1. CSCI 3333 Data Structures Algorithm Analysis
by Dr. Bun Yue Professor of Computer Science

2. Acknowledgement Mr. Charles Moen Dr. Wei Ding Ms. Krishani Abeysekera
Dr. Michael Goodrich

3. Algorithm
“A finite set of instructions that specify a sequence of operations to be carried out in order to solve a specific problem or class of problems.”

4. Example Algorithm What do you think about this algorithm?
Input, output and side effects can be more clearly specified.
No discussion on how this is actually implemented in a given programming language.

5. Some Properties of Algorithms
A deterministic sequential algorithm should be:
Finiteness: Algorithm must complete after a finite number of instructions.
Absence of ambiguity: Each step must be clearly defined, having only one interpretation.
Definition of sequence: Each step must have a unique defined preceding & succeeding step. The first step (start step) & last step (halt step) must be clearly noted.
Input/output: There must be a specified number of input values and result values.
Feasibility: It must be possible to perform each instruction.

6. Specifying Algorithms
There are no universal formal.
By English.
By Pesudocode
By a programming language (not usually desirable.)

7. Specifying Algorithms
Must be included:
Input: type and meaning
Output: type and meaning
Instruction sequences.
May be included:
Assumptions
Side effects
Pre-conditions and post-conditions

8. Specifying Algorithms
Like sub-programs, algorithms can be broken down into sub-algorithms to reduce complexity.
High level algorithms may use English more.
Low level algorithms may use pseudocode more.

9. Pseudocode High-level description of an algorithm
More structured than English prose
Less detailed than a program
Preferred notation for describing algorithms
Hide program design issues

10. Example: ArrayMax Algorithm arrayMax(A, n) Input array A of n integers
Output maximum element value of A
currentMax  A[0]
for i  1 to n  1 do
if A[i]  currentMax then
currentMax  A[i]
return currentMax
What do you think?
What happens when n is 0?

11. Example Algorithm: BogoSort
Algorithm BogoSort(A);
Do while (A is not sorted)
randomize(A);
End do;
What do you think?

12. Better Specification Algorithm BogoSort(A)
Input: A: an array of comparable elements.
Output: none
Side Effect: A is sorted.
while not sorted(A)
randomize(A);
end while;

13. Sorted(A) Algorithm Sorted(A)
Input: A: an array of comparable elements.
Output: a boolean value indicating whether A is sorted (in non-descending order);
for (i=0; i<A.size-2; i++)
if A[i] > A[i+1] return false;
end for;
return true;

14. BogoSort
Is correct!
Terrible performance!

15. Criteria for Judging Algorithms
Correctness
Accuracy (especially for numerical methods and approximation problems)
Performance
Simplicity
Ease of implementations

16. How Bogosort Fare? Correctness: yes
Accuracy: yes (not really applicable)
Performance: terrible even for small arrays
Simplicity: yes
Ease of implementation: medium

17. Summation Algorithm Correctness: ok.
Performance: bad, proportional to N.
Simplicity: not too bad
Ease: not too bad

18. Summation Algorithm A better summation algorithm
Algorithm Summation(N)
Input: N: Natural Number
Output: Summation of 1 to N.
return N * (N+1) / 2;
Performance: constant time independent of N.

19. Performance of Algorithms
Performance of algorithms can be measured by runtime experiments, called benchmark analysis.

20. Experimental Studies 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

21. Limitations of Experiments
It is necessary to implement the algorithm, which may:
be difficult,
introduce implementation related performance issues.
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.

22. Theoretical Analysis
Uses a high-level description of the algorithm instead of an implementation
Characterizes running time as a function of the input sizes (usually n)
Takes into account all possible inputs
Allows us to evaluate the speed of an algorithm independent of the hardware/software environment

23. Analysis Tools (Goodrich, 166)‏
Asymptotic Analysis
Developed by computer scientists for analyzing the running time of algorithms
Describes the running time of an algorithm in terms of the size of the input – in other words, how well does it “scale” as the input gets larger
The most common metric used is:
Upper bound of an algorithm
Called the Big-Oh of an algorithm, O(n)‏

24. Algorithm Analysis Use asymptotic analysis
Analysis Tools (Goodrich web page)‏
Algorithm Analysis
Use asymptotic analysis
Assume that we are using an idealized computer called the Random Access Machine (RAM)‏
CPU
Unlimited amount of memory
Accessing a memory cell takes one unit of time
Primitive operations take one unit of time
Perform the analysis on the pseudocode

25. Pseudocode High-level description of an algorithm Structured
Analysis Tools (Goodrich, 48, 166)‏
Pseudocode
High-level description of an algorithm
Structured
Not as detailed as a program
Easier to understand (written for a human)‏
Algorithm arrayMax(A,n):
Input: An array A storing n >= 1 integers.
Output: The maximum element in A.
currentMax  A[0]
for i  1 to n – 1 do
if currentMax < A[i] then
currentMax  A[i]
return currentMax

26. Pseudocode Details Declaration Control flow Return value Expressions
Analysis Tools (Goodrich, 48)‏
Pseudocode Details
Declaration
Algorithm functionName(arg,…)‏
Input:…
Output:…
Control flow
if…then…[else…]
while…do…
repeat…until…
for…do…
Indentation instead of braces
Return value
return expression
Expressions
Assignment (like  in Java)‏
Equality testing (like  in Java)‏

27. Tips for writing an algorithm
Analysis Tools (Goodrich, 48)‏
Pseudocode Details
Declaration
Algorithm functionName(arg,…)‏
Input:…
Output:…
Control flow
if…then…[else…]
while…do…
repeat…until…
for…do…
Indentation instead of braces
Return value
return expression
Expressions
Assignment (like  in Java)‏
Equality testing (like  in Java)‏
Tips for writing an algorithm
Use the correct data structure
Use the correct ADT operations
Use object-oriented syntax
Indent clearly
Use a return statement at the end

28. Analysis Tools (Goodrich, 164–165)‏
Primitive Operations
To do asymptotic analysis, we can start by counting the primitive operations in an algorithm and adding them up (need not be very accurate).
Primitive operations that we assume will take a constant amount of time:
Assigning a value to a variable
Calling a function
Performing an arithmetic operation
Comparing two numbers
Indexing into an array
Returning from a function
Following a pointer reference

29. Counting Primitive Operations
Analysis Tools (Goodrich, 166)‏
Counting Primitive Operations
Inspect the pseudocode to count the primitive operations as a function of the input size (n)‏
Algorithm arrayMax(A,n):
currentMax  A[0]
for i  1 to n – 1 do
if currentMax < A[i] then
currentMax  A[i]
return currentMax
Count
Array indexing + Assignment 2
Initializing i 1
Verifying i<n n
Array indexing + Comparing 2(n-1)‏
Array indexing + Assignment 2(n-1) worst
Incrementing the counter 2(n-1)‏
Returning 1
Best case: 2+1+n+4(n–1)+1 = 5n
Worst case: 2+1+n+6(n–1)+1 = 7n-2

30. Best, Worst, or Average Case
Analysis Tools (Goodrich, 165)‏
Best, Worst, or Average Case
A program may run faster on some input data than on others
Best case, worst case, and average case are terms that characterize the input data
Best case – the data is distributed so that the algorithm runs fastest
Worst case – the data distribution causes the slowest running time
Average case – very difficult to calculate
We will concentrate on analyzing algorithms by identifying the running time for the worst case data

31. Running Time of arrayMax
Analysis Tools (Goodrich, 166)‏
Running Time of arrayMax
Worst case running time of arrayMax
f(n) = (7n – 2) primitive operations
Actual running time depends on the speed of the primitive operations—some of them are faster than others
Let t = speed of the slowest primitive operation
Let f(n) = the worst-case running time of arrayMax
f(n) = t (7n – 2)‏

32. Growth Rate of arrayMax
Analysis Tools (Goodrich, 166)‏
Growth Rate of arrayMax
Growth rate of arrayMax is linear
Changing the hardware alters the value of t, so that arrayMax will run faster on a faster computer
Growth rate does not change.
Slow PC
10(7n-2)‏
Fast PC
5(7n-2)‏
Fastest PC
1(7n-2)‏

33. Tests of arrayMax Does it really work that way?
Analysis Tools
Tests of arrayMax
Does it really work that way?
Used arrayMax algorithm from p.166, Goodrich
Tested on two PCs
Yes, results both show a linear growth rate
Pentium III, 866 MHz
Pentium 4, 2.4 GHz

34. Big-Oh Notation f(n)  cg(n) for n  n0
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

35. Big-Oh Meaning
Big-O expresses an upper bound on the growth rate of a function, for sufficiently large values of n.
It shows the asymptotic behavior.
Example:
3n2 + n lg n = O(n2)
3n2 + n lg n = O(n3)

36. Big-Oh Example Example: the function n2 is not O(n) n2  cn n  c
The above inequality cannot be satisfied since c must be a constant

37. Big-Oh Proof Example 50n2+100n lgn + 1500 is O(n2) Proof:
Need to find c such that
50n2+100n lgn < cn2
For example, pick c = 151 and n0 = 40. You may complete the proof.
You may also pick c = 1650 and n0 = 1

38. 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
f(n) is O(g(n))
g(n) is O(f(n))
g(n) grows more
Yes No
f(n) grows more
Same growth

39. Big-O Theorems
For all the following theorems, assume that f(n) is a function of n and that K is an arbitrary constant.
Theorem1: K is O(1)
Theorem 2: A polynomial is O(the term containing the highest power of n)
f(n) = 7n4 + 3n2 + 5n is O(n4)
Theorem 3: K*f(n) is O(f(n)) [that is, constant coefficients can be dropped]
g(n) = 7n4 is O(n4)
Theorem 4: If f(n) is O(g(n)) and g(n) is O(h(n)) then f(n) is O(h(n)). [transitivity]

40. Big-O Theorems
Theorem 5: Each of the following functions is strictly big-O of its successors:
K [constant]
logb(n) [always log base 2 if no base is shown] n
nlogb(n) n2
n to higher powers 2n 3n
larger constants to the n-th power
n! [n factorial] nn
For Example: f(n) = 3nlog(n) is O(nlog(n)) and O(n2) and O(2n)
smaller
larger

41. Big-O Theorems
Theorem 6: In general, f(n) is big-O of the dominant term of f(n), where “dominant” may usually be determined from Theorem 5.
f(n) = 7n2+3nlog(n)+5n+1000 is O(n2)
g(n) = 7n4+3n+106 is O(3n)
h(n) = 7n(n+log(n)) is O(n2)
Theorem 7: For any base b, logb(n) is O(log(n)).

42. Asymptotic Algorithm Analysis
The asymptotic analysis of an algorithm determines the running time in big-Oh notation
To perform the asymptotic analysis
We find the worst-case number of primitive operations executed as a function of the input size
We express this function with big-Oh notation
Example:
We determine that algorithm arrayMax executes at most 7n  2 primitive operations
We say that algorithm arrayMax “runs in O(n) time”
Since constant factors and lower-order terms are eventually dropped anyhow, we can disregard them when counting primitive operations

43. Example: 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])/(i+1)
Computing the array A of prefix averages of another array X has applications to financial analysis

44. Prefix Average (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

45. 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

46. 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

47. Math you 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

48. Relatives of Big-Oh big-Omega (Lower bound)
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
big-Theta (order)
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

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)

50. Relatives of Big-Oh Examples
5n2 is (n2)
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
let c = 5 and n0 = 1
5n2 is also (n)
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
let c = 1 and n0 = 1
5n2 is (n2)
f(n) is (g(n)) if it is (n2) and O(n2). We have already seen the former, for the latter recall that f(n) is O(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
Let c = 5 and n0 = 1

51. Algorithm Complexity
Complexity analysis (using Big-O) applies to algorithms, not problems.
May be applied to measure:
Best case: not very useful.
Worst case: most useful and common.
Average case: very useful but can be very difficult:
Depend on the input distribution.

52. What can be analyzed? Primitive operations: Primary memory usage
CPU/primary memory
I/O
Function call
Communication
Primary memory usage
Secondary memory usage
Number of processor used …

53. Questions and Comments?