1. Invitation to Computer Science 5th Edition
Chapter 3
The Efficiency of Algorithms

2. Objectives In this chapter, you will learn about:
Attributes of algorithms
Measuring efficiency
Analysis of algorithms
When things get out of hand
Invitation to Computer Science, 5th Edition

3. Introduction An important part of computer science
Finding algorithms to solve problems of interest
Are some algorithms better than others?
Invitation to Computer Science, 5th Edition

4. Attributes of Algorithms
We expect correctness from our algorithms
Program maintenance
Fixes errors uncovered through repeated use
Extends program to meet new requirements
Ease of understanding
Desirable characteristic of an algorithm
Elegance
Algorithmic equivalent of style
Invitation to Computer Science, 5th Edition

5. Attributes of Algorithms (continued)
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
Invitation to Computer Science, 5th Edition

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

7. 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

8. Figure 3.1 Sequential Search Algorithm
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9. Figure 3.2 Number of Comparisons to Find NAME in a List of n Names Using Sequential Search
Invitation to Computer Science, 5th Edition

10. 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

11. Figure 3.3 Work = 2n
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12. Figure 3.4 Work = cn for Various Values of c
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13. Figure 3.5 Growth of Work = cn for Various Values of c
Invitation to Computer Science, 5th Edition

14. Selection Sort Selection sort algorithm Subtask within selection sort
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

15. Figure 3.6 Selection Sort Algorithm
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16. Figure 3.7 Comparisons Required by Selection Sort
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17. 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

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

19. Figure 3.9 Exchanging the Values at X and Y
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20. 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

21. Figure 3.10 Work 5 cn2 for Various Values of c
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22. Figure 3.11 A Comparison of n and n2
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23. Figure 3.12 For Large Enough n, 0.25n2 Has Larger Values Than 10n
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24. Order of Magnitude - Order n2 (continued)
Part of the job of program documentation
To make clear any assumptions or restrictions about the input size the program was designed to handle
Comparing algorithm efficiency
Only makes sense if there is a choice of algorithms for the task at hand
Invitation to Computer Science, 5th Edition

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

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

27. 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

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

29. 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

30. 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

31. 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
Invitation to Computer Science, 5th Edition

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

33. 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

34. 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

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

36. Figure 3.17 Analysis of Three Data Cleanup Algorithms
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37. 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

38. 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

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

40. 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:
lg n increases by only 1, so lg n grows much more slowly than n
Worst case and average case
Θ(lg n) comparisons
Works only on a list that has already been sorted
Invitation to Computer Science, 5th Edition

41. Figure 3.19 Binary Search Tree for a 7-Element List
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42. Figure 3.20 Values for n and lg n
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43. Figure 3.21 A Comparison of n and lg n
Invitation to Computer Science, 5th Edition

44. 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 3 n)
Invitation to Computer Science, 5th Edition

45. 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

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

47. Figure 3.23 Four Connected Cities
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48. Figure 3. 24 Hamiltonian Circuits among All Paths from A in Figure 3
Figure 3.24 Hamiltonian Circuits among All Paths from A in Figure 3.23 with Four Links
Invitation to Computer Science, 5th Edition

49. When Things Get Out of Hand (continued)
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
Invitation to Computer Science, 5th Edition

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

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

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

53. Summary of Level 1 Level 1 (Chapters 2 and 3) explored algorithms
Pseudocode
Sequential, conditional, and iterative operations
Algorithmic solutions to various practical problems
Chapter 3
Desirable properties for algorithms
Time and space efficiencies of a number of algorithms
Invitation to Computer Science, 5th Edition

54. Summary Desirable attributes in algorithms Correctness
Ease of understanding (clarity)
Elegance
Efficiency
An algorithm’s careful use of resources is extremely important
Invitation to Computer Science, 5th Edition