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

In practice, it is O(1), but this actually is a terrible and mathematically non-sense simplification. The O() notation says how the algorithm behaves when the size of the problem tends to infinity. Hashmap get/put works like an O(1) algorithm for a limited size. The limit is fairly large from the computer memory and from the addressing point of view, but far from infinity.

When one says that hashmap get/put is O(1) it should really say that the time needed for the get/put is more or less constant and does not depend on the number of elements in the hashmap so far as the hashmap can be presented on the actual computing system. If the problem goes beyond that size and we need larger hashmaps then, after a while, certainly the number of the bits describing one element will also increase as we run out of the possible describable different elements. For example, if we used a hashmap to store 32bit numbers and later we increase the problem size so that we will have more than 2^32 bit elements in the hashmap, then the individual elements will be described with more than 32bits.

The number of the bits needed to describe the individual elements is log(N), where N is the maximum number of elements, therefore get and put are really O(log N).

If you compare it with a tree set, which is O(log n) then hash set is O(long(max(n)) and we simply feel that this is O(1), because on a certain implementation max(n) is fixed, does not change (the size of the objects we store measured in bits) and the algorithm calculating the hash code is fast.

Finally, if finding an element in any data structure were O(1) we would create information out of thin air. Having a data structure of n element I can select one element in n different way. With that, I can encode log(n) bit information. If I can encode that in zero bit (that is what O(1) means) then I created an infinitely compressing ZIP algorithm.