Time Complexity of a Loop when Loop variable “Expands or Shrinks” exponentially
For such cases, time complexity of the loop is O(log(log(n))).The following cases analyse different aspects of the problem.
Case 1 :
In this case, i takes values 2, 2k, (2k)k = 2k2, (2k2)k = 2k3, …, 2klogk(log(n)). The last term must be less than or equal to n, and we have 2klogk(log(n)) = 2log(n) = n, which completely agrees with the value of our last term. So there are in total logk(log(n)) many iterations, and each iteration takes a constant amount of time to run, therefore the total time complexity is O(log(log(n))).
Case 2 :
In this case, i takes values n, n1/k, (n1/k)1/k = n1/k2, n1/k3, …, n1/klogk(log(n)), so there are in total logk(log(n)) iterations and each iteration takes time O(1), so the total time complexity is O(log(log(n))).
Refer below article for analysis of different types of loops.
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