Introduction to Asymptotic Notation
Dr. Russ Miller
December 30, 1996

We use to mean `order exactly', O to mean `order at most', and to mean `order at least'.

Definition of : Given nonnegative functions f and g defined on the positive integers, we write if and only if there are positive constants , , and a positive integer N such that , whenever n>N.

Definition of O: Given nonnegative functions f and g defined on the positive integers, we write f=O(g) if and only if there is a positive constant C and an integer N such that , for all n>N.

Definition of : Given nonnegative functions f and g defined on the positive integers, we write if and only if there is a positive constant C and an integer N such that , for all n>N.

When using order notation, one wants to use the simplest function possible within the , O, or . The idea is to use the notation to reduce complicated functions to simpler ones whose behavior is easier to understand.

Proposition 1: Suppose f and g are positive functions defined on the positive integers, and that , for some nonzero a. Then .

Proof: means that for any c>0 there is an integer N such that , for all . If we chose then there is an N such that for all . So, , for all .

Proposition 1 is useful because there are many techniques, such as L'Hopital's rule, for evaluating limits. However, there are some instances where Proposition 1 cannot be used.

For example, , even though is undefined.

Examples of : If f(n) = 3n + 6 and g(n) = n, then , or , or , as can be seen by using , and N=5.

Notice that if then , since if there are positive constants , and N such that for all n>N, then , for all n>N.

. . .

Useful Properties of :

1. If and , then .

2. If and , then .

3. .

4. for any a,b>1.

5.

6. .

7. .

8. .

9. If , and , then .

O and Notation

It is not always possible to determine the behavior of an algorithm using -notation. For example, given a problem with n inputs, we may have an algorithm to solve the problem which takes time when n is even and Cn time when n is odd. Or, we may only be able to prove that an algorithm never uses more than time and never less than Fn time. In either circumstance we can claim neither nor to be the order of the time usage of the algorithm. O and notation will allow us to give at least partial information.

Proposition 2: Suppose f and g are positive functions defined on the positive integers, and that , then f=O(g).

Examples of O: 3 n + 2 = O(n), since , for all . 100 n + 6 = O(n), since , for all . O, since , for all . O(1), since 3 n + 2 is not less than or equal to c for any constant c and all . O(n). O.

Proposition 3: Suppose f and g are positive functions defined on the positive integers, and that , then .

Examples of : , since for all . , since for all . since for all . . .

Useful Properties of O and :

1. if and only if f=O(g) and .

2. If f=O(h) and g=O(h) then f+g=O(h).

3. If and then .

4. If f=O(g) and g=O(h) then f=O(h).

5. If and then .

6. Suppose 0<a<b. Then O and .

7. Suppose 0<a and 1<b, then O and .

8. Suppose 1<c and 0<a, then O and .