Example of Big O of 2^n

Question

So I can picture what an algorithm is that has a complexity of n^c, just the number of nested for loops.

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Answer 1

Consider that you want to guess the PIN of a smartphone, this PIN is a 4-digit integer number. You know that the maximum number of bits to hold a 4-digit number is 14 bits, 2^14 is 16384. So, you will have to guess the value, the 14-bit correct combination let's say, of this PIN.

One way is to ask your friend who trusts you and give it to you. The other way is to brute force. So, for simplicity, consider this simple 2-bit word that you want to guess right, each bit has 2 possible values, 0 or 1. So, all the possibilities are:

00
01
10
11

We know from logic circuits design that all possibilities of an n-bit word will be 2^n possible combinations. So, 2^2 is 4 possible combinations as we saw earlier.

The same applies to the 14-bit integer PIN, so guessing the PIN would require you to solve a 2^14 possible outcome puzzle, hence an algorithm of time complexity O(2^n).

So, those types of problems, where combinations of elements in a set S differs, and you will have to try to solve the problem by trying all possible combinations, will have this O(2^n) time complexity. But, the exponentiation base does not have to be 2. In the example above it's of base 2 because each element, each bit, has two possible values which will not be the case in other problems.

Another good example of O(2^n) algorithms is the recursive knapsack. Where you have to try different combinations to maximize the value, where each element in the set, has two possible values, whether we take it or not.

The Edit Distance problem is an O(3^n) time complexity since you have 3 decisions to choose from for each of the n characters string, deletion, insertion, or replace.