cout << "size: " << basic.size() << endl;
    for (auto& p : basic) {
        cout << *(p.first) << ": " << p.second << endl;
    }

    with_decltype.insert(pair<vector<int>::iterator, int>({v.end(), 0}));
    with_decltype.insert(pair<vector<int>::iterator, int>({v.begin(), 0}));
    with_decltype.insert(pair<vector<int>::iterator, int>({v.begin(), 1}));
    with_decltype.insert(pair<vector<int>::iterator, int>({v.begin() + 1, 1}));

    cout << "size: " << with_decltype.size() << endl;
    for (auto& p : with_decltype) {
        cout << *(p.first) << ": " << p.second << endl;
    }

    with_typedef.insert(pair<vector<int>::iterator, int>({v.end(), 0}));
    with_typedef.insert(pair<vector<int>::iterator, int>({v.begin(), 0}));
    with_typedef.insert(pair<vector<int>::iterator, int>({v.begin(), 1}));
    with_typedef.insert(pair<vector<int>::iterator, int>({v.begin() + 1, 1}));

    cout << "size: " << with_typedef.size() << endl;
    for (auto& p : with_typedef) {
        cout << *(p.first) << ": " << p.second << endl;
    }
}