DOC PREVIEW
CMU CS 15251 - lecture

This preview shows page 1-2-3-4-26-27-28-53-54-55-56 out of 56 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 56 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 56 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 56 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 56 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 56 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 56 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 56 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 56 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 56 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 56 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 56 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 56 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Grade School Multiplication: Quadratic timeHow much time does it take?Worst Case TimeWhat is T(n)?Slide 25Slide 26Slide 27Factoring The Number N By Trial DivisionSlide 29Notation to Discuss Growth RatesSlide 31Other Useful Notation: ΩSlide 33Yet More Useful Notation: ΘSlide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Names For Some Growth RatesLarge Growth RatesSmall Growth RatesSome Big OnesSlide 49Slide 50Slide 51Ackermann’s FunctionSlide 53Slide 54Slide 55Slide 5615-251Great Theoretical Ideas in Computer ScienceThis is The Big Oh!Lecture 21 (March 31, 2009)** ** ** ** ** ** ** ** ** ** ** +How to add 2 n-bit numbers*** ** ** ** ** ** * ** ** ** ** ** +How to add 2 n-bit numbers*** ** ** ** ** * ** * *** ** ** ** ** +How to add 2 n-bit numbers*** ** ** ** * ** * *** * *** ** ** ** ** +How to add 2 n-bit numbers*** ** ** * ** * *** * *** * *** ** ** ** ** +How to add 2 n-bit numbers*** * *** * *** * *** * *** * *** * *** * *** * *** * *** *** * +* * “Grade school addition”How to add 2 n-bit numbers+T(n) = amount of time grade school addition uses to add two n-bit numbers* * * * * * * * * ** * * * * * * * * ** * * * * * * * * ** * * * * * * * * * *Time complexity of grade school additionWhat do we mean by “time”?Our GoalWe want to define “time” in a way that transcends implementation details and allows us to make assertions about grade school addition in a very general yet useful way.A given algorithm will take different amounts of time on the same inputs depending on such factors as:–Processor speed–Instruction set–Disk speed–Brand of compilerRoadblock ???On any reasonable computer, adding 3 bits and writing down the two bit answer can be done in constant timePick any particular computer M and define c to be the time it takes to perform on that computer. Total time to add two n-bit numbers using grade school addition: cn [i.e., c time for each of n columns]On another computer M’, the time to perform may be c’.Total time to add two n-bit numbers using grade school addition: c’n [c’ time for each of n columns]The fact that we get a line is invariant under changes of implementations. Different machines result in different slopes, but the time taken grows linearly as input size increases. # of bits in the numberstimeMachine M: cnMachine M’: c’nThus we arrive at an implementation-independent insight: Grade School Addition is a linear time algorithmThis process of abstracting away details and determining the rate of resource usage in terms of the problem size n is one of the fundamental ideas in computer science.Time vs Input SizeFor any algorithm, define Input Size = # of bits to specify its inputs.Define TIMEn = the worst-case amount of time used by the algorithmon inputs of size nWe often ask: What is the growth rate of Timen ?X* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *n2How to multiply 2 n-bit numbers.X* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *n2How to multiply 2 n-bit numbers.The total time is bounded by cn2 (abstracting away the implementation details).# of bits in the numberstimeGrade School Addition: Linear timeGrade School Multiplication: Quadratic timeNo matter how dramatic the difference in the constants, the quadratic curve will eventually dominate the linear curveHow much time does it take to square the number n using grade school multiplication?Grade School Multiplication:Quadratic time# of bits in numberstimeInput size is measured in bits, unless we say otherwise.c(log n)2 time to square the number nHow much time does it take?Nursery School Addition Input: Two n-bit numbers, a and b Output: a + bStart at a and increment (by 1) b timesT(n) = ?If b = 000…0000, then NSA takes almost no timeIf b = 1111…11111, then NSA takes cn2n timeWorst Case TimeWorst Case Time T(n) for algorithm A:T(n) = Max[all permissible inputs X of size n] Runtime(A,X)Runtime(A,X) = Running time of algorithm A on input X.What is T(n)?Kindergarten Multiplication Input: Two n-bit numbers, a and b Output: a * bStart with a and add a, b-1 timesRemember, we always pick the WORST CASE input for the input size n. Thus, T(n) = cn2nThus, Nursery School addition and Kindergarten multiplication are exponential time. They scale HORRIBLY as input size grows.Grade school methods scale polynomially: just linear and quadratic. Thus, we can add and multiply fairly large numbers.If T(n) is not polynomial, the algorithm is not efficient: the run time scales too poorly with the input size.This will be the yardstick with which we will measure “efficiency”.Multiplication is efficient, what about “reverse multiplication”?Let’s define FACTORING(N) to be any method to produce a non-trivial factor of N, or to assert that N is prime.Factoring The Number N By Trial DivisionTrial division up to Nfor k = 2 to  N doif k | N thenreturn “N has a non-trivial factor k”return “N is prime”cN (logN)2 time if division is c(logN)2 timeIs this efficient?No! The input length n = log N. Hence we’re using c 2n/2 n2 time.Can we do better?We know of methods for FACTORING that are sub-exponential (about 2n1/3 time) but nothing efficient.Notation to Discuss Growth RatesFor any monotonic function f from the positive integers to the positive integers, we say “f = O(n)” or “f is O(n)”If some constant times n eventually dominates f[Formally: there exists a constant c such that for all sufficiently large n: f(n) ≤ cn ]# of bits in numberstimef = O(n) means that there is a line that can be drawn that stays above f from some point onFor any monotonic function f from the positive integers to the positive integers, we say “f = Ω(n)” or “f is Ω(n)”If f eventually dominates some constant times n[Formally: there exists a constant c such that for all sufficiently large n: f(n) ≥ cn ]Other Useful Notation: Ω# of …


View Full Document

CMU CS 15251 - lecture

Documents in this Course
lecture

lecture

66 pages

lecture

lecture

79 pages

lecture

lecture

111 pages

lecture

lecture

85 pages

lecture17

lecture17

64 pages

Lecture

Lecture

85 pages

Lecture

Lecture

71 pages

Lecture

Lecture

70 pages

Lecture

Lecture

11 pages

Lecture

Lecture

45 pages

Lecture

Lecture

50 pages

Lecture

Lecture

93 pages

Lecture

Lecture

93 pages

Lecture

Lecture

35 pages

Lecture

Lecture

98 pages

Lecture

Lecture

74 pages

Lecture

Lecture

13 pages

Lecture

Lecture

15 pages

Lecture

Lecture

66 pages

Lecture

Lecture

82 pages

Lecture

Lecture

15 pages

Lecture

Lecture

47 pages

Lecture

Lecture

69 pages

Lecture

Lecture

13 pages

Lecture

Lecture

67 pages

Lecture

Lecture

68 pages

Lecture

Lecture

69 pages

lecture03

lecture03

44 pages

Lecture

Lecture

69 pages

Lecture

Lecture

68 pages

Lecture

Lecture

55 pages

Lecture

Lecture

79 pages

Lecture

Lecture

85 pages

Lecture

Lecture

87 pages

Lecture

Lecture

85 pages

Lecture

Lecture

103 pages

Lecture

Lecture

9 pages

Lecture

Lecture

83 pages

Lecture

Lecture

8 pages

lecture03

lecture03

68 pages

lecture24

lecture24

78 pages

lecture03

lecture03

72 pages

Thales

Thales

129 pages

lecture13

lecture13

81 pages

Lecture

Lecture

64 pages

lecture01

lecture01

59 pages

lecture11

lecture11

105 pages

Lecture

Lecture

89 pages

Lecture

Lecture

74 pages

lecture25

lecture25

57 pages

Lecture

Lecture

99 pages

lecture

lecture

50 pages

lecture

lecture

14 pages

Lecture

Lecture

78 pages

lecture

lecture

8 pages

Lecture

Lecture

98 pages

lecture

lecture

83 pages

lecture23

lecture23

88 pages

lecture

lecture

64 pages

lecture

lecture

72 pages

Lecture

Lecture

88 pages

lecture

lecture

79 pages

Lecture

Lecture

60 pages

lecture

lecture

74 pages

lecture19

lecture19

72 pages

lecture25

lecture25

86 pages

lecture

lecture

13 pages

lecture17

lecture17

79 pages

lecture

lecture

91 pages

lecture

lecture

78 pages

Lecture

Lecture

11 pages

Lecture

Lecture

54 pages

lecture

lecture

72 pages

lecture

lecture

119 pages

lecture

lecture

167 pages

lecture

lecture

73 pages

lecture

lecture

73 pages

lecture

lecture

83 pages

lecture

lecture

49 pages

lecture

lecture

16 pages

lecture

lecture

67 pages

lecture

lecture

81 pages

lecture

lecture

72 pages

lecture

lecture

57 pages

lecture16

lecture16

82 pages

lecture21

lecture21

46 pages

Lecture

Lecture

92 pages

Lecture

Lecture

14 pages

Lecture

Lecture

49 pages

Lecture

Lecture

132 pages

Lecture

Lecture

101 pages

Lecture

Lecture

98 pages

Lecture

Lecture

59 pages

Lecture

Lecture

64 pages

Lecture

Lecture

106 pages

Lecture

Lecture

70 pages

Lecture

Lecture

80 pages

Lecture

Lecture

76 pages

Lecture

Lecture

91 pages

Lecture

Lecture

112 pages

Lecture

Lecture

91 pages

Lecture

Lecture

10 pages

Lecture

Lecture

39 pages

Lecture

Lecture

79 pages

Lecture

Lecture

74 pages

Lecture

Lecture

44 pages

Lecture

Lecture

39 pages

Lecture

Lecture

99 pages

Lecture

Lecture

44 pages

Lecture

Lecture

59 pages

Lecture

Lecture

36 pages

lecture17

lecture17

36 pages

lecture

lecture

71 pages

lecture

lecture

79 pages

lecture

lecture

12 pages

lecture

lecture

43 pages

lecture

lecture

87 pages

lecture

lecture

35 pages

lecture03

lecture03

23 pages

lecture

lecture

68 pages

lecture

lecture

74 pages

lecture

lecture

21 pages

lecture

lecture

79 pages

lecture

lecture

15 pages

lecture

lecture

83 pages

lecture

lecture

13 pages

Lecture

Lecture

53 pages

lecture

lecture

55 pages

lecture

lecture

49 pages

lecture

lecture

10 pages

lecture

lecture

70 pages

lecture

lecture

12 pages

Lecture

Lecture

105 pages

Lecture

Lecture

9 pages

Lecture

Lecture

72 pages

Lecture

Lecture

66 pages

Lecture

Lecture

54 pages

Lecture

Lecture

98 pages

Lecture

Lecture

57 pages

Lecture

Lecture

75 pages

Lecture

Lecture

48 pages

lecture

lecture

53 pages

Lecture

Lecture

72 pages

Lecture

Lecture

53 pages

Lecture

Lecture

84 pages

Lecture

Lecture

55 pages

Lecture

Lecture

15 pages

Lecture

Lecture

6 pages

Lecture

Lecture

38 pages

Lecture

Lecture

71 pages

Lecture

Lecture

110 pages

Lecture

Lecture

70 pages

lecture

lecture

48 pages

lecture

lecture

76 pages

lecture

lecture

48 pages

lecture

lecture

52 pages

lecture

lecture

43 pages

lecture

lecture

81 pages

lecture

lecture

82 pages

lecture

lecture

83 pages

lecture

lecture

64 pages

lecture

lecture

71 pages

lecture

lecture

65 pages

lecture

lecture

12 pages

lecture

lecture

66 pages

lecture

lecture

50 pages

lecture

lecture

86 pages

lecture

lecture

70 pages

Lecture

Lecture

74 pages

Lecture

Lecture

54 pages

Lecture

Lecture

90 pages

lecture

lecture

78 pages

lecture

lecture

87 pages

Lecture

Lecture

55 pages

Lecture

Lecture

12 pages

lecture21

lecture21

66 pages

Lecture

Lecture

11 pages

lecture

lecture

83 pages

Lecture

Lecture

53 pages

Lecture

Lecture

69 pages

Lecture

Lecture

12 pages

lecture04

lecture04

97 pages

Lecture

Lecture

14 pages

lecture

lecture

75 pages

Lecture

Lecture

74 pages

graphs2

graphs2

8 pages

lecture

lecture

82 pages

Lecture

Lecture

8 pages

lecture

lecture

47 pages

lecture

lecture

91 pages

lecture

lecture

76 pages

lecture

lecture

73 pages

lecture

lecture

10 pages

lecture

lecture

63 pages

lecture

lecture

91 pages

lecture

lecture

79 pages

lecture

lecture

9 pages

lecture

lecture

70 pages

lecture

lecture

86 pages

lecture

lecture

102 pages

lecture

lecture

145 pages

lecture

lecture

91 pages

Lecture

Lecture

87 pages

lecture

lecture

87 pages

Notes

Notes

19 pages

Lecture

Lecture

50 pages

Lecture

Lecture

13 pages

Lecture

Lecture

97 pages

Lecture

Lecture

98 pages

Lecture

Lecture

83 pages

Lecture

Lecture

77 pages

Lecture

Lecture

102 pages

Lecture

Lecture

63 pages

Lecture

Lecture

104 pages

lecture

lecture

41 pages

lecture

lecture

14 pages

Lecture

Lecture

87 pages

Lecture

Lecture

94 pages

lecture

lecture

9 pages

Lecture

Lecture

96 pages

Lecture

Lecture

72 pages

Lecture

Lecture

35 pages

Lecture

Lecture

77 pages

Lecture

Lecture

98 pages

Lecture

Lecture

48 pages

Lecture

Lecture

66 pages

Lecture

Lecture

53 pages

lecture18

lecture18

101 pages

Lecture

Lecture

10 pages

Lecture

Lecture

70 pages

Lecture

Lecture

12 pages

Lecture

Lecture

74 pages

graphs

graphs

10 pages

Lecture

Lecture

62 pages

Lecture

Lecture

11 pages

Lecture

Lecture

71 pages

Lecture

Lecture

42 pages

lecture15

lecture15

72 pages

Lecture

Lecture

82 pages

Load more
Download lecture
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view lecture and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view lecture 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?