8.1: Big O

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Big O

The idea of Big O is to characterize functions according to their growth rates. The O refers to the order of a function. In computer science, Big O is used to classify algorithms for their running time or space requirements.

Notice in the figure below that $f(x)>g(x)$ right before $x=1$. However, for $x>1$, we see that $g(x)>f(x).$ In the long run (namely after $x>1$) $g(x)$ overtakes $f(x).$

We say "$f$ is of order at most $g$" or "$f(x)$ is Big O of $g(x)$" .

We write: $f(x)= O\big(g(x)\big).$

$bigO.png$

In our definition of Big O notation, there are certain parameters.

We use $x$ greater than a certain initial value, $n$; in the diagram above, $n=1$.
We use absolute value for both functions.
We use $k$ as a constant multiplied by the function inside the O.

Definition: Big O Notation

\[f(x)= O\big(g(x)\big).\]

if and only if there exist real numbers $k,n $ with $k>0 , n\geq 0 $ such that

\[|f(x)| \leq k|g(x)| \qquad \forall x>n.\]

Example $\PageIndex{1}$

Take this statement and express it in Big O notation: $|7x^5+4x^3+x| \leq 14|x^5|$ for $x>1.$

Solution: $(7x^5+4x^3+x) \mbox{ is } O(x^5)$

Comparing orders of common functions

A constant function, such as $f(x)=6$ does not grow at all. Logarithmic functions grow very slowly. Here is a list of some common functions in increasing order of growth rates.

constant function, logarithmic function, polynomial function, exponential function

Example $\PageIndex{2}$

Put these functions in order of increasing growth rates:

\[\log_6x,\qquad x^5,\qquad 2^x,\qquad x^2,\qquad \log_{15}x,\qquad 100x^4,\qquad 64x+1000,\qquad x^5 \log_6x,\qquad 5^x, \qquad 6\]

Solution: \[6, \qquad \log_6x,\qquad \log_{15}x,\qquad 64x+1000,\qquad x^2,\qquad 100x^4,\qquad x^5,\qquad x^5 \log_6x, \qquad 2^x,\qquad 5^x\]

Proofs

We will be using the Triangle Inequality Theorem which is \[|x+y|\ \leq |x|+|y|.\]

Example $\PageIndex{3}$

Prove: $4x^3-11x^2+3x-2=O(x^3)$

Proof: Choose $n=1$, i.e. $x \geq 1.$
$|4x^3-11x^2+3x-2 |\leq |4x^3|+|-11x^2|+|3x|+|-2| \qquad \mbox{ by the Triangle Inequality Theorem}$
$\qquad \qquad \qquad \qquad \qquad=4x^3+11x^2+3x+2 \qquad \mbox{    applying absolute value; note: }x \mbox{ is positive}$
$\qquad \qquad \qquad \qquad \qquad\leq 4x^3+11x^3+3x^3+2x^3 \qquad \mbox{    since }x \mbox{ is positive and greater than 1}$
$\qquad \qquad \qquad \qquad \qquad =20x^3$
$\qquad \qquad \qquad \qquad \qquad=20|x^3|\qquad \mbox{         since }x \mbox{ is positive and greater than 1}$

Thus  for all $x \geq 1,$ $|4x^3-11x^2+3x-2 | \leq 20|x^3|$

Therefore, using $n=1$ and $k=20$, $4x^3-11x^2+3x-2=O(x^3)$ by the definition of Big O.

Summary and Review

Big O is used to compare the growth rates of functions.
Be sure to understand the examples here.

Exercises

exercise $\PageIndex{1}$

True or False?

(a) $11x^3=O(87x^2)$

(b) $x^{13}=O(3^x)$

Answer: (a) false
(b) true
(c) false

Exercise $\PageIndex{2}$

True or False?

(a) $4x^3+12x^2+36=O(x^3)$

(b) $.01x^5=O(48x^4)$

(d) $3x\log_2x=O(25x)$

Exercise $\PageIndex{3}$

True or False?

(a) $23\ln x=O(3x)$

(b) $7x^5=O(x^5)$

Answer: all true

Exercise $\PageIndex{4}$

Prove: $2x^5+3x^4-x^3+5x=O(x^5)$

Example $\PageIndex{5}$

Put these functions in order of increasing growth rates:

\[x^7,\qquad 6^x,\qquad 78x^2,\qquad x^2\log x,\qquad 1000x,\qquad 7, \qquad \log_{11}x\]

Answer: \[7, \qquad \log_{11}x, \qquad 1000x, \qquad 78x^2,\qquad x^2\log x, \qquad x^7,\qquad 6^x\]

Exercise $\PageIndex{6}$

Take this statement and express it in Big O notation: $|2x^4-5x^3+x^2-5| \leq 13|x^4|$ for $x>1.$