CSCE 235

Handout 24: Complexity

Assigned April 30, 2003

When we study algorithms, we are interested in characterizing them according to their efficiency. We are usually interested in the order of growth of the running time of an algorithm, not in the exact running time. Asymptotic notation gives us a method for classifying functions according to their rate of growth. (From Prof. Chuck Cusack’s Notes)

An Asymptotic Upper Bound O-Notation

Definition: if and only if there are two positive constants c and such that

for all .

If is nonnegative, we can simplify the above to

for all .

We say that “ is big-O of .” As n increases, grows no faster than . is an asymptotic upperbound on .

Example: . Proof: Let and . Then

.

An Asymptotic Lower Bound Ω-Notation

Definition: if and only if there are two positive constants c and such that

for all .

If is nonnegative, we can simplify the above to

for all .

We say that “ is omega of .” As n increases, grows no slower than . is an asymptotic lowerbound on .

Example: . Proof: Let and . Then we know that . So

.

An Asymptotic Tight Bound Θ-Notation

Definition: if and only if there are three positive constants , , and such that

for all .

If is nonnegative, we can simplify the above to

for all .

We say that “ is theta of .” As n increases, grows at the same rate as . is an asymptotic tight bound on .

Example: . Proof: Let , , and . Then we know that and . So

.

An Application

Let for N. We claim that is . For small values of n, is actually negative. But since we only care about large n, we follow standard practice and allow f to have a few negative values. To check that is we observe that for ; for all N, for ; and so:

for all large enough n.

Observations

If is and is , then

is

and

is

It follows that if is a polynomial in n in which is the highest power of n that appears, then is .

Transitivity:

If and , then .

If and , then .

If and , then .

Based on Chuck Cusack’s Notes, (Ross and Wright 1988,) and (Goodaire and Parmenter 2002).