This notation gives an upper bound for a function to within a constant factor. Given Figure illustrates the plot of f(n) = O(g(n)) depend on big O notation. We write f(n) = O(g(n)) if there are positive constants n_{0} & c such that to the right of n_{0}, the value of f(n) always lies on or below cg(n).
Figure: Plot of f(n) = O(g(n))
Mathematically specking, for a given function g(n), we specified a set of functions through O(g(n)) by the following notation:
O(g(n)) = {f(n) : There exists a positive constant c and n0 such that 0 ≤ f(n) ≤ cg(n)
for all n ≥ n0 }
Obviously, we employ O-notation to describe the upper bound onto a function using a constant factor c.
We can view from the earlier definition of Θ that Θ is a tighter notation in comparison of big-O notation. f(n) = an + c is O(n) is also O(n^{2}), but O (n) is asymptotically tight while O(n^{2}) is notation.
While in terms of Θ notation, the above function f(n) is Θ (n). Because of the reason big-O notation is upper bound of function, it is frequently used to define the worst case running time of algorithms.