Discrrete mathematics for computer science asymptotic

Three Concrete Examples• Polynomials • Logarithmic functions • Exponential functions 3/26/12... Logarithmic Functions• A function f is logarithmic if it is Θlogbn for some constant b.. •

Growth Rates of Functions

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Asymptotic Equivalence

• Def:

For example,

Note that n2 +1 is being used to name the

function f such that f(n) = n2 +1 for every

n

f (n) : g(n) iff limn    

   

⎛

⎝⎜

⎞

n2     

 



 



 



 

An example: Stirling’s

formula

n! : n

e

⎛

⎝⎜

⎞

⎠⎟



 

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Little-Oh: f = o(g)

• Def: f(n) = o(g(n)) if

lim

n

   

     

• For example, n2 = o(n3) since

lim

n



  

 



  

= o( ∙ ) is “all one symbol”

• “f = o(g)” is really a strict partial

order on functions

• NEVER write “o(g) = f”, etc

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Big-Oh: O(∙)

• Asymptotic Order of Growth:

• “f grows no faster than g”

• A Weak Partial Order

f (n)      

 





⎛

⎝⎜

⎞

⎠⎟  

Growth Order



 

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f = o(g) implies f = O(g)

because if  lim

n

   

     

 

   

     

     

• f = Ω(g) means g = O(f)

• “f grows at least as quickly as g”

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Big-Theta: (∙) �

“Same order of growth”

f (n)  



     



     

So, for example,

3n2      

Rough Paraphrase

• f∼g: f and g grow to be roughly

equal

• f=o(g): f grows more slowly than g

• f=O(g): f grows at most as quickly

as g

• f=Ω(g): f grows at least as quickly

as g

• f=�(g): f and g grow at the same rate

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Equivalent Defn of O(∙)

f (n)       

        

          

“From some point on, the

value of f is at most a

constant multiple of the value

of g”

Three Concrete Examples

• Polynomials

• Logarithmic functions

• Exponential functions

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• A (univariate) polynomial is a function such as f(n) = 3n 5 +2n 2 -n+2 (for all natural numbers n)

• This is a polynomial of degree 5 (the largest exponent)

• Or in general

• Theorem:

– If a<b then any polynomial of degree a is o(any

polynomial of degree b )

– If a≤b then any polynomial of degree a is O(any

polynomial of degree b )

f (n)   







Logarithmic Functions

• A function f is logarithmic if it is

Θ(logbn) for some constant b

• Theorem: All logarithmic functions

are Θ() of each other, and are Θ(any logarithmic function of a polynomial)

• Theorem: Any logarithmic function is o(any polynomial)

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Exponential Functions

• A function is exponential if it is Θ(cn) for some constant c>1

• Theorem: Any polynomial is o(any

exponential)

• If c<d then cn=o(dn)

Growth Rates and Analysis of

Algorithms

• Let f(n) measure the amount of time taken by

an algorithm to solve a problem of size n.

• Most practical algorithms have polynomial

running times

• E.g sorting algorithms generally have running times that are quadratic (polynomial or degree 2) or less (for example, O(n log n)).

• Exhaustive search over an exponentially

growing set of possible answers requires

exponential time.

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Another way to look at it

• Suppose an algorithm can solve a problem of size S

in time T and you give it twice as much time.

• If the running time is f(n)=n 2 , so that T=S 2 , then in time 2T you can solve a problem of size (2 1/2 )∙S

• If the running time is f(n)=2 n , so that T=2 S , then in time 2T you can solve a problem of size S+1

• In general doubling the time available to a

polynomial algorithm results in a MULTIPLICATIVE

increase in the size of the problem that can be solved

• But doubling the time available to an exponential

algorithm results in an ADDITIVE increase to the size

of the problem that can be solved.

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