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The Efficiency of Algorithms

The Efficiency of Algorithms. Chapter 9. Slides by Steve Armstrong LeTourneau University Longview, TX ã 2007, Prentice Hall. Chapter Contents. Motivation Measuring an Algorithm's Efficiency Big Oh Notation Formalities Picturing Efficiency

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The Efficiency of Algorithms

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  1. The Efficiency of Algorithms Chapter 9 Slides by Steve ArmstrongLeTourneau UniversityLongview, TX ã 2007,Prentice Hall

  2. Chapter Contents • Motivation • Measuring an Algorithm's Efficiency • Big Oh Notation • Formalities • Picturing Efficiency • The Efficiency of Implementations of the ADT List • An Array-Based Implementation • A Linked Implementation • Comparing Implementations Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  3. Motivation 1 • Even a simple program can be noticeably inefficient • When the 423 is changed to 100,000,000 there is a significant delay in seeing the result Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  4. Motivation • We seek the design of a Java class Huge to represent extremely large integers. • Consider an inefficient approach to add numbers that result in a large sum • View source code • Even a simple program can be noticeably inefficient Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  5. Measuring Algorithm Efficiency 6 • Types of complexity • Space complexity • Time complexity • Analysis of algorithms • The measuring of the complexity of an algorithm • Cannot compute actual time for an algorithm • We usually measure worst-case time Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  6. Measuring Algorithm Efficiency 9 Fig. 9-1 Three algorithms for computing 1 + 2 + … n for an integer n > 0 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  7. Measuring Algorithm Efficiency Fig. 9-2 The number of operations required by the algorithms for Fig 9-1 Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  8. Measuring Algorithm Efficiency 10 Fig. 9-3 The number of operations required by the algorithms in Fig. 9-1 as a function of n Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  9. Big Oh Notation 12 • To say "Algorithm A has a worst-case time requirement proportional to n" • We say A is O(n) • Read "Big Oh of n" • For the other two algorithms • Algorithm B is O(n2) • Algorithm C is O(1) Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  10. Big Oh Notation 15 Fig. 9-4 Typical growth-rate functions evaluated at increasing values of n Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  11. Big Oh Notation 16 Fig. 9-5 The number of digits in an integer n compared with the integer portion of log10n Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  12. Big Oh Notation Fig. 9-6 The values of two logarithmic growth-rate functions for various ranges of n. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  13. Formalities 17 • Formal definition of Big OhAn algorithm's time requirement f(n) is of order at most g(n) • f(n) = O(g(n)) • For a positive real number c and positive integer N exist such that f(n) ≤ c•g(n) for all n ≥ N Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  14. Formalities Fig. 9-7 An illustration of the definition of Big Oh Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  15. Formalities 21 • The following identities hold for Big Oh notation: • O(k f(n)) = O(f(n)) • O(f(n)) + O(g(n)) = O(f(n) + g(n)) • O(f(n)) O(g(n)) = O(f(n) g(n)) Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  16. Picturing Efficiency 24 Fig. 9-8 an O(n) algorithm. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  17. Picturing Efficiency 25 Fig. 9-9 An O(n2) algorithm. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  18. Picturing Efficiency 27 Fig. 9-10 Another O(n2) algorithm. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  19. Picturing Efficiency Fig. 9-11 The effect of doubling the problem size on an algorithm's time requirement. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  20. Picturing Efficiency 28 Fig. 9-12 The time to process one million items by algorithms of various orders at the rate of one million operations per second. Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  21. Comments on Efficiency • A programmer can use O(n2), O(n3) or O(2n) as long as the problem size is small • At one million operations per second it would take 1 second … • For a problem size of 1000 with O(n2) • For a problem size of 1000 with O(n3) • For a problem size of 20 with O(2n) Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  22. Efficiency of Implementations of ADT List 29 • Reference AListfrom Chapter 5 • For array-based implementation • Add to end of list O(1) • Add to list at given position O(n) • Reference LListfrom Chapter 6 • For linked implementation • Add to end of list O(n) • Add to list at given position O(n) • Retrieving an entry O(n) Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

  23. Comparing Implementations 34 Fig. 9-13 The time efficiencies of the ADT list operations for two implementations, expressed in Big Oh notation Carrano, Data Structures and Abstractions with Java, Second Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-237045-X

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