What if the keys are not unique?

Solution 1: Insert implies Replace

Solution 2: Array elements are pointers to lists.

If we assume a uniform distribution over keys, a $\Theta(1)$ search is maintained.

If we can maintain $\Theta(1)$ performance for multiple entries for the same key, perhaps we can do the same while mapping multiple keys into the same array element.

In other words, use Hash Tables.