Question

Order notation question, big-o notation and the like:

What does the max and min of a function mean in terms of order notation?

for example:

DEFINITION:

“Maximum” rules: Suppose that f(n) and g(n) are positive functions for all n > n₀.

Then:

• O[f(n) + g(n)] = O[max (f(n),g(n)) ]

• etc…

I need to use these definitions to prove something for homework.. thanks for the help!

EDIT: f(n) and g(n) are supposed to represent running times of algorithms with respect to input size

Answer 1

With Big-O notation, you’re talking about upper bounds on computation. This means that you are only interested in the largest term of the combined function as n (the variable) tends to infinity. What’s more, you drop any constant multipliers as well since the formal definition of the notation allows you to discard those parts, which is important as it allows you to focus on the algorithm’s behavior and not on the implementation of the algorithm.

So, we are combining two functions through summation. Well, there are two cases (strictly three, but it’s symmetric):

1. One function is of higher order than the other. At that point, the higher order function dominates the lesser; you can pretend that the lesser doesn’t exist.
2. Both functions are of the same order. This is just like you are doing some kind of ratio-ed sum of the two (since we’ve already thrown away the scaling factors) but then you just end up with the same factor and have just changed the scaling factors a bit.

The net result looks very much like the max() function, even though it’s really not (it’s actually a generalization of max over the space of functions), so it’s very convenient to use the notation.