How to choose between map and unordered_map?

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.

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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.

                       | 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).