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Mathematics LibreTexts

8.1: Big O

  • Page ID
    28955
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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) \(x^{13}=O(3^x)\)

    (c) \(-2x=O(58\log_{35}x)\)

    Answer

    (a) false
    (b) true
    (c) false

    Exercise \(\PageIndex{2}\)

    True or False?

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

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

    (c) \(4^x=O(x^7)\)

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

    Exercise \(\PageIndex{3}\)

    True or False?

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

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

    (c) \(x^5=O(7x^5)\)

    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 \(x>1.\)