Analysis of Algorithms Lecture 2 Algorithm Input Output
Jan 07, 2016
Analysis of AlgorithmsLecture 2
Algorithm
Input Output
Analysis of Algorithms 2
Outline
Running timePseudo-codeCounting primitive operationsAsymptotic notationAsymptotic analysis
Analysis of Algorithms 3
How good is Insertion-Sort?
How can you answer such questions?
What is “goodness”?
1. Measure2. Count3. Estimate
Analysis of Algorithms 4
How can we quantify it?
1. Correctness2. Minimum use of “time” + “space”
Analysis of Algorithms 5
1) Measure it – do an experiment!
Write a program implementing the algorithmRun the program with inputs of varying size and compositionUse a method like System.currentTimeMillis() to get an accurate measure of the actual running timePlot the results
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Input Size
Tim
e (
ms)
Analysis of Algorithms 6
Analysis of Algorithms 7
Limitations of Experiments
Must implement the algorithm, which may be difficult!Results may not reflect the running time on other inputsIn order to compare two algorithms, the same hardware and software environments must be used
2) Count Primitive Operations
The Idea Write down the pseudocode Count the number of “primitive
operations”
Analysis of Algorithms 8
Analysis of Algorithms 9
PseudocodeHigh-level description of an algorithmMore structured than English proseLess detailed than a programPreferred notation for describing algorithmsHides program design issues
Algorithm arrayMax(A, n)Input array A of n integersOutput maximum element
of A
currentMax A[0]for i 1 to n 1 do
if A[i] currentMax then
currentMax A[i]
return currentMax
Example: find max element of an array
Analysis of Algorithms 10
Pseudocode Details
Control flow if … then … [else …] while … do … repeat … until … for … do … Indentation replaces
braces
Method declarationAlgorithm method (arg [, arg…])
Input …
Output …
Method callvar.method (arg [, arg…])
Return valuereturn expression
Expressions¬ Assignment
(like in Java)= Equality testing
(like in Java)n2 Superscripts and
other mathematical formatting allowed
Analysis of Algorithms 11
The Random Access Machine (RAM) Model
A CPU
An potentially unbounded bank of memory cells, each of which can hold an arbitrary number or character
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2
Memory cells are numbered and accessing any cell in memory takes unit time.
Random Access Model (RAM)
Time complexity (running time) = number of instructions executedSpace complexity = the number of memory cells accessed
Analysis of Algorithms 12
Analysis of Algorithms 13
Primitive Operations
Basic computations performed by an algorithmIdentifiable in pseudocodeLargely independent from the programming languageExact definition not important (we will see why later)
Examples: Evaluating an
expression Assigning a
value to a variable
Indexing into an array
Calling a method Returning from a
method
Analyzing pseudocode (by counting)
1. For each line of pseudocode, count the number of primitive operations in it. Pay attention to the word "primitive" here; sorting an array is not a primitive operation.
2. Multiply this count with the number of times this line is executed.
3. Sum up over all lines.Analysis of Algorithms 14
Analysis of Algorithms 15
Counting Primitive Operations
By inspecting the pseudocode, we can determine the maximum number of primitive operations executed by an algorithm, as a function of the input size
Algorithm arrayMax(A, n) Cost Times
currentMax A[0] 2 1for i 1 to n 1 do 2 n
if A[i] currentMax then 2 (n 1)currentMax A[i] 2 (n 1)
{ increment counter i } 2 (n 1)return currentMax 1 1
Total 8n 3
operations
Analysis of Algorithms 16
Estimating Running Time
Algorithm arrayMax executes 8n 3 primitive operations in the worst case Definea Time taken by the fastest primitive
operationb Time taken by the slowest
primitive operation
Let T(n) be the actual worst-case running time of arrayMax. We have
a (8n 3) T(n) b(8n 3)
Hence, the running time T(n) is bounded by two linear functions
Insertion Sort
Analysis of Algorithms 17
Hint: 1. observe each line i can be implemented using a constant number
of RAM instructions, Ci
2. for each value of j in the outer loop, we let tj be the number of times that the while loop test in line 4 is executed.
Insertion Sort
Analysis of Algorithms 18
Question: So what is the running time?
Time Complexity may depend on the input!
Best Case:?
Worst Case:?
Average Case:?
Analysis of Algorithms 19
Complexity may depend on the input!
Best Case: Already sorted. tj=1. Running time = C * N (a linear function of N)
Worst Case: Inverse order. tj=j.
Analysis of Algorithms 20
Analysis of Algorithms 21
Arithmetic Progression
Neglecting the constants, the worst-base running time of insertion sort is proportional to1 + 2 + …+ n
The sum of the first n integers is n(n + 1) / 2
Thus, algorithm insertion sort required almost n2 operations.
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What about the Average Case?
Idea: Assume that each of the n!
permutations of A is equally likely. Compute the average over all
possible different inputs of length N.
Difficult to compute!In this course we focus on Worst Case Analysis!
Analysis of Algorithms 22
See TimeComplexity Spreadsheet
Analysis of Algorithms 23
Analysis of AlgorithmsLecture 3
Algorithm
Input Output
What is “goodness”?
1. Measure wall clock time2. Count operations3.Estimate Computational
ComplexityAnalysis of Algorithms 25
How can we quantify it?
1. Correctness2. Minimum use of “time” + “space”
Analysis of Algorithms 26
Asymptotic Analysis
Uses a high-level description of the algorithm instead of an implementationAllows us to evaluate the speed of an algorithm independent of the hardware/software environmentEstimate the growth rate of T(n)A back of the envelope calculation!!!
Analysis of Algorithms 27
The idea
Write down an algorithm Using Pseudocode In terms of a set of primitive operations
Count the # of steps In terms of primitive operations Considering worst case input
Bound or “estimate” the running time Ignore constant factors Bound fundamental running time
Analysis of Algorithms 28
Growth Rate of Running Time
Changing the hardware/ software environment Affects T(n) by a constant factor, but Does not alter the growth rate of T(n)
The linear growth rate of the running time T(n) is an intrinsic property of algorithm arrayMax
Analysis of Algorithms 29
Which growth rate is best?
T(n) = 1000n + n2 or T(n) = 2n + n3
Analysis of Algorithms 30
Growth Rates
Growth rates of functions: Linear n Quadratic n2
Cubic n3
In a log-log chart, the slope of the line corresponds to the growth rate of the function
1E+01E+21E+41E+61E+8
1E+101E+121E+141E+161E+181E+201E+221E+241E+261E+281E+30
1E+0 1E+2 1E+4 1E+6 1E+8 1E+10n
T(n
)
Cubic
Quadratic
Linear
Functions Graphed Using “Normal” Scale
31Analysis of Algorithms
g(n) = 2ng(n) = 1
g(n) = lg n
g(n) = n lg n
g(n) = n
g(n) = n2
g(n) = n3
Analysis of Algorithms 32
Constant Factors
The growth rate is not affected by constant factors
or lower-order terms
Examples 102n + 105 is a
linear function 105n2 + 108n is a
quadratic function 1E+01E+21E+41E+61E+8
1E+101E+121E+141E+161E+181E+201E+221E+241E+26
1E+0 1E+2 1E+4 1E+6 1E+8 1E+10n
T(n
)
Quadratic
Quadratic
Linear
Linear
Analysis of Algorithms 33
Big-Oh NotationGiven functions f(n) and g(n), we say that f(n) is O(g(n)) if there are positive constantsc and n0 such that
f(n) cg(n) for n n0
Example: 2n + 10 is O(n) 2n + 10 cn (c 2) n 10 n 10/(c 2) Pick c = 3 and n0 = 10
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f(n) is O(g(n)) iff f(n) cg(n) for n n0
Analysis of Algorithms 35
Big-Oh Notation (cont.)
Example: the function n2 is not O(n) n2 cn n c The above
inequality cannot be satisfied since c must be a constant
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Analysis of Algorithms 36
More Big-Oh Examples
7n-27n-2 is O(n)need c > 0 and n0 1 such that 7n-2 c•n for n n0
this is true for c = 7 and n0 = 1
3n3 + 20n2 + 53n3 + 20n2 + 5 is O(n3)need c > 0 and n0 1 such that 3n3 + 20n2 + 5 c•n3 for n
n0
this is true for c = 4 and n0 = 21 3 log n + 53 log n + 5 is O(log n)need c > 0 and n0 1 such that 3 log n + 5 c•log n for n n0
this is true for c = 8 and n0 = 2
Analysis of Algorithms 37
Big-Oh and Growth Rate
The big-Oh notation gives an upper bound on the growth rate of a functionThe statement “f(n) is O(g(n))” means that the growth rate of f(n) is no more than the growth rate of g(n)
We can use the big-Oh notation to rank functions according to their growth rate
Analysis of Algorithms 38
Questions
Is T(n) = 9n4 + 876n = O(n4)? Is T(n) = 9n4 + 876n = O(n3)? Is T(n) = 9n4 + 876n = O(n27)?
T(n) = n2 + 100n = O(?)T(n) = 3n + 32n3 + 767249999n2 = O(?)
Analysis of Algorithms 39
Big-Oh Rules (shortcuts)
If is f(n) a polynomial of degree d, then f(n) is O(nd), i.e.,
1. Drop lower-order terms2. Drop constant factors
Use the smallest possible class of functions
Say “2n is O(n)” instead of “2n is O(n2)”
Use the simplest expression of the class Say “3n + 5 is O(n)” instead of “3n + 5 is
O(3n)”
Analysis of Algorithms 40
Rank From Fast to Slow…
T(n) = O(n4) T(n) = O(n log n)
T(n) = O(n2)T(n) = O(n2 log n)
T(n) = O(n)T(n) = O(2n)T(n) = O(log n)
T(n) = O(n + 2n)
Note: Assume base of log is 2 unless
otherwise instructedi.e. log n = log2 n
Analysis of Algorithms 41
Computing Prefix Averages
We further illustrate asymptotic analysis with two algorithms for prefix averagesThe i-th prefix average of an array X is average of the first (i + 1) elements of XA[i] = X[0] + X[1] + … + X[i]
Computing the array A of prefix averages of another array X has applications to financial analysis
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Analysis of Algorithms 42
Prefix Averages (Quadratic)
The following algorithm computes prefix averages in quadratic time by applying the definition
Algorithm prefixAverages1(X, n)Input array X of n integersOutput array A of prefix averages of X #operations A new array of n integers nfor i 0 to n 1 do n
s X[0] nfor j 1 to i do 1 + 2 + …+ (n
1)s s + X[j] 1 + 2 + …+ (n
1)A[i] s / (i + 1) n
return A 1
Analysis of Algorithms 43
Arithmetic Progression
The running time of prefixAverages1 isO(1 + 2 + …+ n)
The sum of the first n integers is n(n + 1) / 2 There is a simple
visual proof of this fact
Thus, algorithm prefixAverages1 runs in O(n2) time
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Analysis of Algorithms 44
Prefix Averages (Linear)The following algorithm computes prefix averages in linear time by keeping a running sumAlgorithm prefixAverages2(X, n)
Input array X of n integersOutput array A of prefix averages of X #operationsA new array of n integers ns 0 1for i 0 to n 1 do n
s s + X[i] nA[i] s / (i + 1) n
return A 1Algorithm prefixAverages2 runs in O(n) time
Analysis of Algorithms 45
properties of logarithms:logb(xy) = logbx + logbylogb (x/y) = logbx - logbylogbxa = alogbxlogba = logxa/logxb
properties of exponentials:a(b+c) = aba c
abc = (ab)c
ab /ac = a(b-c)
b = a logab
bc = a c*logab
SummationsLogarithms and Exponents
Proof techniquesBasic probability
Math you may need to Review
Analysis of Algorithms 46
Big-Omega Definition: f(n) is (g(n)) if there is a constant c
> 0 and an integer constant n0 1 such that f(n) c•g(n) for n n0
An asymptotic lower Bound
Analysis of Algorithms 47
Big-Theta Definition: f(n) is (g(n)) if there are constants c’ > 0 and c’’ > 0 and an integer constant n0 1 such that c’•g(n) f(n) c’’•g(n) for n n0
Here we say that g(n) is an asymptotically tight bound for f(n).
Big-Theta (cont)
Based on the definitions, we have the following theorem: f(n) = Θ(g(n)) if and only if f(n) = O(g(n)) and f(n) =
Ω(g(n)).
For example, the statement n2/2 + lg n = Θ(n2) is equivalent to n2/2 + lg n = O(n2) and n2/2 + lg n = Ω(n2).
Note that asymptotic notation applies to asymptotically positive functions only, which are functions whose values are positive for all sufficiently large n.
Analysis of Algorithms 48
Prove n2/2 + lg n = Θ(n2).
Proof. To prove this claim, we must determine positive constants c1, c2 and n0, s.t. c1 n2<= n2/2 + lg n <= c2 n2
c1 <= ½ + (lg n) / n2 <= c2 (divide thru by n2)
Pick c1 = ¼ , c2 = ¾ and n0 = 2 For n0 = 2, 1/4 <= ½ + (lg 2) / 4 <= ¾,
TRUE When n0 > 2, the (½ + (lg 2) / 4) term
grows smaller but never less than ½, therefore n2/2 + lg n = Θ(n2).
Analysis of Algorithms 49
Analysis of Algorithms 50
Intuition for Asymptotic Notation
Big-Oh f(n) is O(g(n)) if f(n) is
asymptotically less than or equal to g(n)
big-Omega f(n) is (g(n)) if f(n) is asymptotically
greater than or equal to g(n)big-Theta f(n) is (g(n)) if f(n) is asymptotically
equal to g(n)
A word of caution…
Asymptotic analysis is fragile Constants do matter! Don’t play
games!
Modern processors are not RAM
Analysis of Algorithms 51
A word of caution…In practice, performance comes from
Analysis of Algorithms 52
Careful Engineeri
ng
Systematic
Evaluation
Algorithmic Analysis