Hash table runtime complexity (insert, search and delete)

Hash tables are O(1) average and amortized case complexity, however it suffers from O(n) worst case time complexity. [And I think this is where your confusion is]

Hash tables suffer from O(n) worst time complexity due to two reasons:

1. If too many elements were hashed into the same key: looking inside this key may take O(n) time.
2. Once a hash table has passed its load balance - it has to rehash [create a new bigger table, and re-insert each element to the table].

However, it is said to be O(1) average and amortized case because:

1. It is very rare that many items will be hashed to the same key [if you chose a good hash function and you don't have too big load balance.
2. The rehash operation, which is O(n), can at most happen after n/2 ops, which are all assumed O(1): Thus when you sum the average time per op, you get : (n*O(1) + O(n)) / n) = O(1)

Note because of the rehashing issue - a realtime applications and applications that need low latency - should not use a hash table as their data structure.

EDIT: Annother issue with hash tables: cache
Another issue where you might see a performance loss in large hash tables is due to cache performance. Hash Tables suffer from bad cache performance, and thus for large collection - the access time might take longer, since you need to reload the relevant part of the table from the memory back into the cache.

Ideally, a hashtable is O(1). The problem is if two keys are not equal, however they result in the same hash.

For example, imagine the strings "it was the best of times it was the worst of times" and "Green Eggs and Ham" both resulted in a hash value of 123.

When the first string is inserted, it's put in bucket 123. When the second string is inserted, it would see that a value already exists for bucket 123. It would then compare the new value to the existing value, and see they are not equal. In this case, an array or linked list is created for that key. At this point, retrieving this value becomes O(n) as the hashtable needs to iterate through each value in that bucket to find the desired one.

For this reason, when using a hash table, it's important to use a key with a really good hash function that's both fast and doesn't often result in duplicate values for different objects.

Make sense?