Course information

(1)

Introduction to Algorithms 6.046J/18.401J

LECTURE1

Analysis of Algorithms

•Insertion sort

•Asymptotic analysis

•Merge sort

•Recurrences

Prof. Charles E. Leiserson

Course information

1.Staff

2.Distance learning 3.Prerequisites 4.Lectures 5.Recitations 6.Handouts 7.Textbook

8.Course website 9.Extra help

10.Registration 11.Problem sets

12.Describing algorithms 13.Grading policy

14.Collaboration policy

September 7, 2005 Introduction to Algorithms L1.2

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Analysis of algorithms

The theoretical study of computer-program performance and resource usage.

What’s more important than performance?

•modularity

•correctness

•maintainability

•functionality

•robustness

•user-friendliness

•programmer time

•simplicity

•extensibility

•reliability

September 7, 2005

Introduction to Algorithms L1.3

Why study algorithms and performance?

‧Algorithms help us to understand scalability.

‧Performance often draws the line between what is feasible and what is impossible.

‧Algorithmic mathematics provides a language for talking about program behavior.

‧Performance is the currency of computing.

‧The lessons of program performance generalize to other computing resources.

‧Speed is fun!

September 7, 2005

(3)

The problem of sorting

Input: sequence 〈a ₁ , a ₂ , …, a _n 〉 of numbers.

Output: permutation 〈a' ₁ , a' ₂ , …, a' _n 〉 Such that a' ₁ ≤a' ₂ ≤…≤a' _n .

Example:

Input: 8 2 4 9 3 6 Output: 2 3 4 6 8 9

September 7, 2005 L1.5

INSERTION-SORT

I

NSERTION

-S

ORT

(A, n) ⊳ A[1 . . n]

for j ←2 to n do key ← A[ j]

i ← j –1

while i > 0 and A[i] > key do A[i+1] ← A[i]

i ← i –1 A[i+1] = key

September 7, 2005

“pseudocode”

(4)

September 7, 2005

I

NSERTION

-S

ORT

(A, n) ⊳ A[1 . . n]

for j ←2 to n do key ← A[ j]

i ← j –1

while i > 0 and A[i] > key do A[i+1] ← A[i]

i ← i –1 A[i+1] = key

“pseudocode”

INSERTION-SORT

i

A:

sorted key

n 1 j

Example of insertion sort

8 2 4 9 3 6

September 7, 2005

(5)

Example of insertion sort

8 2 4 9 3 6

8 2 4 9 3 6 2 8 4 9 3 6

Example of insertion sort

(6)

September 7, 2005

Example of insertion sort

8 2 4 9 3 6 2 8 4 9 3 6

September 7, 2005

Example of insertion sort

8 2 4 9 3 6

2 8 4 9 3 6

2 4 8 9 3 6

(7)

Example of insertion sort

8 2 4 9 3 6 2 8 4 9 3 6 2 4 8 9 3 6

September 7, 2005

Example of insertion sort

8 2 4 9 3 6

2 8 4 9 3 6

2 4 8 9 3 6

(8)

Example of insertion sort

8 2 4 9 3 6 2 8 4 9 3 6 2 4 8 9 3 6 2 4 8 9 3 6

Example of insertion sort

8 2 4 9 3 6

2 8 4 9 3 6

2 4 8 9 3 6

2 3 4 8 9 6

(9)

Example of insertion sort

8 2 4 9 3 6 2 8 4 9 3 6 2 4 8 9 3 6 2 4 8 9 3 6 2 3 4 8 9 6

Example of insertion sort

8 2 4 9 3 6 2 8 4 9 3 6 2 4 8 9 3 6 2 4 8 9 3 6 2 3 4 8 9 6

2 3 4 6 8 9 ^done

(10)

Running time

•The running time depends on the input: an already sorted sequence is easier to sort.

•Parameterize the running time by the size of the input, since short sequences are easier to sort than long ones.

•Generally, we seek upper bounds on the running time, because everybody likes a guarantee.

Kinds of analyses

Worst-case: (usually)

• T(n) =maximum time of algorithm on any input of size n.

Average-case: (sometimes)

• T(n) =expected time of algorithm over all inputs of size n.

• Need assumption of statistical distribution of inputs.

Best-case: (bogus)

• Cheat with a slow algorithm that works fast on some input.

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“Asymptotic Analysis”

Machine-independent time

What is insertion sort’s worst-case time?

•It depends on the speed of our computer:

•relative speed (on the same machine),

•absolute speed (on different machines).

BIG IDEA:

•Ignore machine-dependent constants.

•Look at growth of T(n) as n→∞.

“Asymptotic Analysis”

Θ-notation

Math:

Θ(g(n)) = { f (n): there exist positive constants c ₁ , c ₂ , and n ₀

such that 0 ≤c ₁ g(n) ≤f (n) ≤c ₂ g(n) for all n≥n ₀ }

Engineering:

•Drop low-order terms; ignore leading constants.

•Example: 3n ³ + 90n ² –5n+ 6046 = Θ(n ³ )

(12)

Asymptotic performance

When n gets large enough, a Θ(n ² )algorithm always beats a Θ(n ³ )algorithm.

•We shouldn’t ignore asymptotically slower algorithms, however.

•Real-world design situations often call for a careful balancing of engineering objectives.

•Asymptotic analysis is a useful tool to help to structure our thinking.

T(n)

n n

₀

Insertion sort analysis

Worst case: Input reverse sorted.

[arithmetic series ] Average case: All permutations equally likely.

Is insertion sort a fast sorting algorithm?

•Moderately so, for small n.

•Not at all, for large n.

) ( ) ( )

(

²

2

n j

n T

n

j

Θ

= Θ

= ∑

=

) ( ) 2 / ( )

(

²

2

n j

n T

n

j

Θ

= Θ

= ∑

=

(13)

Merge sort

MERGE-SORT A[1 . . n]

1.If n= 1, done.

2.Recursively sort A[ 1 . . .n/2.]and A[ [n/2]+1 . . n ] .

3.“Merge” the 2 sorted lists.

Key subroutine: MERGE

Merging two sorted arrays

20 12 13 11 7 9 2 1

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September 7, 2005 L1. 27

Merging two sorted arrays

20 12 13 11 7 9 2 1

1 Merging two sorted arrays

20 12 13 11 7 9 2 1

1 20 12

13 11

7 9

2

(15)

Merging two sorted arrays

20 12 13 11 7 9 2 1

1 20 12 13 11 7 9 2

2 Merging two sorted arrays

20 12 13 11 7 9 2 1

1 20 12 13 11 7 9 2

2 20 12

13 11

7 9

(16)

Merging two sorted arrays

20 12 13 11 7 9 2 1

1 20 12 13 11 7 9 2

20 12 13 11 7 9

2 7

Merging two sorted arrays

20 12 13 11 7 9 2 1

1 20 12 13 11 7 9 2

20 12 13 11 7 9

2 7

20 12

13 11

9

(17)

Merging two sorted arrays

20 12 13 11 7 9 2 1

1 20 12 13 11 7 9 2

20 12 13 11 7 9

20 12 13 11 9

2 7 9

Merging two sorted arrays

20 12 13 11 7 9 2 1

20 12 13 11 7 9 2

20 12 13 11 7 9

20 12 13 11 9

20 12 13 11

1 2 7 9

(18)

Merging two sorted arrays

20 12 13 11 7 9 2 1

20 12 13 11 7 9 2

20 12 13 11 7 9

20 12 13 11 9

20 12 13 11

1 2 7 9 11

Merging two sorted arrays

20 12 13 11 7 9 2 1 1

20 12 13 11 7 9 2

2 20 12 13 11 7 9

20 12 13 11 9

20 12 13 11

20 12 13

7 9 11

(19)

Merging two sorted arrays

20 12 13 11 7 9 2 1

20 12 13 11 7 9 2

20 12 13 11 7 9

20 12 13 11 9

20 12 13 11

20 12 13

1 2 7 9 11 12

Merging two sorted arrays

20 12 13 11 7 9 2 1

20 12 13 11 7 9 2

20 12 13 11 7 9

20 12 13 11 9

20 12 13 11

20 12 13

1 2 7 9 11 12

Time = Θ(n) to merge a total

of n elements (linear time).

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Analyzing merge sort

MERGE-SORTA[1 . . n]

1.If n= 1, done.

2.Recursively sort A[ 1 . . ^「 n/2 ^」 ] and A[ ^「 n/2 ^」 +1 . . n ] .

3.“Merge”the 2sorted lists Sloppiness: Should be T( 「 n/2 」 ) + T( 「 n/2 」 ) , but it turns out not to matter asymptotically.

T(n) Θ(1) 2T(n/2) Θ(n) Abuse

Recurrence for merge sort

• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.

• CLRS and Lecture 2 provide several ways to find a good upper bound on T(n).

Θ(1) if n= 1;

2T(n/2)+ Θ(n) if n> 1.

T(n) =

(21)

Recursion tree

Solve T(n) = 2T(n/2) + cn, where c > 0 is constant.

Recursion tree

Solve T(n) = 2T(n/2) + cn, where c > 0 is constant.

T(n)

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Recursion tree

Solve T(n) = 2T(n/2) + cn, where c > 0 is constant.

cn

T(n/2) T(n/2)

Recursion tree

Solve T(n) = 2T(n/2) + cn, where c > 0 is constant.

cn

cn/2 cn/2

T(n/4) T(n/4) T(n/4) T(n/4)

(23)

Recursion tree

Solve T(n) = 2T(n/2) + cn, where c > 0 is constant.

cn

cn/2 cn/2

cn/4 cn/4 cn/4 cn/4

Θ(1)

．

Recursion tree

Solve T(n) = 2T(n/2) + cn, where c > 0 is constant.

cn

cn/2 cn/2 cn/4 cn/4 cn/4 cn/4

Θ(1)

．

h= lgn

(24)

Recursion tree

Solve T(n) = 2T(n/2) + cn, where c > 0 is constant.

cn

cn/2 cn/2

cn/4 cn/4 cn/4 cn/4

Θ(1)

．

h= lgn

cn

Recursion tree

Solve T(n) = 2T(n/2) + cn, where c > 0 is constant.

cn

cn/2 cn/2 cn/4 cn/4 cn/4 cn/4

Θ(1)

．

h= lgn

cn

(25)

Recursion tree

Solve T(n) = 2T(n/2) + cn, where c > 0 is constant.

cn

cn/2 cn/2

cn/4 cn/4 cn/4 cn/4

Θ(1)

．

h= lgn

cn

．．．

Recursion tree

Solve T(n) = 2T(n/2) + cn, where c > 0 is constant.

cn

cn/2 cn/2 cn/4 cn/4 cn/4 cn/4

Θ(1)

．

h= lgn

cn

．．．

#leaves = n Θ(n)

(26)

Recursion tree

Solve T(n) = 2T(n/2) + cn, where c > 0 is constant.

cn

cn/2 cn/2

cn/4 cn/4 cn/4 cn/4

Θ(1)

．

h= lgn

cn

．．．

#leaves = n Θ(n)

Total= Θ( n lg n)

Conclusions

• Θ(n lg n) grows more slowly than Θ(n ² ).

• Therefore, merge sort asymptotically beats insertion sort in the worst case.

• In practice, merge sort beats insertion sort for n> 30 or so.

• Go test it out for yourself!

Course information

Introduction to Algorithms 6.046J/18.401J

LECTURE1

Analysis of Algorithms

•Insertion sort

•Asymptotic analysis

•Merge sort

•Recurrences

Prof. Charles E. Leiserson

Course information

1.Staff

2.Distance learning 3.Prerequisites 4.Lectures 5.Recitations 6.Handouts 7.Textbook

8.Course website 9.Extra help

10.Registration 11.Problem sets

12.Describing algorithms 13.Grading policy

14.Collaboration policy

Analysis of algorithms

The theoretical study of computer-program performance and resource usage.

What’s more important than performance?

•modularity

•correctness

•maintainability

•functionality

•robustness

•user-friendliness

•programmer time

•simplicity

•extensibility

•reliability

Why study algorithms and performance?

‧Algorithms help us to understand scalability.

‧Performance often draws the line between what is feasible and what is impossible.

‧Algorithmic mathematics provides a language for talking about program behavior.

‧Performance is the currency of computing.

‧The lessons of program performance generalize to other computing resources.

‧Speed is fun!

The problem of sorting

Input: sequence 〈a 1 , a 2 , …, a n 〉 of numbers.

Output: permutation 〈a' 1 , a' 2 , …, a' n 〉 Such that a' 1 ≤a' 2 ≤…≤a' n .

Example:

Input: 8 2 4 9 3 6 Output: 2 3 4 6 8 9

INSERTION-SORT

I

-S

(A, n) ⊳ A[1 . . n]

for j ←2 to n do key ← A[ j]

i ← j –1

while i > 0 and A[i] > key do A[i+1] ← A[i]

i ← i –1 A[i+1] = key

“pseudocode”

I

-S

(A, n) ⊳ A[1 . . n]

for j ←2 to n do key ← A[ j]

i ← j –1

while i > 0 and A[i] > key do A[i+1] ← A[i]

i ← i –1 A[i+1] = key

“pseudocode”

INSERTION-SORT

A:

sorted key

Example of insertion sort

8 2 4 9 3 6

Example of insertion sort

8 2 4 9 3 6

8 2 4 9 3 6 2 8 4 9 3 6

Example of insertion sort

Example of insertion sort

8 2 4 9 3 6 2 8 4 9 3 6

Example of insertion sort

8 2 4 9 3 6

2 8 4 9 3 6

2 4 8 9 3 6

Example of insertion sort

8 2 4 9 3 6 2 8 4 9 3 6 2 4 8 9 3 6

Example of insertion sort

8 2 4 9 3 6

2 8 4 9 3 6

2 4 8 9 3 6

2 4 8 9 3 6

Input: sequence 〈a ₁ , a ₂ , …, a _n 〉 of numbers.

Output: permutation 〈a' ₁ , a' ₂ , …, a' _n 〉 Such that a' ₁ ≤a' ₂ ≤…≤a' _n .

2 3 4 6 8 9 ^done

Θ(g(n)) = { f (n): there exist positive constants c ₁ , c ₂ , and n ₀

such that 0 ≤c ₁ g(n) ≤f (n) ≤c ₂ g(n) for all n≥n ₀ }

•Example: 3n ³ + 90n ² –5n+ 6046 = Θ(n ³ )

When n gets large enough, a Θ(n ² )algorithm always beats a Θ(n ³ )algorithm.