## Language Complexity (Part 5)

David A. Tanzer

Big O analysis

Recall our function f(n) from Part 4, which gives the values 2, 13, 14, 25, 26, 37, …

Using ‘big O’ notation, we write f(n) = O(n) to say that f is linearly bounded.

This means that f(n) will eventually become less than some linear function.

As we said, f(n) has a “rough slope” of 6. So f could never be bounded by e.g. the linear function 2n. On the other hand it looks like f should be bounded by any linear function with slope greater than 6, e.g., g(n) = 10n. However, g is not a perfect bound on f, as f(0)=2 > g(0)=0, and f(1)=13 > g(1)=10.

But once n reaches 2 we have that f(n) < g(n). So we say that f is eventually bounded by g.

Now let’s recap.

Definition. $f(n) = O(n)$ means that for some $r > 0$ and $n_1$, we have that $n > n_1 \implies |f(n)| < r n$.

Now let’s apply big O to the analysis of this function from the previous post:

def approximately_linear(text):
counter = 0
for i = 1 to length(text):
if i is odd:
for j = 1 to 10:
counter = counter + 1
return counter


The function approximately_linear has definite linear time complexity because: $f(n) \equiv \text{MaxSteps}(\mathit{approximately\_linear}, n) = O(n)$