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![Page 1: Introduction to Algorithms 6.046J/18 - MIT OpenCourseWare · PDF fileProblem sets 12. Describing algorithms ... September 7, 2005 Introduction to Algorithms L1.4 ... Input: 8 2 4 9](https://reader033.fdocuments.us/reader033/viewer/2022050901/5a71fef37f8b9aa2538d4e29/html5/thumbnails/1.jpg)
Introduction to Algorithms6.046J/18.401J
LECTURE 1Analysis of Algorithms• Insertion sort• Asymptotic analysis• Merge sort• Recurrences
Prof. Charles E. LeisersonCopyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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September 7, 2005 Introduction to Algorithms L1.2
Course information
1. Staff2. Distance learning3. Prerequisites4. Lectures5. Recitations6. Handouts7. Textbook
8. Course website9. Extra help10. Registration 11. Problem sets12. Describing algorithms13. Grading policy14. Collaboration policy
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September 7, 2005 Introduction to Algorithms L1.3Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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
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September 7, 2005 Introduction to Algorithms L1.4Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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!
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September 7, 2005 Introduction to Algorithms L1.5Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
The problem of sorting
Input: sequence ⟨a1, a2, …, an⟩ of numbers.
Output: permutation ⟨a'1, a'2, …, a'n⟩ suchthat a'1 ≤ a'2 ≤ … ≤ a'n .
Example:Input: 8 2 4 9 3 6
Output: 2 3 4 6 8 9
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September 7, 2005 Introduction to Algorithms L1.6Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Insertion sortINSERTION-SORT (A, n) ⊳ A[1 . . n]
for j ← 2 to ndo key ← A[ j]
i ← j – 1while i > 0 and A[i] > key
do A[i+1] ← A[i]i ← i – 1
A[i+1] = key
“pseudocode”
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September 7, 2005 Introduction to Algorithms L1.7Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Insertion sortINSERTION-SORT (A, n) ⊳ A[1 . . n]
for j ← 2 to ndo key ← A[ j]
i ← j – 1while i > 0 and A[i] > key
do A[i+1] ← A[i]i ← i – 1
A[i+1] = key
“pseudocode”
sorted
i j
keyA:
1 n
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September 7, 2005 Introduction to Algorithms L1.8Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Example of insertion sort8 2 4 9 3 6
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September 7, 2005 Introduction to Algorithms L1.9Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Example of insertion sort8 2 4 9 3 6
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September 7, 2005 Introduction to Algorithms L1.10Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Example of insertion sort8 2 4 9 3 6
2 8 4 9 3 6
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September 7, 2005 Introduction to Algorithms L1.11Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Example of insertion sort8 2 4 9 3 6
2 8 4 9 3 6
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September 7, 2005 Introduction to Algorithms L1.12Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Example of insertion sort8 2 4 9 3 6
2 8 4 9 3 6
2 4 8 9 3 6
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September 7, 2005 Introduction to Algorithms L1.13Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Example of insertion sort8 2 4 9 3 6
2 8 4 9 3 6
2 4 8 9 3 6
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September 7, 2005 Introduction to Algorithms L1.14Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Example of insertion sort8 2 4 9 3 6
2 8 4 9 3 6
2 4 8 9 3 6
2 4 8 9 3 6
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September 7, 2005 Introduction to Algorithms L1.15Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Example of insertion sort8 2 4 9 3 6
2 8 4 9 3 6
2 4 8 9 3 6
2 4 8 9 3 6
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September 7, 2005 Introduction to Algorithms L1.16Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Example of insertion sort8 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
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September 7, 2005 Introduction to Algorithms L1.17Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Example of insertion sort8 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
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September 7, 2005 Introduction to Algorithms L1.18Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Example of insertion sort8 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
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September 7, 2005 Introduction to Algorithms L1.19Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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.
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September 7, 2005 Introduction to Algorithms L1.20Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Kinds of analysesWorst-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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September 7, 2005 Introduction to Algorithms L1.21Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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”“Asymptotic Analysis”
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September 7, 2005 Introduction to Algorithms L1.22Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Θ-notation
• Drop low-order terms; ignore leading constants.• Example: 3n3 + 90n2 – 5n + 6046 = Θ(n3)
Math:Θ(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:
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September 7, 2005 Introduction to Algorithms L1.23Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Asymptotic performance
n
T(n)
n0
• 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.
When n gets large enough, a Θ(n2) algorithm always beats a Θ(n3) algorithm.
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September 7, 2005 Introduction to Algorithms L1.24Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Insertion sort analysisWorst case: Input reverse sorted.
( )∑=
Θ=Θ=n
jnjnT
2
2)()(
Average case: All permutations equally likely.
( )∑=
Θ=Θ=n
jnjnT
2
2)2/()(
[arithmetic series]
Is insertion sort a fast sorting algorithm?• Moderately so, for small n.• Not at all, for large n.
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September 7, 2005 Introduction to Algorithms L1.25Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
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
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September 7, 2005 Introduction to Algorithms L1.26Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
12
11
9
1
20
13
7
2
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September 7, 2005 Introduction to Algorithms L1.27Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
13
7
2
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September 7, 2005 Introduction to Algorithms L1.28Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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1
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September 7, 2005 Introduction to Algorithms L1.29Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.30Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.31Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.32Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
20
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September 7, 2005 Introduction to Algorithms L1.33Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.34Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.35Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.36Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.37Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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September 7, 2005 Introduction to Algorithms L1.38Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Merging two sorted arrays
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Time = Θ(n) to merge a total of n elements (linear time).
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September 7, 2005 Introduction to Algorithms L1.39Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Analyzing 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
T(n)Θ(1)2T(n/2)
Θ(n)Abuse
Sloppiness: Should be T( ⎡n/2⎤ ) + T( ⎣n/2⎦ ) , but it turns out not to matter asymptotically.
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September 7, 2005 Introduction to Algorithms L1.40Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recurrence for merge sort
T(n) =Θ(1) if n = 1;2T(n/2) + Θ(n) if n > 1.
• 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).
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September 7, 2005 Introduction to Algorithms L1.41Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
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September 7, 2005 Introduction to Algorithms L1.42Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
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September 7, 2005 Introduction to Algorithms L1.43Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2) T(n/2)
cn
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September 7, 2005 Introduction to Algorithms L1.44Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
T(n/4) T(n/4) T(n/4) T(n/4)
cn/2 cn/2
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September 7, 2005 Introduction to Algorithms L1.45Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
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September 7, 2005 Introduction to Algorithms L1.46Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
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September 7, 2005 Introduction to Algorithms L1.47Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
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September 7, 2005 Introduction to Algorithms L1.48Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
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September 7, 2005 Introduction to Algorithms L1.49Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
…
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September 7, 2005 Introduction to Algorithms L1.50Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
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September 7, 2005 Introduction to Algorithms L1.51Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Recursion treeSolve T(n) = 2T(n/2) + cn, where c > 0 is constant.
cn
cn/4 cn/4 cn/4 cn/4
cn/2 cn/2
Θ(1)
…
h = lg n
cn
cn
cn
#leaves = n Θ(n)
…
Total = Θ(n lg n)
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September 7, 2005 Introduction to Algorithms L1.52Copyright © 2001-5 Erik D. Demaine and Charles E. Leiserson
Conclusions
• Θ(n lg n) grows more slowly than Θ(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!