1. Analysis of Algorithms
Input
Output
Algorithm

2. Analysis of Algorithms
Outline
Running time
Pseudo-code
Counting primitive operations
Asymptotic notation
Asymptotic analysis
Analysis of Algorithms

3. How good is Insertion-Sort?
How can you answer such questions?
Analysis of Algorithms

4. Analysis of Algorithms
What is “goodness”?
Correctness
Minimum use of “time” + “space”
How can we quantify it?
Measure
Count
Estimate
Analysis of Algorithms

5. Analysis of Algorithms
1) Measure it
Write a program implementing the algorithm
Run the program with inputs of varying size and composition
Use a method like System.currentTimeMillis() to get an accurate measure of the actual running time
Plot the results
Analysis of Algorithms

6. Analysis of Algorithms

7. Limitations of Experiments
It is necessary to implement the algorithm, which may be difficult
Results may not be indicative of the running time on other inputs not included in the experiment.
In order to compare two algorithms, the same hardware and software environments must be used
Analysis of Algorithms

8. 2) Count Primitive Operations
The Idea
Write down the pseudocode
Count the number of “primitive operations”
Analysis of Algorithms

9. Analysis of Algorithms
Pseudocode
Algorithm arrayMax(A, n)
Input array A of n integers
Output 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
High-level description of an algorithm
More structured than English prose
Less detailed than a program
Preferred notation for describing algorithms
Hides program design issues
Analysis of Algorithms

10. Analysis of Algorithms
Pseudocode Details
Control flow
if … then … [else …]
while … do …
repeat … until …
for … do …
Indentation replaces braces
Method declaration
Algorithm method (arg [, arg…])
Input …
Output …
Method call
var.method (arg [, arg…])
Return value
return 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 1 2
Memory cells are numbered and accessing any cell in memory takes unit time.
Analysis of Algorithms

12. Random Access Model (RAM)
Time complexity (running time) = number of instructions executed
Space complexity = the number of memory cells accessed
Analysis of Algorithms

13. Analysis of Algorithms
Primitive Operations
Basic computations performed by an algorithm
Identifiable in pseudocode
Largely independent from the programming language
Exact 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
Analysis of Algorithms

14. Analyzing pseudocode (by counting)
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.
Multiply this count with the number of times this line is executed.
Sum up over all lines.
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)
currentMax  A[0]
Cost Times
for i  1 to n  1 do n
if A[i]  currentMax then (n  1)
currentMax  A[i] (n  1)
{ increment counter i } (n  1)
return currentMax
Total 8n  3
Analysis of Algorithms

16. Estimating Running Time
Algorithm arrayMax executes 8n  2 primitive operations in the worst case
Define
a Time taken by the fastest primitive operation
b Time taken by the slowest primitive operation
Let T(n) be the actual worst-case running time of arrayMax. We have a (8n  2)  T(n)  b(8n  2)
Hence, the running time T(n) is bounded by two linear functions
Analysis of Algorithms

17. Analysis of Algorithms
Insertion Sort
Hint:
observe each line i can be implemented using a constant number of RAM instructions, Ci
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. 
Analysis of Algorithms

18. Analysis of Algorithms
Insertion Sort
Question: So what is the running time?
Analysis of Algorithms

19. Time Complexity may depend on the input!
Best Case:?
Worst Case:?
Average Case:?
Analysis of Algorithms

20. 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

21. Arithmetic Progression
Neglecting the constants, the worst-base running time of insertion sort is proportional to …+ n
The sum of the first n integers is n(n + 1) / 2
Thus, algorithm insertion sort required almost n2 operations.
Analysis of Algorithms

22. 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

23. See TimeComplexity Spreadsheet
Analysis of Algorithms

24. Analysis of Algorithms
What is “goodness”?
Correctness
Minimum use of “time” + “space”
How can we quantify it?
Measure wall clock time
Count operations
Estimate Computational Complexity
Analysis of Algorithms

25. Analysis of Algorithms
Asymptotic Analysis
Uses a high-level description of the algorithm instead of an implementation
Allows us to evaluate the speed of an algorithm independent of the hardware/software environment
Estimate the growth rate of T(n)
A back of the envelope calculation!!!
Analysis of Algorithms

26. Analysis of Algorithms
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

27. 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

28. Which growth rate is best?
T(n) = 1000n + n2 or T(n) = 2n + n3
Analysis of Algorithms

29. Analysis of Algorithms
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
Analysis of Algorithms

30. Functions Graphed Using “Normal” Scale
g(n) = 2n
g(n) = 1
g(n) = n lg n
g(n) = lg n
g(n) = n2
g(n) = n
g(n) = n3
Analysis of Algorithms

31. Analysis of Algorithms
Constant Factors
The growth rate is not affected by
constant factors or
lower-order terms
Examples
102n is a linear function
105n n is a quadratic function
Analysis of Algorithms

32. Analysis of Algorithms
Big-Oh Notation
Given functions f(n) and g(n), we say that f(n) is O(g(n)) if there are positive constants c 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
Analysis of Algorithms

33. f(n) is O(g(n)) iff f(n)  cg(n) for n  n0
Analysis of Algorithms

34. 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
Analysis of Algorithms

35. Analysis of Algorithms
More Big-Oh Examples
7n-2
7n-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 + 5
3n3 + 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 + 5
3 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

36. Analysis of Algorithms
Big-Oh and Growth Rate
The big-Oh notation gives an upper bound on the growth rate of a function
The 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

37. Analysis of Algorithms
Questions
Is T(n) = 9n n = O(n4)?
Is T(n) = 9n n = O(n3)?
Is T(n) = 9n n = O(n27)?
T(n) = n n = O(?)
T(n) = 3n + 32n n2 = O(?)
Analysis of Algorithms

38. Big-Oh Rules (shortcuts)
If is f(n) a polynomial of degree d, then f(n) is O(nd), i.e.,
Drop lower-order terms
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

39. Analysis of Algorithms
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 instructed
i.e. log n = log2 n
Analysis of Algorithms

40. Computing Prefix Averages
We further illustrate asymptotic analysis with two algorithms for prefix averages
The i-th prefix average of an array X is average of the first (i + 1) elements of X
A[i] = X[0] + X[1] + … + X[i]
Computing the array A of prefix averages of another array X has applications to financial analysis
Analysis of Algorithms

41. Analysis of Algorithms
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 integers
Output array A of prefix averages of X #operations
A  new array of n integers n
for i  0 to n  1 do n
s  X[0] n
for j  1 to i do …+ (n  1)
s  s + X[j] …+ (n  1)
A[i]  s / (i + 1) n
return A
Analysis of Algorithms

42. Arithmetic Progression
The running time of prefixAverages1 is O( …+ 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
Analysis of Algorithms

43. Analysis of Algorithms
Prefix Averages (Linear)
The following algorithm computes prefix averages in linear time by keeping a running sum
Algorithm prefixAverages2(X, n)
Input array X of n integers
Output array A of prefix averages of X #operations
A  new array of n integers n
s 
for i  0 to n  1 do n
s  s + X[i] n
A[i]  s / (i + 1) n
return A
Algorithm prefixAverages2 runs in O(n) time
Analysis of Algorithms

44. Math you may need to Review
Summations
Logarithms and Exponents
Proof techniques
Basic probability
properties of logarithms:
logb(xy) = logbx + logby
logb (x/y) = logbx - logby
logbxa = alogbx
logba = 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
Analysis of Algorithms

45. Analysis of Algorithms
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

46. Analysis of Algorithms
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).
Analysis of Algorithms

47. Analysis of Algorithms
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. Analysis of Algorithms
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. 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)
Analysis of Algorithms