The document discusses the CS 321 Algorithm Analysis and Design course. It covers announcements about homework, grading, and clarifications. Algorithm analysis is introduced, covering running time and correctness. Insertion sort is presented, including a loop invariant and examples. Best-case, average-case, and worst-case running times are defined. Asymptotics and big O notation are introduced. Merge sort is mentioned along with induction.

1. CS 321. Algorithm Analysis & Design
Lecture 2
Sorting

2. Announcements

3. Announcements
The first homework will be online this Wednesday
with a deadline of next Wednesday.

4. Announcements
The first homework will be online this Wednesday
with a deadline of next Wednesday.
Will deal with clarifications on Friday.

5. Announcements
The first homework will be online this Wednesday
with a deadline of next Wednesday.
Please take the informal survey on the website,
or send me an email with the subject [CS321].
Will deal with clarifications on Friday.

6. Graded Work
Google Code Jam Qualifiers (20%)
Self-assessment quizzes via MCQs online (25%)
Assignments on the Hacker Rank Platform (25%)
Mid-sem and End-sem Exams (30%)

7. Algorithm
Analysis

8. Algorithm
Analysis
Running TimeCorrectness

9. Insertion Sort

10. Loop Invariants

12. At the start of each iteration of the for loop of lines 1–8,
the subarray A[1 . . j − 1] consists of the elements
originally in A[1 . . j − 1] but in sorted order.

13. Initialization
Maintenance
Termination

14. Initialization
Maintenance
Termination
The statement is true at the beginning.

15. Initialization
Maintenance
Termination
The statement is true at the beginning.
If the statement was true before the interation,
it is true after the iteration.

16. Initialization
Maintenance
Termination
The statement is true at the beginning.
If the statement was true before the interation,
it is true after the iteration.
When the loop ends, the invariant gives us something useful.

17. At the start of each iteration of the for loop of lines 1–8,
the subarray A[1 . . j − 1] consists of the elements
originally in A[1 . . j − 1] but in sorted order.

18. At the start of each iteration of the for loop of lines 1–8,
the subarray A[1 . . j − 1] consists of the elements
originally in A[1 . . j − 1] but in sorted order.

19. At the start of each iteration of the for loop of lines 1–8,
the subarray A[1 . . j − 1] consists of the elements
originally in A[1 . . j − 1] but in sorted order.

20. Insertion Sort

25. [1,2,3]
[1,3,2]
[3,1,2]
[2,1,3]
[2,3,1]
[3,2,1]

26. Best-case running time
Average-case running time
Worst-case running time

27. Best-case running time
Average-case running time
Worst-case running time
There is some input on which the algorithm enjoys
this running time.

28. Best-case running time
Average-case running time
Worst-case running time
Work averaged over the space of all inputs.
There is some input on which the algorithm enjoys
this running time.

29. Best-case running time
Average-case running time
Worst-case running time
Work averaged over the space of all inputs.
No matter what the input, the algorithm will not need more
time than this.
There is some input on which the algorithm enjoys
this running time.

30. Worst-case running time
No matter what the input, the algorithm will not need more
time than this.
We’ll typically obtain lower and upper bounds
on the worst case running time.

31. Worst-case is sometimes too
pessimistic an approach.
Real-world inputs almost always
have some structure!

32. The best case is O(n) and the worst case is O(n2).

33. Asymptotics

34. T(n) is O(f(n))
There exist constants
c > 0 and no> 0 so that for all n >no, we
have T(n) ≤ c f(n).

35. [Chapter 2, KT]
Running times on processors performing a
million high-level operations per second.

36. [Chapter 2, KT]
Running times on processors performing a
million high-level operations per second.

37. Allows for coarse running time analysis,
rather than being bogged down with
details.
Ignoring constants can be dangerous in
the real world.

38. Merge Sort

39. Induction

47. Merge Sort