CS 321. Algorithm Analysis & Design Lecture 2

Sorting

Announcements

Announcements

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

Announcements

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

Will deal with clarifications on Friday.

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.

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%)

Algorithm Analysis

Algorithm Analysis

Running TimeCorrectness

Insertion Sort

Loop Invariants

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.

Initialization

Maintenance

Termination

Initialization

Maintenance

Termination

The statement is true at the beginning.

Initialization

Maintenance

Termination

The statement is true at the beginning.

If the statement was true before the interation,it is true after the iteration.

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.

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.

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.

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.

Insertion Sort

[1,2,3]

[1,3,2]

[3,1,2]

[2,1,3]

[2,3,1]

[3,2,1]

Best-case running time

Average-case running time

Worst-case running time

Best-case running time

Average-case running time

Worst-case running time

There is some input on which the algorithm enjoysthis running time.

Best-case running time

Average-case running time

Worst-case running time

Work averaged over the space of al l inputs.

There is some input on which the algorithm enjoysthis running time.

Best-case running time

Average-case running time

Worst-case running time

Work averaged over the space of al l inputs.

No matter what the input, the algorithm wil l not need moretime than this.

There is some input on which the algorithm enjoysthis running time.

Worst-case running time

No matter what the input, the algorithm wil l not need more time than this.

We’ll typically obtain lower and upper bounds on the worst case running time.

Worst-case is sometimes too pessimistic an approach.

Real-world inputs almost always have some structure!

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

Asymptotics

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).

[Chapter 2, KT]

Running times on processors performing a million high-level operations per second.

[Chapter 2, KT]

Running times on processors performing a million high-level operations per second.

Allows for coarse running time analysis, rather than being bogged down with

details.

Ignoring constants can be dangerous in the real world.

Merge Sort

Induction

Merge Sort