introduction to cs 4104courses.cs.vt.edu/~cs4104/murali/fall09/lectures/... · i prerequisites: cs...

About the Course Data and Algorithm Analysis Proof by Induction

Introduction to CS 4104

T. M. Murali

August 24, 2009

T. M. Murali August 24, 2009 Introduction to CS 4104


Course InformationI Instructor

I T. M. Murali, 2160B Torgerson, 231-8534, [email protected] O�ce Hours: 9:30am�11:30am Mondays and Wednesdays

I Teaching assistantI Chris Poirel, [email protected] O�ce Hours: to be decided

I Class meeting timeI MW 2:30pm�3:45pm, McBryde 134

I Keeping in TouchI Course web site

http://courses.cs.vt.edu/~cs4104/spring2009, updated

regularly through the semesterI Listserv: [email protected]

I Prerequisites: CS 2604 or CS 2606, MATH 3134 or MATH 3034


http://courses.cs.vt.edu/~cs4104/spring2009


Course InformationI Instructor

I T. M. Murali, 2160B Torgerson, 231-8534, [email protected] O�ce Hours: 9:30am�11:30am Mondays and Wednesdays

I Teaching assistantI Chris Poirel, [email protected] O�ce Hours: to be decided

I Class meeting timeI MW 2:30pm�3:45pm, McBryde 134

I Keeping in TouchI Course web site

http://courses.cs.vt.edu/~cs4104/spring2009, updated

regularly through the semesterI Listserv: [email protected]

I Prerequisites: CS 2604 or CS 2606, MATH 3134 or MATH 3034T. M. Murali August 24, 2009 Introduction to CS 4104

http://courses.cs.vt.edu/~cs4104/spring2009


Required Course Textbook

I Algorithm DesignI Jon Kleinberg and Éva TardosI Addison-WesleyI 2006I ISBN: 0-321-29535-8



Course Goals

I Learn methods and principles to construct algorithms.I Learn techniques to analyze algorithms mathematically for

correctness and e�ciency (e.g., running time and space used).I Course roughly follows the topics suggested in textbook

I Measures of algorithm complexityI GraphsI Greedy algorithmsI Divide and conquerI Dynamic programmingI Network �ow problemsI NP-completeness



Required Readings

I Reading assignment available on the website.I Read before class.



Lecture Slides

I Will be available on class web site.I Usually posted just before class.I Class attendance is extremely important.

Lecture in classcontains signi�cant and substantial additions to material on theslides.



Lecture Slides

I Will be available on class web site.I Usually posted just before class.I Class attendance is extremely important. Lecture in class

contains signi�cant and substantial additions to material on theslides.



Homeworks

I Posted on the web site ≈ one week before due date.I Announced on the class listserv.I Prepare solutions digitally but hand in hard-copy.

I Solution preparation recommended in LATEX.



Examinations

I Take-home midterm.I Take-home �nal (comprehensive).I Prepare digital solutions (recommend LATEX).

I Examinations may change to be in class.



Examinations

I Take-home midterm.I Take-home �nal (comprehensive).I Prepare digital solutions (recommend LATEX).I Examinations may change to be in class.



Grades

I Homeworks: ≈ 8, 60% of the grade.I Take-home midterm: 15% of the grade.I Take-home �nal: 25% of the grade.



What is an Algorithm?

Chamber's A set of prescribed computational procedures for solvinga problem; a step-by-step method for solving a problem.

Knuth, TAOCP An algorithm is a �nite, de�nite, e�ective procedure,with some input and some output.



Origin of the word �Algorithm�

1. From the Arabic al-Khwarizmi, a native of Khwarazm, a namefor the 9th century mathematician, Abu Ja'far Mohammed benMusa.

He wrote �Kitab al-jabr wa'l-muqabala,� which evolvedinto today's high school algebra text.

2. From Al Gore, the former U.S. vice-president who invented theinternet.

3. From the Greek algos (meaning �pain,� also a root of�analgesic�) and rythmos (meaning ��ow,� also a root of�rhythm�). �Pain �owed through my body whenever I worked onCS 4104 homeworks.� � former student.







3. From the Greek algos (meaning �pain,� also a root of�analgesic�) and rythmos (meaning ��ow,� also a root of�rhythm�).

�Pain �owed through my body whenever I worked onCS 4104 homeworks.� � former student.




1. From the Arabic al-Khwarizmi, a native of Khwarazm, a namefor the 9th century mathematician, Abu Ja'far Mohammed benMusa. He wrote �Kitab al-jabr wa'l-muqabala,� which evolvedinto today's high school algebra text.





Problem Example

Find Minimum

INSTANCE: Nonempty list x1, x2, . . . , xn of integers.

SOLUTION: Pair (i , xi) such that xi = min{xj | 1 ≤ j ≤ n}.



Algorithm Example

Find-Minimum(x1, x2, . . . , xn)1 i ← 1

2 for j ← 2 to n

3 do if xj < xi4 then i ← j

5 return (i , xi)



Running Time of Algorithm


2 for j ← 2 to n

3 do if xj < xi4 then i ← j

5 return (i , xi)

I At most 2n − 1 assignments and n − 1 comparisons.



Correctness of Algorithm: Proof 1


2 for j ← 2 to n

3 do if xj < xi

4 then i ← j

5 return (i , xi )

I Proof by contradiction: Suppose algorithm returns (k , xk) but thereexists 1 ≤ l ≤ n such that xl < xk and xl = min{xj | 1 ≤ j ≤ n}.

I Is k < l? No. Since the algorithm returns (k , xk), xk ≤ xj , for allk < j ≤ n. Therefore l < k .

I What does the algorithm do when j = l? It must set i to l , since we

have been told that xl is the smallest element.I What does the algorithm do when j = k (which happens after j = l)?

Since xl < xk , the value of i does not change.I Therefore, the algorithm does not return (k , xk) yielding a

contradiction.





2 for j ← 2 to n

3 do if xj < xi

4 then i ← j

5 return (i , xi )

I Proof by contradiction:

Suppose algorithm returns (k , xk) but thereexists 1 ≤ l ≤ n such that xl < xk and xl = min{xj | 1 ≤ j ≤ n}.





contradiction.





2 for j ← 2 to n

3 do if xj < xi

4 then i ← j

5 return (i , xi )






contradiction.





2 for j ← 2 to n

3 do if xj < xi

4 then i ← j

5 return (i , xi )


I Is k < l?

No. Since the algorithm returns (k , xk), xk ≤ xj , for allk < j ≤ n. Therefore l < k .




contradiction.





2 for j ← 2 to n

3 do if xj < xi

4 then i ← j

5 return (i , xi )






contradiction.





2 for j ← 2 to n

3 do if xj < xi

4 then i ← j

5 return (i , xi )






contradiction.T. M. Murali August 24, 2009 Introduction to CS 4104




2 for j ← 2 to n

3 do if xj < xi

4 then i ← j

5 return (i , xi )

I Proof by induction: What is true at the end of each iteration?

I Claim: xi = min{xm | 1 ≤ m ≤ j}, for all 1 ≤ j ≤ n.I Claim is true ⇒ algorithm is correct (set j = n).I Proof of the claim involves three steps.

1. Base case: j = 1 (before loop). xi = min{xm | 1 ≤ m ≤ 1} is triviallytrue.

2. Inductive hypothesis: Assume xi = min{xm | 1 ≤ m ≤ j}.3. Inductive step: Prove xi = min{xm | 1 ≤ m ≤ j + 1}.

I In the loop, i is set to be j + 1 if and only if xj+1 < xi .I Therefore, xi is the smallest of x1, x2, . . . , xj+1 after the loop

ends.





2 for j ← 2 to n

3 do if xj < xi

4 then i ← j

5 return (i , xi )

I Proof by induction: What is true at the end of each iteration?I Claim: xi = min{xm | 1 ≤ m ≤ j}, for all 1 ≤ j ≤ n.I Claim is true

⇒ algorithm is correct (set j = n).I Proof of the claim involves three steps.




ends.





2 for j ← 2 to n

3 do if xj < xi

4 then i ← j

5 return (i , xi )

I Proof by induction: What is true at the end of each iteration?I Claim: xi = min{xm | 1 ≤ m ≤ j}, for all 1 ≤ j ≤ n.I Claim is true ⇒ algorithm is correct (set j = n).

I Proof of the claim involves three steps.




ends.





2 for j ← 2 to n

3 do if xj < xi

4 then i ← j

5 return (i , xi )

I Proof by induction: What is true at the end of each iteration?I Claim: xi = min{xm | 1 ≤ m ≤ j}, for all 1 ≤ j ≤ n.I Claim is true ⇒ algorithm is correct (set j = n).I Proof of the claim involves three steps.




ends.T. M. Murali August 24, 2009 Introduction to CS 4104


Format of Proof by Induction

I Goal: prove some proposition P(n) is true for all n.I Strategy: prove base case, assume inductive hypothesis, prove

inductive step.

I Base case: prove that P(1) or P(2) (or P(small number)) istrue.

I Inductive hypothesis: assume P(n − 1) is true.I Inductive step: prove that P(n − 1)⇒ P(n).I Why does this strategy work?





inductive step.I Base case: prove that P(1) or P(2) (or P(small number)) is

true.I Inductive hypothesis: assume P(n − 1) is true.I Inductive step: prove that P(n − 1)⇒ P(n).

I Why does this strategy work?





inductive step.I Base case: prove that P(1) or P(2) (or P(small number)) is

true.I Inductive hypothesis: assume P(n − 1) is true.I Inductive step: prove that P(n − 1)⇒ P(n).I Why does this strategy work?



Sum of �rst n natural numbers

P(n) =n∑

i=1

i =

n(n + 1)2

.



Sum of �rst n natural numbers

P(n) =n∑

i=1

i =n(n + 1)

2.



Recurrence Relation

Given

P(n) =

{P(bn

2c) + 1 if n > 1

1 if n = 1

prove that

P(n) ≤

1 + log2 n.

I Use strong induction: In the inductive hypothesis, assume thatP(i) is true for all i < n.



Recurrence Relation

Given

P(n) =

{P(bn

2c) + 1 if n > 1

1 if n = 1

prove that

P(n) ≤ 1 + log2 n.

I Use strong induction: In the inductive hypothesis, assume thatP(i) is true for all i < n.


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