DOC PREVIEW
CMU CS 15251 - Lecture

This preview shows page 1-2-3-4-5-32-33-34-35-65-66-67-68-69 out of 69 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 69 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 69 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 69 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 69 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 69 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 69 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 69 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 69 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 69 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 69 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 69 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 69 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 69 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 69 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 69 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

On Raising A Number To A PowerSo Far We Have Seen:Slide 3Let’s Articulate A New One:Slide 5Compiler Translationb:=a8General VersionPowering By Repeated MultiplicationExampleDefinition of M(n)What is M(n)? Can we calculate it exactly? Can we approximate it?Some Very Small ExamplesM(8) = ?Slide 15Slide 16Slide 17Slide 18Applying Two IdeasWhat is the more essential representation of M(n)?The a is a red herring.Addition ChainsExamplesAddition Chains Are A Simpler To Represent The Original ProblemSlide 25Some Addition Chains For 30Slide 27Binary RepresentationBinary Method Repeated Squaring Method Repeated Doubling MethodBinary Method Applied To 30Rhind Papyrus (1650 BC) What is 30 times 5?Rhind Papyrus (1650 BC) Actually used faster chain for 30*5.The Egyptian ConnectionSlide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40GENERALIZATIONSlide 42A Lower Bound IdeaInduction ProofLet Sk be the statement that no k stage addition chain will contain a number greater than 2kProof By Invariant (Induction)Change Of VariableTheorem: 2i is the largest number that can be made in i stages, and can only be made by repeated doublingSlide 495 < M(30)Suppose M(15) = 4Slide 52Slide 53Rhind Papyrus (1650 BC)Factoring BoundSlide 56Slide 57Corollary (Using Induction)More CorollariesM(33) < M(3) + M(11)Conjecture: M(2n) = M(n) +1 (A. Goulard)Open ProblemConjectureSlide 64High Level PointStudy BeeSlide 67Slide 68REFERENCESOn Raising A Number To A PowerGreat Theoretical Ideas In Computer ScienceSteven RudichCS 15-251 Spring 2004Lecture 5 Jan 27, 2004 Carnegie Mellon University1515aaSo Far We Have Seen:Don’t let the representation choose you. CHOOSE THE REPRESENTATION!Representation:Representation:Understand the relationship betweenUnderstand the relationship betweendifferent representations of the samedifferent representations of the sameinformation or ideainformation or idea123So Far We Have Seen:Induction is how we define and manipulate mathematical ideas.Let’s Articulate A New One:AbstractionAbstraction: : Abstract away the inessential Abstract away the inessential features of a problem or solutionfeatures of a problem or solution=Even very simple computational problems can be surprisingly subtle.Compiler TranslationA compiler must translate a high level language (e.g., C) with complex operations (e.g., exponentiation) into a lower level language (e.g., assembly) that can only support simpler operations (e.g., multiplication).b:=a8b:=a*ab:=b*ab:=b*ab:=b*ab:=b*ab:=b*ab:=b*ab:=a*ab:=b*bb:=b*bThis method costs only 3 multiplications. The savings are significant if b:=a8 is executed often.General VersionGiven a constant k, how do we implement b:=ak with the fewest number of multiplications?Powering By Repeated MultiplicationInput: a,nOutput:A sequence starting A sequence starting with a, ending with awith a, ending with ann, , and such that each and such that each entry other than the entry other than the first is the product of first is the product of previous entries.previous entries.ExampleInput: a,5Output: a, a2, a3, a4, a5orOutput: a, a2, a3, a5 orOutput: a, a2, a4, a5Definition of M(n)M(n) = The minimum number of multiplications required to produce an by repeated multiplicationWhat is M(n)? Can we calculate it exactly? Can we approximate it?Exemplification:Exemplification:Try out a problem orTry out a problem or solution on small examples. solution on small examples.Some Very Small ExamplesWhat is M(1)?–M(1) = 0 [a]•What is M(0)?–M(0) is not clear how to define•What is M(2)?–M(2) = 1 [a, a2]M(8) = ?a, a2, a4, a8 is a way to make a8 in 3 multiplications. What does this tell us about the value of M(8)?M(8) = ?a, a2, a4, a8 is a way to make a8 in 3 multiplications. What does this tell us about the value of M(8)?M( )8 3Upper Bound? ( ) M 8 3Lower Bound? ( ) M 8 3Lower Bound3 8 M( )Exhaustive Search. There are only two sequences with 2 multiplications. Neither of them make 8: a, a2, a3 & a, a2, a43 8 3 M( )Upper BoundLower BoundM(8) = 3Applying Two IdeasAbstractionAbstraction: : Abstract away the inessential Abstract away the inessential features of a problem or solutionfeatures of a problem or solution=Representation:Representation:Understand the relationship betweenUnderstand the relationship betweendifferent representations of the samedifferent representations of the sameinformation or ideainformation or idea123What is the more essential representation of M(n)?AbstractionAbstraction: : Abstract away the inessential Abstract away the inessential features of a problem or solutionfeatures of a problem or solution=Representation:Representation:Understand the relationship betweenUnderstand the relationship betweendifferent representations of the samedifferent representations of the sameinformation or ideainformation or idea123 ((( )))The a is a red herring.ax times ay is ax+yEverything besides the exponent is inessential. This should be viewed as a problem of repeated addition, rather than repeated multiplication.Addition ChainsM(n) = Number of stages required to make n, where we start at 1 and in each subsequent stage we add two previously constructed numbers.ExamplesAddition Chain for 8:1 2 3 5 8Minimal Addition Chain for 8:1 2 4 8Addition Chains Are A Simpler To Represent The Original ProblemAbstractionAbstraction: : Abstract away the inessential Abstract away the inessential features of a problem or solutionfeatures of a problem or solution=Representation:Representation:Understand the relationship betweenUnderstand the relationship betweendifferent representations of the samedifferent representations of the sameinformation or ideainformation or idea123M(30) = ?1515aaSome Addition Chains For 301 2 4 8 16 24 28 301 2 4 5 10 20 301 2 3 5 10 15 301 2 4 8 10 20 30? ( )? ( ?  MM n30 6)Binary RepresentationLet Bn be the number of 1s in the binary representation of n. Ex: B5 = 2 since 101 is the binary representation of 5Proposition: Bn 6 b log2 (n) c + 1 The length of the binary representation of n is bounded by this quantity.Binary MethodRepeated Squaring MethodRepeated Doubling MethodPhase I (Repeated Doubling)For Add largest so far to itself(1, 2, 4, 8, 16, . . . )Phase II (Make n from bits and pieces)Expand n in binary to see how n is the sum of Bn powers of 2. Use Bn-1 stages to make n from the powers of 2 created in phase Ilog2n stages:Total Cost: log21n Bn Binary Method Applied To 30Binary30 11110Phase I1 12 104


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

56 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

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?