introduction to algorithms 6.046j/18.401j/sma5503 lecture 1 prof. charles e. leiserson

Introduction to Algorithms

6.046J/18.401J/SMA5503

Lecture 1Prof. Charles E. Leiserson

Analysis of algorithms

The theoretical study of computer-programperformance and resource usage.What’s more important than performance?• modularity • user-friendliness• correctness • programmer time• maintainability • simplicity• functionality • extensibility• robustness • reliability

Why study algorithms andperformance?

• 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!

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 6Output: 2 3 4 6 8 9

Insertion sort

Example of insertion sort

8 2 4 9 3 6


8 2 4 9 3 62 8 4 9 3 6


8 2 4 9 3 62 8 4 9 3 62 4 8 9 3 6


8 2 4 9 3 62 8 4 9 3 62 4 8 9 3 62 4 8 9 3 6


8 2 4 9 3 62 8 4 9 3 62 4 8 9 3 62 4 8 9 3 62 3 4 8 9 6


8 2 4 9 3 62 8 4 9 3 62 4 8 9 3 62 4 8 9 3 62 3 4 8 9 6

2 3 4 6 8 9 done

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.

Insertion SortInsertion Sort

Statement Statement EffortEffortInsertionSort(A, n) {InsertionSort(A, n) {

for i = 2 to n { for i = 2 to n { cc11nn

key = A[i]key = A[i] cc22(n-1)(n-1)

j = i - 1;j = i - 1; cc33(n-1)(n-1)

while (j > 0) and (A[j] > key) {while (j > 0) and (A[j] > key) { cc44TT

A[j+1] = A[j]A[j+1] = A[j] cc55(T-(n-1))(T-(n-1))

j = j - 1j = j - 1 cc66(T-(n-1))(T-(n-1))

}} 00A[j+1] = keyA[j+1] = keycc77(n-1)(n-1)

}} 00}}T = T = tt22 + t + t33 + … + t + … + tnn where t where tii is number of while expression evaluations for the i is number of while expression evaluations for the ithth for loop iteration for loop iteration

David Luebke 19 05/05/23

Analyzing Insertion SortAnalyzing Insertion Sort• T(n)T(n) == cc11n + cn + c22(n-1) + c(n-1) + c33(n-1) + c(n-1) + c44T + cT + c55(T - (n-1)) + c(T - (n-1)) + c66(T - (n-1)) + c(T - (n-1)) + c77(n-1) (n-1)

= = cc88T + cT + c99n + cn + c1010

• What can T be?What can T be?• Best case -- inner loop body never executedBest case -- inner loop body never executed

• ttii = 1 = 1 T(n) is a linear function T(n) is a linear function• Worst case -- inner loop body executed for all previous Worst case -- inner loop body executed for all previous

elementselements• ttii = i = i T(n) is a quadratic function T(n) is a quadratic function

• Average caseAverage case• ??????

Kinds of analyses

Worst-case: (usually)• T(n) = maximum time of algorithmon any input of size n.Average-case: (sometimes)• T(n) = expected time of algorithmover all inputs of size n.• Need assumption of statisticaldistribution of inputs.Best-case: (bogus)• Cheat with a slow algorithm thatworks fast on some input.

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 PerformanceAsymptotic Performance

• In this course, we care most about In this course, we care most about asymptotic performanceasymptotic performance• How does the algorithm behave as the problem How does the algorithm behave as the problem

size gets very large?size gets very large?• Running timeRunning time• Memory/storage requirementsMemory/storage requirements• Bandwidth/power requirements/logic gates/etc.Bandwidth/power requirements/logic gates/etc.

Analysis of AlgorithmsAnalysis of Algorithms• Analysis is performed with respect to a computational Analysis is performed with respect to a computational

modelmodel• We will usually use a generic uniprocessor random-We will usually use a generic uniprocessor random-

access machine (RAM)access machine (RAM)• All memory equally expensive to accessAll memory equally expensive to access• No concurrent operationsNo concurrent operations• All reasonable instructions take unit timeAll reasonable instructions take unit time

• Except, of course, function callsExcept, of course, function calls• Constant word sizeConstant word size

• Unless we are explicitly manipulating bitsUnless we are explicitly manipulating bits


Input SizeInput Size

• Time and space complexityTime and space complexity• This is generally a function of the input sizeThis is generally a function of the input size

• E.g., sorting, multiplicationE.g., sorting, multiplication• How we characterize input size depends:How we characterize input size depends:

• Sorting: number of input itemsSorting: number of input items• Multiplication: total number of bitsMultiplication: total number of bits• Graph algorithms: number of nodes & edgesGraph algorithms: number of nodes & edges• EtcEtc


AnalysisAnalysis

• SimplificationsSimplifications• Ignore actual and abstract statement costsIgnore actual and abstract statement costs• Order of growthOrder of growth is the interesting measure: is the interesting measure:

• Highest-order term is what countsHighest-order term is what counts• Remember, we are doing asymptotic analysisRemember, we are doing asymptotic analysis• As the input size grows larger it is the high order term that As the input size grows larger it is the high order term that

dominatesdominates

Θ-notation

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:• Drop low-order terms; ignore leading constants.• Example: 3n3 + 90n2 – 5n + 6046 = Θ(n3)

Asymptotic performance

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


Practical ComplexityPractical Complexity

0

250

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

f(n) = n

f(n) = log(n)

f(n) = n log(n)f(n) = n 2̂

f(n) = n 3̂

f(n) = 2 n̂



0

500

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

f(n) = n

f(n) = log(n)

f(n) = n log(n)f(n) = n 2̂

f(n) = n 3̂

f(n) = 2 n̂



0

1000

1 3 5 7 9 11 13 15 17 19

f(n) = n

f(n) = log(n)

f(n) = n log(n)f(n) = n 2̂

f(n) = n 3̂

f(n) = 2 n̂



0

1000

2000

3000

4000

5000

1 3 5 7 9 11 13 15 17 19

f(n) = n

f(n) = log(n)

f(n) = n log(n)f(n) = n 2̂

f(n) = n 3̂

f(n) = 2 n̂



1

10

100

1000

10000

100000

1000000

10000000

1 4 16 64 256 1024 4096 16384 65536

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.

Merge sort

Merge SortMerge Sort

MERGEMERGE

Merging two sorted

20 1220 12 13 1113 11 7 97 9 2 12 1

Merging two sorted

20 1220 12 13 1113 11 7 97 9 2 12 1

11

Merging two sorted

20 12 20 1220 12 20 12 13 11 13 1113 11 13 11 7 9 7 97 9 7 9 2 1 2 2 1 2

11

Merging two sorted

20 12 20 1220 12 20 12 13 11 13 1113 11 13 11 7 9 7 97 9 7 9 2 1 2 2 1 2

1 1 2 2

Merging two sorted

20 12 20 12 20 1220 12 20 12 20 12 13 11 13 11 13 1113 11 13 11 13 11 7 9 7 9 7 97 9 7 9 7 9 2 1 2 2 1 2

1 21 2

Merging two sorted arrays

20 12 20 12 20 1220 12 20 12 20 12 13 11 13 11 13 1113 11 13 11 13 11 7 9 7 9 7 97 9 7 9 7 9 2 1 2 2 1 2

1 1 2 2 7 7


20 12 20 12 20 12 20 1220 12 20 12 20 12 20 12 13 11 13 11 13 11 13 1113 11 13 11 13 11 13 11 7 9 7 9 7 97 9 7 9 7 9 9 9 2 1 2 2 1 2

1 1 2 2 7 7

○


20 12 20 12 20 12 20 1220 12 20 12 20 12 20 12 13 11 13 11 13 11 13 1113 11 13 11 13 11 13 11 7 9 7 9 7 97 9 7 9 7 9 9 9 2 1 2 2 1 2

1 1 2 2 7 7 99


20 12 20 12 20 12 20 12 20 12 20 12 20 12 20 12 20 12 20 12 13 11 13 11 13 11 13 11 13 1113 11 13 11 13 11 13 11 13 11 7 9 7 9 7 97 9 7 9 7 9 9 9 2 1 2 2 1 2

1 1 2 2 7 9 7 9


20 12 20 12 20 12 20 12 20 12 20 12 20 12 20 12 20 12 20 12 13 11 13 11 13 11 13 11 13 1113 11 13 11 13 11 13 11 13 11 7 9 7 9 7 97 9 7 9 7 9 9 9 2 1 2 2 1 2

1 1 2 2 7 9 7 9 11 11


20 12 20 12 20 12 20 12 20 12 20 1220 12 20 12 20 12 20 12 20 12 20 12 13 11 13 11 13 11 13 11 13 11 13 13 11 13 11 13 11 13 11 13 11 13 7 9 7 9 7 9 97 9 7 9 7 9 9 2 1 2 2 1 2

1 1 22 7 9 7 9 11 11


20 12 20 12 20 12 20 12 20 12 20 1220 12 20 12 20 12 20 12 20 12 20 12 13 11 13 11 13 11 13 11 13 11 13 13 11 13 11 13 11 13 11 13 11 13 7 9 7 9 7 9 97 9 7 9 7 9 9 2 1 2 2 1 2

1 1 22 7 9 7 9 11 11 12 12


20 12 20 12 20 12 20 12 20 12 20 1220 12 20 12 20 12 20 12 20 12 20 12 13 11 13 11 13 11 13 11 13 11 13 13 11 13 11 13 11 13 11 13 11 13 7 9 7 9 7 9 97 9 7 9 7 9 9 2 1 2 2 1 2

1 1 22 7 9 7 9 11 11 12 12

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

Analyzing merge sort

T(n) MERGE-SORT A[1 . . n]Θ(1) 1. If n = 1, done.2T(n/2) 2. Recursively sort A[ 1 . . n/2 ]

Abuse and A[ n/2 +1 . . n ] .Θ(n) 3. “Merge” the 2 sorted lists

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

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 basecase when T(n) = Θ(1) for sufficientlysmall n, but only when it has no effect onthe asymptotic solution to the recurrence.• CLRS and Lecture 2 provide several waysto find a good upper bound on T(n).

Recursion tree

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

Recursion tree


T(n)

Recursion tree


cn

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

Recursion tree


cn

cn/2 cn/2

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

Recursion tree


cn

cn/2 cn/2

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

Θ(1)

Recursion tree


cn

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

Θ(1)

Recursion tree


cn cn

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

Θ(1)

Recursion tree


cn cn

cn/2 cn/2 cnh = lg n T(n/4) T(n/4) T(n/4) T(n/4)

Θ(1)

Recursion tree


cn cn

cn/2 cn/2 cnh = lg n T(n/4) T(n/4) T(n/4) T(n/4) cn

Θ(1)

Recursion tree


cn cn


Θ(1) Θ(n)#leaves = n

Recursion tree


cn cn


Θ(n) Θ(1) Θ(n lg n)#leaves = n

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!

introduction to algorithms 6.046j/18.401j/sma5503 lecture 1 prof. charles e. leiserson

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