Using std::unordered_map with custom types

If we use std::unordered_map instead of std::map, we have a different degree of freedom for the choice of the key type which shall be used. std::map demands that there is a natural order between all key items. This way, items can be sorted. But what if we want, for example, mathematical vectors as a key type? There is no meaning in a smaller < relation for such types, as a vector (0, 1) is not smaller or larger than (1, 0). They just point in different directions. This is completely fine for std::unordered_map because it will not distinguish items via their smaller/greater ordering relationship but via hash values. The only thing we need to do is to implement a hash function for our own type, and an equal to == operator implementation, which tells whether two objects are identical. This section will demonstrate this in an example.