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How can I specify computational complexity of an algorithm in Big-O notation whose execution follows the following pattern, according to input sizes?
Input size: 4
Number of execution steps: 4 + 3 + 2 + 1 = 10
Input size: 5
Number of execution steps: 5 + 4 + 3 + 2 + 1 = 15
Input size: 6
Number of execution steps: 6 + 5 + 4 + 3 + 2 + 1 = 21
Technically, you have not given enough information to determine the complexity. On the information so far, it could take 21 steps for all input sizes greater than 5. In that case, it would be O(1) regardless of the behavior for sizes 4 and 5. Complexity is about limiting behavior for large input sizes, not about how something behaves for three extremely small sizes.
If the number of steps for size n is, in general, the sum of the numbers from 1 through n then the formula for the number of steps is indeed n(n+1)/2 and it is O(n^2).