CSCI 6212Design and Analysis of

Algorithms

Dr. Juman ByunThe George Washington University

Please drop this course if you have not taken the following prerequisite.

Sometimes enthusiasm alone is not enough.

• CSci 1311: Discrete Structures I (3)• CSci 1112: Algorithms and Data Structures

(3)

Official Course Description

• http://www.cs.gwu.edu/academics/courses/graduate/csci-6212

• Design and analysis of algorithms. Turing machines; NP-Complete theory. Algorithmic techniques: divide-and-conquer, greedy, dynamic programming, graph traversal, backtracking, and branch-and-bound. Applications include sorting and searching, graph algorithms, and optimization. Prerequisite: CSci 1311, 1112. (Fall andspring)

• Textbook: Introduction to Algorithms, Cormen, Leiserson, Rivest, and Stein.

Basic Principles of Algorithm Design and

Analysis

Algorithm Design

How to Start the Design ?

• Understand the problem. What are we trying to achieve ?

• Get a rough idea how to solve the problem using your intuition and imagination

Example: Sorting an Array of integers

55 22 44 66 11 33

Example: Sorting an Array of integers

55

22 44 66 11 33

Example: Sorting an Array of integers

5522

44 66 11 33

Example: Sorting an Array of integers

5522

44 66 11 33

Example: Sorting an Array of integers

552244

66 11 33

Example: Sorting an Array of integers

5522 44

66 11 33

Example: Sorting an Array of integers

5522 4466

11 33

Example: Sorting an Array of integers

5522 44 66

11 33

Example: Sorting an Array of integers

5522 44 6611

33

Example: Sorting an Array of integers

5522 44 6611

33

Example: Sorting an Array of integers

5522 44 6611

33

Example: Sorting an Array of integers

5522 44 6611

33

Example: Sorting an Array of integers

5522 44 6611

33

Example: Sorting an Array of integers

5522 44 6611

33

Example: Sorting an Array of integers

5522 44 661133

Example: Sorting an Array of integers

5522 44 661133

Example: Sorting an Array of integers

5522 44 661133

Example: Sorting an Array of integers

5522 44 661133

Example: Sorting an Array of integers

5522 44 6611 33

Accept Concise Notation

• The previous colorful animation was pretty but it took me a some time to create it.

• In order for a human being to perceive the entire problem, you need to compress the problem or commit it into the long-term storage of your brain.

• Algorithmic notation is an excellent way of compressing a problem so that it is easier to remember and handle.

Accept Concise Notation

55 22 44 66 11 33

Accept Concise Notation

5 2 4 6 1 3

Disambiguate it

5 2 4 6 1 3, , , , ,

Abstract it

5 2 4 6 1 3, , , , ,A = { }

Denote it

Insertion Sort (A)1 for j = 2 to A.length2 key = A[j]3 // Insert A[j] into the sorted sequence A[1..j-1]4 i = j - 15 while i > 0 and A[i] > key6 A[i +1] = A[i]7 i = i - 18 A[i + 1] = key

Pascal NotationInsertion Sort (A)1 for j = 2 to A.length2 begin3 key = A[j]4 // Insert A[j] into the sorted sequence A[1..j-1]5 i = j - 16 while i > 0 and A[i] > key7 begin8 A[i +1] = A[i]9 i = i - 110 end11 A[i + 1] = key12 end

C NotationInsertion Sort (A)1 for j = 2 to A.length2 {3 key = A[j]4 // Insert A[j] into the sorted sequence A[1..j-1]5 i = j - 16 while i > 0 and A[i] > key7 {8 A[i +1] = A[i]9 i = i - 110 }11 A[i + 1] = key12 }

Algorithm Analysis

Example: Running (Execution) Time Analysis

Insertion Sort (A)1 for j = 2 to A.length2 key = A[j]3 // Insert A[j] into the sorted sequence A[1..j-1]4 i = j - 15 while i > 0 and A[i] > key6 A[i +1] = A[i]7 i = i - 18 A[i + 1] = key

How many times does each statement execute ?

Insertion Sort (A)1 for j = 2 to A.length2 key = A[j]3 // Insert A[j] into the sorted sequence A[1..j-1]4 i = j - 15 while i > 0 and A[i] > key6 A[i +1] = A[i]7 i = i - 18 A[i + 1] = key

Factors to Consider• Input Size

• Running Time

• Worst-Case

• Best-Case

• Average

• Rate of Growth

Assumptions

• Technologies

• Memory Model: RAM (Random Access Machine/Memory)

• Processor: Single Processor