HashMap get/put complexity

Question

We are used to saying that HashMap get/put operations are O(1). However it depends on the hash implementation. The default object hash is actually

Answer 1

It has already been mentioned that hashmaps are O(n/m) in average, if n is the number of items and m is the size. It has also been mentioned that in principle the whole thing could collapse into a singly linked list with O(n) query time. (This all assumes that calculating the hash is constant time).

However what isn't often mentioned is, that with probability at least 1-1/n (so for 1000 items that's a 99.9% chance) the largest bucket won't be filled more than O(logn)! Hence matching the average complexity of binary search trees. (And the constant is good, a tighter bound is (log n)*(m/n) + O(1)).

All that's required for this theoretical bound is that you use a reasonably good hash function (see Wikipedia: Universal Hashing. It can be as simple as a*x>>m). And of course that the person giving you the values to hash doesn't know how you have chosen your random constants.

TL;DR: With Very High Probability the worst case get/put complexity of a hashmap is O(logn).