Lecture 15 Complexity of Functions

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Lecture 15Complexity of Functions

• CSCI 1900 Mathematics for Computer Science
• Spring 2012
• Bill Pine

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Lecture Introduction

• Reading
• Kolman Sections 5.2, 5.3
• Examine some functions occurring frequently in
  Computer Science
• Characterize the efficiency of an algorithm
• Big-Oh
• Big-Theta

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Functions in CS

• mod n function
• fn( m ) m mod n
• Example
• f3( 7 ) 1 f3( 8 ) 2 f3( 111 ) 0
• Factorial function
• f ( n ) n! n! n (n-1)!
• f ( 0 ) 1 f ( 1 ) 1 f ( 2 ) 2 f ( 3 ) 6f
  ( 10 )gt 106 f ( 20 )gt1018 f (70) gt 10100

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Functions in CS (cont)

• Floor function
• f ( x ) largest integer x
• f (2.1) 2
• f (2.9) 2
• f (2.999999) 2
• f (3) 3
• f (-2.3) -3 Nota Bene -2 gt -2.3

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Functions in CS (cont)

• Ceiling function
• f( x ) smallest integer x
• f( 2.000001 ) 3
• f( 2.9 ) 3
• f( 2.999999 ) 3
• f( 3 ) 3
• f( 3.000001 ) 4
• f( -2.3 ) -2 Nota Bene -2 gt -2.3

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Functions in CS (cont)

• Exponential
• f ( x )2x
• f ( 1 )2
• f ( 2 )4
• f ( 2.2 )4.4957
• f ( 0 )1
• f ( -1 )1/2
• f ( -2 )1/4

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Functions in CS (cont)

• Logarithm
• fn( x )logn(x) the power to which n must
  be raised to yield x
• f2( 4 ) 2
• f2( 8 ) 3
• f2( 2 ) 1
• f2( 1 ) 0
• f2( 1/2 ) -1
• f2( 1/16 ) -4
• f2( 0 ) undefined

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Growth of Functions

• Previous example show that functions grow at
  different rates
• Specifically, let f (x) x, g(x) 2x
• f (1) 1, g(1) 2
• f (10) 10, but g (10) 1024
• Polynomial functions
• n n2 n2.2 n3

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Growth of Functions (cont)

• All log functions grow at the same rate,
  regardless of the base
• log2 ( n ) grows at the same rate as log100 ( n )
• We usually write log2 as lg
• Recall logn( k ) logm( k ) / logm( n )
• Powers of the log function
• (log n ) (log n)2 (log n)304 n

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Growth of Functions (cont)

• log compared to polynomials
• lg n n n lg( n ) n ( lg( n ) )2 n
  (lg( n )1670 n2
• Polynomials compare to exponentials
• n2 n3 n2876 2n
• Exponentials compared to factorial
• 2n n!

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Growth of Functions (cont)

• In general, the classes of functions, in order of
  increasing running time are
• constant log( n ) n n log( n )
  n2 n3 2n n!

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Big-Oh Notation

• f ? O( g ) if there exist two constants, c and n0
  such that f (n) ? c g(n) for all n ? n0
• This means f grows no faster than g does
• What is in O(n3)?
• Anything that grows no faster than n3
• f ( n ) 10n3
• f ( n ) n3 100n2 n
• f ( n ) n2
• f ( n ) n lg n
• f ( n ) lg n
• f ( n ) 2

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Big-Oh Example

• Show that n3 100n2 20 ? O( n3 )
• Need to find a c and n0 such that c n3 gt n3
  100n2 n for all n gt n0
• Two important consequences of f ? O( g )
• g serves as an upper bound on f
• We ignore the behavior for small values of n

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Big-Theta Notation

• f ? ? ( g ), if f and g have same order
• f ? O( g ) and g ? O( f )
• What is in ?( n3 )
• f ( n ) n3 100n2 n is in ?( n3 )
• f ( n ) 2 is not in ?( n3 )
• f ( n ) n2 is not in ?( n3 )
• f ( n ) lg n is not in ?( n3 )
• f ( n ) n lg n is not in ?( n3 )

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Running Time

• The ?-class of a function that describes the
  number of steps performed by an algorithm is
  called the running time of the algorithm
• This allows us to compare algorithms for a
  specific task
• For example
• bubble sort ? ?( n2 )
• quicksort ? ?( n lg n)
• What is the fastest running-time for a
  comparison-based sorting algorithm?

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Determining Running Time

• To compare two algorithms running times
• Select an operation that is central to the
  algorithm
• For searching, we might choose comparison
• For sorting, we might choose comparison or
  perhaps data saving
• Examine the algorithm to determine how the count
  of the key operation depends upon input size
• To evaluate a single algorithm
• Count the total number of operations
• Examine the algorithm to determine how the count
  of the key operation depends upon input size

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Running Time Example

• Consider the following function
• function meanOf( n )
• sum ? 0
• for i 1 thru n
• sum ? sum i
• mean sum / n
• return mean

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Running Time Example (cont)

• What is the input size?
• Identify and count the characterizing operation
• Number of pre-loop operations
• Number of operations in the loop
• Number of post-loop operations
• What is the running time?
• Is there a more efficient way to determine meanOf
  ?

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Key Concepts Summary

• Examined some functions useful in Computer
  Science
• Characterize the efficiency of an algorithm
• Big-Oh
• Big-Theta
• Reading for next time
• Kolman Sections 7.1, 7.2