大欧米茄和大西塔

Question

A function like f(n)=3n^2+2 is O(n^2) because n^2 is the biggest exponent in the function. However, the function 1f(n)= n^31 is not O(n^2) because the biggest exponent is 3, not 2.

So in order to make a guess like this on Big Omega or Big Theta, what should we look for in the function? Can we do something analogous to what we did for Big O notation above?

For example, let's say the questions asks us to find the Big Omega or Big Theta of the function f(n)= 3n^2 +1 . Is f(n)= O(n) , Big Omega(n) or a Big Theta(n) ? If I am about to take an educated guess on whether this function is Big O(n), I would say no (because the biggest exponent of the function is 2, not 1). I would prove this more formally using induction.

So, can we do something analogous to what we did with Big O notation in the first example? What should I look for in the function to guess what the Big Omega and Theta will be, and to determine if the "educated guess" is correct?

Answer 1

Your example uses polynomials, so I will assume that.

• your polynomial is O(n^k) if k is greater than or equal to the order of your polynomial.

• your polynomial is Omega(n^k) if k is less than or equal to the order of your polynomial.

• your polynomial is Theta(n^k) if it is both O(n^k) and Omega(n^k).

Answer 2

So, can we do something analogous to what we did with Big O notation in the first example?

If you're looking for something that allows you to eyeball if something is Big Omega, Big O, or Big Theta for polynomials, you can use the Theorem of Polynomial Orders (pretty much what Patrick87 said).

Basically, the Theorem of Polynomial Orders allows you to solely look at the highest order term and use that as your desired bound for Big O, Big Omega, and Big Theta.

What should I look for in the function to guess what the Big Omega and Theta will be, and to determine if the "educated guess" is correct?

Ideally, your function would be a polynomial as that would make it the problem much simpler. But, it can also be a logarithmic function or an exponential function.

To determine if the "educated guess" is correct, you have to first understand what kind of runtime you are looking for. Ask yourself: am I looking for the worst case running time for this algorithm? or am I looking for the best case running time for this algorithm? or am I looking for the general running time of the algorithm?

If you are looking at the worst-case running time of the algorithm, you can simply prove Big Omega using the theorem of polynomial order (if it's a polynomial function) or through an example. However, you must analyze the algorithm to be able to prove Big O and Big Theta.

If you are looking at the best-case running time of the algorithm, you can prove Big O using an example or through the theorem of polynomial order (if it's a polynomial). However, Big Omega and Big Theta can only be proved by analyzing the algorithm.

Basically, you can only prove the least informative bounds for best-case running time and worst-case running time of the algorithm with an example.

For proving the general running time of an algorithm, you have to make sure that the function for the algorithm's running time you have been given is for all input - a single example is not sufficient in this case. When you don't have a function for all input, you have to analyze the algorithm to prove any of the three (Big O, Big Omega, Big Theta) for all inputs for the algorithm.