How to choose between map and unordered_map?

                       | map              | unordered_map
---------------------------------------------------------
element ordering       | strict weak      | n/a 
                       |                  |
common implementation  | balanced tree    | hash table
                       | or red-black tree|  
                       |                  |
search time            | log(n)           | O(1) if there are no hash collisions
                       |                  | Up to O(n) if there are hash collisions 
                       |                  | O(n) when hash is the same for any key
                       |                  |     
Insertion time         | log(n)+rebalance | Same as search
                       |                  | 
Deletion time          | log(n)+rebalance | Same as search
                       |                  | 
needs comparators      | only operator <  | only operator ==
                       |                  |
needs hash function    | no               | yes
                       |                  |
common use case        | when good hash is| In most other cases. 
                       | not possible or  | 
                       | too slow. Or when|
                       | order is required| 

In practice, if memory is no issue, unordered_map is always faster if you want single element access.

The worst case is theoretical and bound to a single hash accounting for all of the elements. This is not of practical relevance. The unordered_map gets slower as soon as you have at least log N elements belonging to the same hash. This is also not of practical relevance. In some special scenarios you could use a specific hashing algorithm that ensures a more uniform distribution. For ordinary strings that don't share a specific pattern, the generic hash functions coming with unordered_map are just as good.

If you want to traverse the map (using iterators) in a sorted fashion, you cannot use unordered_map. On the contrary, map not only allows that, but also can provide you with the next element in a map based on an approximation of the key (see lower_bound and upper_bound methods).

What container should I have chosen, map or unordered_map? unordered_map takes up more memory so let's suppose memory isn't an issue, and the concern is speed.

Profile and then decide. unordered_map is generally faster, but it varies per case.

In what cases would it get to O(n)?

When the hashing isn't good and a bunch of elements are being assigned to the same bins.

When does a map get more time efficient than unordered_map? Does it happaen when n is small?

Probably not, but profile it if you really care. Having a container with a small size be the bottleneck of your program seems extremely unlikely. Anyway, a simple vector with linear search may be faster for such cases.

The most important thing when deciding is the requirements of ordering and lack of iterator invalidation. If you need either, you pretty much have to use map. Otherwise, unordered_map.

In what cases would it get to O(n)?

if you have such a bad hash function which produces the same hash value for all input stirngs (i.e. produce collisions)...

What container should I have chosen, map or unordered_map?

It is always the questions of requirements and kind/amount of data do you have.

When does a map get more time efficient than unordered_map?

It is just different structures. You'd better to make a chiose to use one of them depending on your typical use cases (takeing in account what kind of data do you have and its amount)

Does it hppaen when n is small?

In case of small data amount everything depends on particular STL implementation... So sometimes even a plain vector/array could be faster than associative containers...