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Complexity Analysis of Functions in Computer Science

Explore various functions commonly used in Computer Science, analyze efficiency with Big-Oh and Big-Theta notations. Learn growth rates, logarithms, polynomials, exponentials, and compare function classes. Understand running time comparisons, Big-Oh examples, and algorithm efficiency evaluation techniques.

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Complexity Analysis of Functions in Computer Science

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  1. Lecture 16Complexity of Functions CSCI – 1900 Mathematics for Computer Science Spring 2014 Bill Pine

  2. 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 CSCI 1900

  3. 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 )> 106f ( 20 )>1018f (70) > 10100 CSCI 1900

  4. 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 > -2.3 CSCI 1900

  5. 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 > -2.3 CSCI 1900

  6. 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 CSCI 1900

  7. 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 CSCI 1900

  8. Growth of Functions CSCI 1900

  9. Growth of Functions (cont) • 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 CSCI 1900

  10. 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 • (logn ) « (log n)2 « (log n)304 « n CSCI 1900

  11. Growth of Functions (cont) • log compared to polynomials • lgn« n« nlg( n ) « n ( lg( n ) )2 « n (lg( n))1670 « n2 • Polynomials compare to exponentials • n2 « n3 « n2876 « 2n • Exponentials compared to factorial • 2n« n! CSCI 1900

  12. 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! CSCI 1900

  13. 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 CSCI 1900

  14. Big-Oh Example • Show that n3 + 100n2 +20  O( n3 ) • Need to find a c and n0 such that c n3 > n3 + 100n2 –n for all n > 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 CSCI 1900

  15. 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 – nis in ( n3 ) • f ( n ) = 2 is not in ( n3 ) • f ( n ) = n2 is not in ( n3 ) • f ( n ) = lg nis not in ( n3 ) • f ( n ) = n lg nis not in ( n3 ) CSCI 1900

  16. 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? CSCI 1900

  17. Determining Running Time • To compare two algorithm’s 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” CSCI 1900

  18. 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 CSCI 1900

  19. 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? CSCI 1900

  20. 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 CSCI 1900

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