Introduction to Algorithms
6.046J/18.401J/SMA5503
Lecture 1
Prof. Charles E. Leiserson
Day 1 Introduction to Algorithms L1.2
Welcome to Introduction to
Algorithms, Fall 2001
Handouts
1. Course Information
2. Calendar
3. Registration (MIT students only)
4. References
5. Objectives and Outcomes
6. Diagnostic Survey
Day 1 Introduction to Algorithms L1.3
Course information
1. Staff
2. Distance learning
3. Prerequisites
4. Lectures
5. Recitations
6. Handouts
7. Textbook (CLRS)
8. Website
9. Extra help
10.Registration (MIT only)
11.Problem sets
12.Describing algorithms
13.Grading policy
14.Collaboration policy
? Course information handout
Day 1 Introduction to Algorithms L1.4
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
Day 1 Introduction to Algorithms L1.5
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.
? The lessons of program performance generalize
to other computing resources.
? Speed is fun!
Day 1 Introduction to Algorithms L1.6
The problem of sorting
Input: sequence ?a
1
, a
2
, …, a
n
? of numbers.
Example:
Input: 8 2 4 9 3 6
Output: 2 3 4 6 8 9
Output: permutation ?a'
1
, a'
2
, …, a'
n
? such
that a'
1
≤ a'
2
≤
…
≤ a'
n
.
Day 1 Introduction to Algorithms L1.7
Insertion sort
INSERTION-SORT (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”
ij
key
sorted
A:
1 n
Day 1 Introduction to Algorithms L1.8
Example of insertion sort
824936
Day 1 Introduction to Algorithms L1.9
Example of insertion sort
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Day 1 Introduction to Algorithms L1.10
Example of insertion sort
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Day 1 Introduction to Algorithms L1.11
Example of insertion sort
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Day 1 Introduction to Algorithms L1.12
Example of insertion sort
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Day 1 Introduction to Algorithms L1.13
Example of insertion sort
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Day 1 Introduction to Algorithms L1.14
Example of insertion sort
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Day 1 Introduction to Algorithms L1.15
Example of insertion sort
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Day 1 Introduction to Algorithms L1.16
Example of insertion sort
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Day 1 Introduction to Algorithms L1.17
Example of insertion sort
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Day 1 Introduction to Algorithms L1.18
Example of insertion sort
824936
284936
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234689done
Day 1 Introduction to Algorithms L1.19
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.
Day 1 Introduction to Algorithms L1.20
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.
Day 1 Introduction to Algorithms L1.21
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”
Day 1 Introduction to Algorithms L1.22
Θ-notation
? Drop low-order terms; ignore leading constants.
? Example: 3n
3
+ 90n
2
–5n + 6046 = Θ(n
3
)
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:
Day 1 Introduction to Algorithms L1.23
Asymptotic performance
n
T(n)
n
0
? 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(n
2
) algorithm
always beats a Θ(n
3
) algorithm.
Day 1 Introduction to Algorithms L1.24
Insertion sort analysis
Worst case: Input reverse sorted.
()
∑
=
Θ=Θ=
n
j
njnT
2
2
)()(
Average case: All permutations equally likely.
( )
∑
=
Θ=Θ=
n
j
njnT
2
2
)2/()(
Is insertion sort a fast sorting algorithm?
? Moderately so, for small n.
? Not at all, for large n.
[arithmetic series]
Day 1 Introduction to Algorithms L1.25
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
Day 1 Introduction to Algorithms L1.26
Merging two sorted arrays
20
13
7
2
12
11
9
1
Day 1 Introduction to Algorithms L1.27
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
Day 1 Introduction to Algorithms L1.28
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
11
9
Day 1 Introduction to Algorithms L1.29
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
11
9
2
Day 1 Introduction to Algorithms L1.30
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
11
9
2
20
13
7
12
11
9
Day 1 Introduction to Algorithms L1.31
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
11
9
2
20
13
7
12
11
9
7
Day 1 Introduction to Algorithms L1.32
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
11
9
2
20
13
7
12
11
9
7
20
13
12
11
9
Day 1 Introduction to Algorithms L1.33
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
11
9
2
20
13
7
12
11
9
7
20
13
12
11
9
9
Day 1 Introduction to Algorithms L1.34
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
11
9
2
20
13
7
12
11
9
7
20
13
12
11
9
9
20
13
12
11
Day 1 Introduction to Algorithms L1.35
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
11
9
2
20
13
7
12
11
9
7
20
13
12
11
9
9
20
13
12
11
11
Day 1 Introduction to Algorithms L1.36
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
11
9
2
20
13
7
12
11
9
7
20
13
12
11
9
9
20
13
12
11
11
20
13
12
Day 1 Introduction to Algorithms L1.37
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
11
9
2
20
13
7
12
11
9
7
20
13
12
11
9
9
20
13
12
11
11
20
13
12
12
Day 1 Introduction to Algorithms L1.38
Merging two sorted arrays
20
13
7
2
12
11
9
1
1
20
13
7
2
12
11
9
2
20
13
7
12
11
9
7
20
13
12
11
9
9
20
13
12
11
11
20
13
12
12
Time = Θ(n) to merge a total
of n elements (linear time).
Day 1 Introduction to Algorithms L1.39
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.
Day 1 Introduction to Algorithms L1.40
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).
Day 1 Introduction to Algorithms L1.41
Recursion tree
Solve T(n) = 2T(n/2) + cn, where c > 0 is constant.
Day 1 Introduction to Algorithms L1.42
Recursion tree
Solve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n)
Day 1 Introduction to Algorithms L1.43
Recursion tree
Solve T(n) = 2T(n/2) + cn, where c > 0 is constant.
T(n/2)
T(n/2)
cn
Day 1 Introduction to Algorithms L1.44
Recursion tree
Solve 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
Day 1 Introduction to Algorithms L1.45
Recursion tree
Solve 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)
…
Day 1 Introduction to Algorithms L1.46
Recursion tree
Solve 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
Day 1 Introduction to Algorithms L1.47
Recursion tree
Solve 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
Day 1 Introduction to Algorithms L1.48
Recursion tree
Solve 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
Day 1 Introduction to Algorithms L1.49
Recursion tree
Solve 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
…
Day 1 Introduction to Algorithms L1.50
Recursion tree
Solve 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)
…
Day 1 Introduction to Algorithms L1.51
Recursion tree
Solve 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)
…
Day 1 Introduction to Algorithms L1.52
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!
