The.Algorithm.Design.Manual.Springer-Verlag.1998

The Big Oh Notation
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The Big Oh Notation
We have agreed that the best, worst, and average-case complexities of a given algorithm are numerical
functions of the size of the instances. However, it is difficult to work with these functions exactly
because they are often very complicated, with many little up and down bumps, as shown in Figure .
Thus it is usually cleaner and easier to talk about upper and lower bounds of such functions. This is
where the big Oh notation comes into the picture.
Figure: Upper and lower bounds smooth out the behavior of complex functions
We seek this smoothing in order to ignore levels of detail that do not impact our comparison of
algorithms. Since running our algorithm on a machine that is twice as fast will cut the running times of
all algorithms by a multiplicative constant of two, such constant factors would be washed away in
upgrading machines. On the RAM we ignore such constant factors anyway and therefore might as well
ignore them when comparing two algorithms. We use the following notation:
● f(n) = O(g(n)) means is an upper bound on f(n). Thus there exists some constant c such that
f(n) is always , for large enough n.
● means is a lower bound on f(n). Thus there exists some constant c such that
f(n) is always , for large enough n.
● means is an upper bound on f(n) and is a lower bound on f(n), for
large enough n. Thus there exists constants and such that and .
This means that g(n) is a nice, tight bound on f(n).
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