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Chapter 3 The Efficiency of Algorithms

Chapter 3 The Efficiency of Algorithms. Algorithmic problem solving. Attributes of Algorithms. Are some algorithms better than others? We expect correctness from our algorithms Ease of understanding; Elegance Analysis of Algorithms Efficiency

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Chapter 3 The Efficiency of Algorithms

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  1. Chapter 3 The Efficiency of Algorithms Algorithmic problem solving

  2. Attributes of Algorithms • Are some algorithms better than others? • We expect correctness from our algorithms • Ease of understanding; Elegance Analysis of Algorithms • Efficiency • Term used to describe an algorithm’s careful use of resources • Benchmarks • Useful for rating one machine against another and for rating how sensitive a particular algorithm is with respect to variations in input on one particular machine

  3. Measuring Efficiency • Analysis of algorithms • The study of the efficiency of algorithms • An important part of computer science Invitation to Computer Science, 5th Edition

  4. Sequential Search • Search for NAME among a list of n names • Start at the beginning and compare NAME to each entry until a match is found Invitation to Computer Science, 5th Edition

  5. Figure 3.1 Sequential Search Algorithm Invitation to Computer Science, 5th Edition

  6. Figure 3.2 Number of Comparisons to Find NAME in a List of n Names Using Sequential Search Invitation to Computer Science, 5th Edition

  7. Order of Magnitude - Order n • Order of magnitude n • Anything that varies as a constant times n (and whose graph follows the basic shape of n) • Sequential search • An Θ(n) algorithm in both the worst case and the average case Invitation to Computer Science, 5th Edition

  8. Figure 3.3 Work = 2n Invitation to Computer Science, 5th Edition

  9. Figure 3.4Work = cn for Various Values of c Invitation to Computer Science, 5th Edition

  10. Figure 3.5 Growth of Work = cn for Various Values of c Invitation to Computer Science, 5th Edition

  11. Selection Sort • Selection sort algorithm • Sorts in ascending order • Subtask within selection sort • Task of finding the largest number in a list • When selection sort algorithm begins • The largest-so-far value must be compared to all the other numbers in the list • If there are n numbers in the list, n – 1 comparisons must be done Invitation to Computer Science, 5th Edition

  12. Figure 3.6 Selection Sort Algorithm Invitation to Computer Science, 5th Edition

  13. Figure 3.7 Comparisons Required by Selection Sort Invitation to Computer Science, 5th Edition

  14. Selection Sort (continued) • Selection sort algorithm • Does n exchanges, one for each position in the list to put the correct value in that position • Space efficiency of the selection sort • Original list occupies n memory locations • Storage is needed for: • The marker between the unsorted and sorted sections • Keeping track of the largest-so-far value and its location in the list Invitation to Computer Science, 5th Edition

  15. Figure 3.8An Attempt to Exchange the Values at X and Y Invitation to Computer Science, 5th Edition

  16. Figure 3.9 Exchanging the Values at X and Y Invitation to Computer Science, 5th Edition

  17. Order of Magnitude - Order n2 • Order of magnitude n2, or Θ(n2) • An algorithm that does cn2 work for any constant c • Selection sort • An Θ(n2) algorithm (in all cases) • Sequential search • An Θ(n) algorithm (in the worst case) Invitation to Computer Science, 5th Edition

  18. Figure 3.10 Work 5 cn2 for Various Values of c Invitation to Computer Science, 5th Edition

  19. Figure 3.11 A Comparison of n and n2 Invitation to Computer Science, 5th Edition

  20. Figure 3.12 For Large Enough n, 0.25n2 Has Larger Values Than 10n Invitation to Computer Science, 5th Edition

  21. Figure 3.13 A Comparison of Two Extreme Q(n2) and Q(n) Algorithms Invitation to Computer Science, 5th Edition

  22. Analysis of Algorithms • Data cleanup algorithms • The Shuffle-Left Algorithm • The Copy-Over Algorithm • The Converging-Pointers Algorithm Invitation to Computer Science, 5th Edition

  23. The Shuffle-Left Algorithm • Scans list from left to right • When a zero is found, copy each remaining data item in the list one cell to the left • Value of legit • Originally set to the length of the list • Is reduced by 1 every time a 0 is encountered Invitation to Computer Science, 5th Edition

  24. Figure 3.14 The Shuffle-Left Algorithm for Data Cleanup Invitation to Computer Science, 5th Edition

  25. The Shuffle-Left Algorithm (continued) • To analyze time efficiency: • Identify the fundamental units of work the algorithm performs • Copying numbers • Best case occurs when the list has no 0 values because no copying is required • Worst case occurs when the list has all 0 values Invitation to Computer Science, 5th Edition

  26. The Shuffle-Left Algorithm (continued) • Worst case • Occurs when the list has all 0 values • An Θ(n2) algorithm • Space-efficient • Only requires four memory locations to store the quantities n, legit, left,and right Invitation to Computer Science, 5th Edition

  27. The Copy-Over Algorithm • Scans the list from left to right, copying every legitimate (non-zero) value into a new list that it creates • Every list entry is examined to see whether it is 0 • Every non-zero list entry is copied once • Best case • Occurs if all elements are 0 • Θ(n) in time efficiency • No extra space is used

  28. Figure 3.15 The Copy-Over Algorithm for Data Cleanup Invitation to Computer Science, 5th Edition

  29. The Copy-Over Algorithm (continued) • Worst case • Occurs if there are no 0 values in the list • Algorithm copies all n non-zero elements into the new list and doubles the space required • Θ(n) in time efficiency • Time/space tradeoff • You gain something by giving up something else Invitation to Computer Science, 5th Edition

  30. The Converging-Pointers Algorithm • Swap zero values from left with values from right until pointers converge in the middle • Best case • A list containing no 0 elements • Worst case • A list of all 0 entries • Θ(n) in time efficiency • Is space-efficient Invitation to Computer Science, 5th Edition

  31. Figure 3.16 The Converging-Pointers Algorithm for Data Cleanup Invitation to Computer Science, 5th Edition

  32. Figure 3.17 Analysis of Three Data Cleanup Algorithms Invitation to Computer Science, 5th Edition

  33. The Converging-Pointers Algorithm (continued) • In an Θ(n) algorithm: • The work is proportional to n • In an Θ(n2) algorithm: • The work is proportional to the square of n Invitation to Computer Science, 5th Edition

  34. Binary Search • Procedure • First looks for NAME at roughly the halfway point in list • If name equals NAME,search is over • If NAME comes alphabetically before name at halfway point, search is narrowed to the front half of list • If NAME comes alphabetically after name at halfway point, search is narrowed to the back half of the list • Algorithm halts when NAME is found Invitation to Computer Science, 5th Edition

  35. Figure 3.18 Binary Search Algorithm (list must be sorted) Invitation to Computer Science, 5th Edition

  36. Binary Search (continued) • Logarithm of n to the base 2 • Number of times a number n can be cut in half and not go below 1 • As n doubles: • lgn increases by only 1, so lgn grows much more slowly than n • Worst case and average case • Θ(lgn) comparisons • Works only on a list that has already been sorted

  37. Figure 3.19 Binary Search Tree for a 7-Element List Invitation to Computer Science, 5th Edition

  38. Figure 3.20 Values for n and lg n Invitation to Computer Science, 5th Edition

  39. Figure 3.21 A Comparison of n and lg n Invitation to Computer Science, 5th Edition

  40. Pattern-Matching • Usually involves a pattern length that is short compared to the text length • That is, when m is much less than n • Best case • Θ(n) • Worst case • Θ(m * n) Invitation to Computer Science, 5th Edition

  41. When Things Get Out of Hand • Polynomially bound algorithms • Work done is no worse than a constant multiple of n2 • Graph • A collection of nodes and connecting edges • Hamiltonian circuit • A path through a graph that begins and ends at the same node and goes through all other nodes exactly once Invitation to Computer Science, 5th Edition

  42. Figure 3.22 Order-of-Magnitude Time Efficiency Summary Invitation to Computer Science, 5th Edition

  43. Figure 3.23 Four Connected Cities Invitation to Computer Science, 5th Edition

  44. Figure 3.24 Hamiltonian Circuits among All Paths from A in Figure 3.23 with Four Links Invitation to Computer Science, 5th Edition

  45. When Things Get Out of Hand • Exponential algorithm • An Θ(2n) algorithm • Brute force algorithm • One that beats the problem into submission by trying all possibilities • Intractable problem • No polynomially bounded algorithm exists • Approximation algorithms • Do not solve the problem, but provide a close approximation to a solution

  46. Figure 3.25 Comparisons of lg n, n, n2, and 2n Invitation to Computer Science, 5th Edition

  47. Figure 3.26 Comparisons of lg n, n, n2, and 2n for Larger Values of n Invitation to Computer Science, 5th Edition

  48. Figure 3.27 A Comparison of Four Orders of Magnitude Invitation to Computer Science, 5th Edition

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