1. CSE 2010: Algorithms and Data Structures Algorithms

2. Worst case vs. Average case
Next, we will look at the concept of rate of growth (or order of growth).
Average case is often as bad as the worst case.
When analyzing algorithms, we will mostly focus on the worst case.

3. Linear search example

4. Linear/sequential search

5. Linear search
var doLinearSearch = function(array) {
for (var guess = 0; guess < array.length; guess++) {
if (array[guess] === targetValue) {
return guess; // found it! }
return -1; // didn't find it };
Worst case: loop runs n times where n is the length of the input array.
Best case: loop runs 1 time.

6. Linear search
var doLinearSearch = function(array) {
for (var guess = 0; guess < array.length; guess++) {
if (array[guess] === targetValue) {
return guess; // found it! }
return -1; // didn't find it };
Each iteration of the loop performs a fixed number of instructions.
Each of these instructions runs in fixed amount of time, i.e., constant time.
Extra instruction: return -1 also runs in constant time.

7. Linear search
var doLinearSearch = function(array) {
for (var guess = 0; guess < array.length; guess++) {
if (array[guess] === targetValue) {
return guess; // found it! }
return -1; // didn't find it };
Rate of growth is linear
Running time of a program as a function of the size of its input.

8. Search in a matrix
for i = 0 to N
for j = 0 to N
print matrix[i,j]
print “Completed.”
Rate of growth is quadratic
Running time of a program as a function of the size of its input.

9. Asymptotic notations
for i = 0 to N
for j = 0 to N
print matrix[i,j]
print “Completed.”
Rate of growth is quadratic
Drop the slow-growing terms == keep only the fastest growing term.
Running time of a program as a function of the size of its input.

10. Rate of growth: Example functions that often appear in algorithm analysis
Constant » 1 Logarithmic » log n
Linear » n N-Log-N » n log n
Quadratic » n2 Cubic » n3 Exponential » 2n
T(n) = n2
T(n) = n
T(n) = log2 n
T(n) = 1

11. Asymptotic notations
Concept: upper bound and lower bound

12. Asymptotic notations
Concept: upper bound and lower bound

13. Asymptotic notations
upper bound
Concept: upper bound and lower bound

14. Asymptotic notations tight bound lower bound upper bound
Concept: upper bound and lower bound

15. https://youtu.be/ddsP7NecEBk
Asymptotic notations
tight bound
lower bound
upper bound
Concept: upper bound and lower bound
Less or equal
We can then say that:

16. https://youtu.be/ddsP7NecEBk
Asymptotic notations
tight bound
lower bound
upper bound
Concept: upper bound and lower bound
Less or equal
We can then say that:

17. https://youtu.be/ddsP7NecEBk
Asymptotic notations
tight bound
lower bound
upper bound
Concept: upper bound and lower bound
Less or equal
We can then say that:
and

18. https://youtu.be/ddsP7NecEBk
Asymptotic notations
tight bound
lower bound
upper bound
Concept: upper bound and lower bound
Less or equal
We can then say that:
and

19. Asymptotic Notation
f(n) = O(g(n) means f(n) in O(g(n))…. abuse of notation…

20. Asymptotic Notation

21. Asymptotic Notation

22. Asymptotic Notation

23. Asymptotic Notation

24. Asymptotic Notation

25. Asymptotic notations

26. Asymptotic notations
for i = 0 to M
for j = 0 to N
if a == b
count = count + 1
print “Completed.”
Rate of growth is quadratic
Drop the slow-growing terms == keep only the fastest growing term.
Running time of a program as a function of the size of its input.
O( M N )

27. Confusing worst case with upper bound
Upper bound refers to a growth rate.
Worst case refers to the worst input from among the choices for possible inputs of a given size.