How Big-O Analysis Works

11 08 2011

In big-O analysis, input size is assumed to be n. For example n can represent the number of elements in an array. In other problems, n may represent the number of nodes in a linked list, or the number of bits in a data type, or the number of entries in a hash table, and so on. After figuring out what n means in terms of the input, you have to determine how many times the n input items are “examined” in terms of n.  An examination might be something like adding  or comparing an input value to a constant.

When n input items are each examined once the result is n examinations. This is considered O(n), usually referred to as linear time: The time required to run the algorithm increases linearly with the number of input items.

Big-O analysis yields the asymptotic running time, the limit of the running time as n gets very large so you eliminate all but the highest-order term, the term that is largest as n gets very large. So something like O(n + 2) would actually be O(n) because the difference between n and n+2 is so tiny when n is very large.

The fastest-possible running time for any run-time analysis is O(1), commonly referred to as constant running time. An algorithm with constant running time always takes the same amount of time to execute, regardless of the input size.

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