Open Mind Tree: Lecture 1 Algorithms

LECTURE 1

Analysis of Algorithms

•Insertion sort

•Asymptotic analysis

•Merge sort

•Recurrences

Prof. Charles E. Leiserson

Course information

1.	Staff	8.	Course website
2.	Distance learning	9.	Extra help
3.	Prerequisites	10.	Registration
4.	Lectures	11.	Problem sets
5.	Recitations	12.	Describing algorithms
6.	Handouts	13.	Grading policy
7.	Textbook	14.	Collaboration policy

September 7, 2005

Introduction to Algorithms

L1.2

Analysis of algorithms

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

What’s more important than performance?

• modularity	• user-friendliness
• correctness	• programmer time
• maintainability	• simplicity
• functionality	• extensibility
• robustness	• reliability

	Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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 languagefor 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

Introduction to Algorithms

L1.4

The problem of sorting

Input: sequence <a1, a2, …, an> 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

Introduction to Algorithms

L1.5

Insertion sort
	INSERTION-SORT (A, n)	⊳A[1 . . n]
	INSERTION-SORT (A, n)	⊳A[1 . . n]
	for j ←2 to n
	do key ←A[ j]
	i ←j – 1
“pseudocode”	i ←j – 1
“pseudocode”	while i > 0 and A[i] > key
	while i > 0 and A[i] > key
	do A[i+1] ←A[i]
	i ←i – 1
	A[i+1] = key
	A[i+1] = key

September 7, 2005

Introduction to Algorithms

L1.6

Insertion sort
		INSERTION-SORT (A, n)		⊳A[1 . . n]

			for j ←2 to n
			do key ←A[ j]
			i ←j – 1
“pseudocode”
			while i > 0 and A[i] > key

			do A[i+1] ←A[i]
			i ←i – 1
			A[i+1] = key

1	i		j	n
A:

	sorted		key

	Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson

September 7, 2005

Introduction to Algorithms

L1.7

Example of insertion sort

8 2 4 9 3 6

September 7, 2005

Introduction to Algorithms

L1.8

Example of insertion sort

8 2 4 9 3 6

September 7, 2005

Introduction to Algorithms

L1.9

Example of insertion sort

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

September 7, 2005

Introduction to Algorithms

L1.10

Example of insertion sort

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

September 7, 2005

Introduction to Algorithms

L1.11

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

Introduction to Algorithms

L1.12

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

Introduction to Algorithms

L1.13

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

September 7, 2005

Introduction to Algorithms

L1.14

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

September 7, 2005

Introduction to Algorithms

L1.15

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

September 7, 2005

Introduction to Algorithms

L1.16

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

September 7, 2005

Introduction to Algorithms

L1.17

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

September 7, 2005

Introduction to Algorithms

L1.18

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.

September 7, 2005

Introduction to Algorithms

L1.19

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.

September 7, 2005

Introduction to Algorithms

L1.20

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 →8 .

“Asymptotic Analysis”

September 7, 2005

Introduction to Algorithms

L1.21

T-notation

Math:

T(g(n)) = { f (n) : there exist positive constants c1, c2, and

n0 such that 0 = c1 g(n) = f (n) = c2 g(n)for all n = n0 }

Engineering:

•Drop low-order terms; ignore leading constants.

•Example: 3n3 + 90n2 – 5n + 6046 = T(n3)

September 7, 2005

Introduction to Algorithms

L1.22

Asymptotic performance

When n gets large enough, a T(n2) algorithmalways beats a T(n3) algorithm.

		• We shouldn’t ignore
		asymptotically slower
		algorithms, however.
		• Real-world design
T(n)		situations often call for a
T(n)		careful balancing of
		engineering objectives.
		• Asymptotic analysis is a
n	n0	useful tool to help to
	n0	structure our thinking.
		structure our thinking.

September 7, 2005

Introduction to Algorithms

L1.23

Insertion sort analysis

Worst case: Input reverse sorted.

2
T (n) = T( j) = T(n )	[arithmetic series]
T (n) = T( j) = T(n )
j=2

Average case: All permutations equally likely.T (n) = n T( j / 2) = T(n2 )

j=2

Is insertion sort a fast sorting algorithm?

•Moderately so, for small n.

•Not at all, for large n.

September 7, 2005

Introduction to Algorithms

L1.24

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

September 7, 2005

Introduction to Algorithms

L1.25

Merging two sorted arrays

20 12

13 11

7 9

2 1

September 7, 2005

Introduction to Algorithms

L1.26

Merging two sorted arrays

20 12

13 11

7 9

2 1

September 7, 2005

Introduction to Algorithms

L1.27

Merging two sorted arrays

20	12	20	12
13	11	13	11
7	9	7	9
2	1	2

September 7, 2005

Introduction to Algorithms

L1.28

Merging two sorted arrays

20	12	20	12
13	11	13	11
7	9	7	9
2	1	2

September 7, 2005

Introduction to Algorithms

L1.29

Merging two sorted arrays

20	12	20	12	20	12
13	11	13	11	13	11
7	9	7	9	7	9
2	1	2

September 7, 2005

Introduction to Algorithms

L1.30

Merging two sorted arrays

20	12	20	12	20	12
13	11	13	11	13	11
7	9	7	9	7	9
2	1	2

September 7, 2005

Introduction to Algorithms

L1.31

Merging two sorted arrays

20	12	20	12	20	12	20	12
13	11	13	11	13	11	13	11
7	9	7	9	7	9		9
2	1	2

September 7, 2005

Introduction to Algorithms

L1.32

Merging two sorted arrays

20	12	20	12	20	12	20	12
13	11	13	11	13	11	13	11
7	9	7	9	7	9		9
2	1	2

September 7, 2005

Introduction to Algorithms

L1.33

Merging two sorted arrays

20	12	20	12	20	12	20	12	20	12
13	11	13	11	13	11	13	11	13	11
7	9	7	9	7	9		9
2	1	2

September 7, 2005

Introduction to Algorithms

L1.34

Merging two sorted arrays

20	12	20	12	20	12	20	12	20	12
13	11	13	11	13	11	13	11	13	11
7	9	7	9	7	9		9
2	1	2

September 7, 2005

Introduction to Algorithms

L1.35

Merging two sorted arrays

20	12	20	12	20	12	20	12	20	12	20	12
13	11	13	11	13	11	13	11	13	11	13
7	9	7	9	7	9		9
2	1	2

September 7, 2005

Introduction to Algorithms

L1.36

Merging two sorted arrays

20	12	20	12	20	12	20	12	20	12	20	12
13	11	13	11	13	11	13	11	13	11	13
7	9	7	9	7	9		9
2	1	2

September 7, 2005

Introduction to Algorithms

L1.37

Merging two sorted arrays

20	12	20	12	20	12	20	12	20	12	20	12
13	11	13	11	13	11	13	11	13	11	13
7	9	7	9	7	9		9
2	1	2

Time = T(n) to merge a total of n elements (linear time).

September 7, 2005

Introduction to Algorithms

L1.38

Analyzing merge sort

T(n)	MERGE-SORT A[1 . . n]
T(1)	1.	If n = 1, done.
2T(n/2)	1.	If n = 1, done.
2T(n/2)
Abuse	2.	Recursively sort A[ 1 . . n/2	]
Abuse		and A[ n/2 +1 . . n ] .
T(n)		and A[ n/2 +1 . . n ] .
T(n)	3.	“Merge” the 2 sorted lists
	3.	“Merge” the 2 sorted lists

Sloppiness: Should be T( n/2 ) + T( n/2 ) , but it turns out not to matter asymptotically.

September 7, 2005

Introduction to Algorithms

L1.39

Recurrence for merge sort

T(n) =

T(1) if n = 1;

2T(n/2) + T(n) if n > 1.

•We shall usually omit stating the base case when T(n) = T(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).

September 7, 2005

Introduction to Algorithms

L1.40

Recursion tree

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

September 7, 2005

Introduction to Algorithms

L1.41

Recursion tree

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

T(n)

September 7, 2005

Introduction to Algorithms

L1.42

Recursion tree

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

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

September 7, 2005

Introduction to Algorithms

L1.43

Recursion tree

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

		cn
cn/2			cn/2
cn/2
T(n/4)	T(n/4)	T(n/4)	T(n/4)

September 7, 2005

Introduction to Algorithms

L1.44

Recursion tree

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

	cn/2	cn/2
	cn/2
cn/4	cn/4	cn/4	cn/4
…
T(1)
	Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson

September 7, 2005

Introduction to Algorithms

L1.45

Recursion tree

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

		cn/2	cn/2
		cn/2
h = lg n	cn/4	cn/4	cn/4	cn/4
	…
T(1)
		Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson

September 7, 2005

Introduction to Algorithms

L1.46

Recursion tree

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

		cn/2	cn/2
		cn/2
h = lg n	cn/4	cn/4	cn/4	cn/4
	…
T(1)
		Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson

September 7, 2005

Introduction to Algorithms

L1.47

Recursion tree

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

		cn/2	cn/2	cn
		cn/2
h = lg n	cn/4	cn/4	cn/4	cn/4
	…
T(1)
		Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson

September 7, 2005

Introduction to Algorithms

L1.48

Recursion tree

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

		cn/2		cn/2	cn
		cn/2
h = lg n	cn/4	cn/4	cn/4	cn/4	cn
	…				…
	…
T(1)

September 7, 2005

Introduction to Algorithms

L1.49

Recursion tree

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

		cn/2			cn/2	cn
		cn/2
h = lg n	cn/4	cn/4		cn/4	cn/4	cn
	…					…
	…

T(1)			#leaves = n			T(n)

September 7, 2005

Introduction to Algorithms

L1.50

Recursion tree

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

		cn/2			cn/2	cn
		cn/2
h = lg n	cn/4	cn/4		cn/4	cn/4	cn
	…					…
	…

T(1)			#leaves = n			T(n)
T(1)			#leaves = n			T(n)

					Total = T(n lg n)
		Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson

September 7, 2005

Introduction to Algorithms

L1.51

Conclusions

•T(n lg n) grows more slowly than T(n2).

•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!

September 7, 2005

Introduction to Algorithms

L1.52

Open Mind Tree

Lecture 1 Algorithms

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