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Design and Analysis of Algorithms

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Design and Analysis of Algorithms Dr. Ameer Ali What is Algorithm? Algorithm has come to refer: a method that can be used by a computer for the solution of a problem. – PowerPoint PPT presentation

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Title: Design and Analysis of Algorithms


1
Design and Analysis of Algorithms
  • Dr. Ameer Ali

2
What is Algorithm?
  • Algorithm has come to refer a method that can be
    used by a computer for the solution of a problem.
  • Definition An algorithm is a finite set of
    instructions that, if followed, accomplishes a
    particular task.

3
What is Algorithm?
  • Criteria
  • Input Zero or more input
  • Output At least one output
  • Completeness Must be complete within time frame
  • Definiteness Each instruction is clear and
    unambiguous
  • Finiteness Finite number of steps
  • Effectiveness Very basic, feasible.

4
Why Algorithm?
  • Developing reasoning
  • Way to solve a real world problem
  • Effective and efficient implementation in
    computer language
  • Efficient algorithm designing

5
Application Algorithm
  • In all type of computer software as well as in
    some hardware such as ROM.

6
Books
  • Introduction to Algorithms Thomas H. Cormen
  • Fundamentals of Computer Algorithms S. Sahni
  • Algorithm Design Michael T. Goodrich
  • The Art of Computer Programming Donald E. Knuth

7
Overview
  • Introduction to Algorithm
  • Elementary Data Structures
  • Sorting
  • Divide and Conquer
  • Greedy Method
  • Dynamic Programming
  • Elementary Graph Algorithms

8
Overview
  • Minimum Spanning Tree
  • Single Source Shortest Path
  • All Pair Shortest Path
  • Maximum Flow
  • Sorting Networks
  • Computational Geometry
  • NP-Completeness
  • Some Research Papers

9
Example
  • INSERT-SORT (A)
  • For j2 to lengthA do
  • keyAj
  • ij-1
  • while igt0 and Aigtkey do
  • Ai1Ai
  • ii-1
  • Ai1key

10
Example
  • A5,2,4,6,1,3
  • 5, 2, 4, 6, 1, 3
  • 2, 5 4 6 1 3
  • 2 4 5 6 1 3
  • 2 4 5 6 1 3
  • 1 2 4 5 6 3
  • 1 2 3 4 5 6

11
Analyzing Algorithms
  • Space
  • Time

12
Analyzing Insertion Sort
INSERT-SORT (A) For j2 to lengthA do keyAj /comments ij-1 while igt0 and Aigtkey do Ai1Ai ii-1 Ai1key Cost C1 C2 C4 C5 C6 C7 C8 Times n n-1 n-1 n-1 ?nj2tj ?nj2 (tj-1) ?nj2 (tj-1) n-1
13
Analyzing Insertion Sort
  • T(n)c1nc2(n-1)c4(n-1)c5(?nj2tj)c6(?nj2
    (tj-1))c7(?nj2 (tj-1))c8(n-1)
  • For best case already sorted list
  • T(n)c1nc2(n-1)c4(n-1)c5(n-1)c8(n-1)
  • (c1c2c4c5c8)n-(c2c4c5c8)

14
Analyzing Insertion Sort
  • For worst case sorted in reverse order here tjj
  • ?nj2 j(n(n-1)/2) 1
  • ?nj2 (j-1)n(n-1)/2
  • T(n)(c5/2c6/2c7/2)n2 (c1c2c4c5/2-c6/2-c7/2
    c8)n- (c2c4c5c8)
  • Which is quadratic equation

15
Analyzing Insertion Sort
  • For average case is similar to worst case
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