I have seen this term "O(1) access time" used to mean "quickly" but I don't understand what it means. The other term that I see with it in the same context is "O(n) access time". Could someone please explain in a simple way what these terms mean?
See Also
The easiest way to differentiate O(1) and O(n) is comparing array and list.
For array, if you have the right index value, you can access the data instantly. (If you don't know the index and have to loop through the array, then it won't be O(1) anymore)
For list, you always need to loop through it whether you know the index or not.
Introduction to Algorithms: Second Edition by Cormen, Leiserson, Rivest & Stein says on page 44 that
Since any constant is a degree-0 polynomial, we can express any constant function as Theta(n^0), or Theta(1). This latter notation is a minor abuse, however, because it is not clear what variable is tending to infinity. We shall often use the notation Theta(1) to mean either a constant or a constant function with respect to some variable. ... We denote by O(g(n))... the set of functions f(n) such that there exist positive constants c and n0 such that 0 <= f(n) <= c*g(n) for all n >= n0. ... Note that f(n) = Theta(g(n)) implies f(n) = O(g(n)), since Theta notation is stronger than O notation.
If an algorithm runs in O(1) time, it means that asymptotically doesn't depend upon any variable, meaning that there exists at least one positive constant that when multiplied by one is greater than the asymptotic complexity (~runtime) of the function for values of n above a certain amount. Technically, it's O(n^0) time.
According to my perspective,
O(1) means time to execute one operation or instruction at a time is one, in time complexity analysis of algorithm for best case.