1. Chapter 3: The Efficiency of Algorithms
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C++ Version, Fourth Edition

2. Objectives In this chapter, you will learn about
Attributes of algorithms
Measuring efficiency
Analysis of algorithms
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3. Introduction Desirable characteristics in an algorithm Correctness
Ease of understanding (clarity)
Elegance
Efficiency
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4. Attributes of Algorithms
Correctness
Does the algorithm solve the problem it is designed for?
Does the algorithm solve the problem correctly?
Ease of understanding (clarity)
How easy is it to understand or alter the algorithm?
Important for program maintenance
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5. Attributes of Algorithms (continued)
Elegance
How clever or sophisticated is the algorithm?
Sometimes elegance and ease of understanding work at cross-purposes
Efficiency
How much time and/or space does the algorithm require when executed?
Perhaps the most important desirable attribute
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6. Quiz
1.(T/F)The correctness of algorithms is important for program maintenance.
2.(T/F)The clarity of alogrithms is the most important desirable attribute.
3.(T/F)The elegance of algorithms describes the time and space of algorithms execution.
4.(T/F)The efficiency of algorithms describes the results of algorithms execution.
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7. Measuring Efficiency Analysis of algorithms
Study of the efficiency of various algorithms
For one input size, best case, worst case, and average case behavior must be considered
The  notation captures the order of magnitude of the efficiency function
(n): order of magnitude n.
Order of magnitude is used to measure the time efficiency of algorithms.
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8. What is Order of Magnitude?
Kim makes $1 dollar an hour and asks her coworker John about his pay. John told her that his pay is 2 order of magnitude higher than her pay. What exactly does John make?
A. $1
B. $2
C. $10
D. $100
E. $20
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9. What is Order of Magnitude?
Kim makes $10 dollar an hour and asks her coworker John about his pay. John told her that his pay is 2 order of magnitude higher than her pay. What exactly does John make?
A. $1
B. $2
C. $10
D. $100
E. $20
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10. What is () or Big 0?
Orders of magnitude are generally used to make very approximate comparisons. If two numbers differ by one order of magnitude, one is about ten times larger than the other. If they differ by two orders of magnitude, they differ by a factor of about 100.
1 - (1)
10 - (2)
100 - (3)
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11. How quickly the values grow!
Orders of magnitude (n) determines how quickly the values grow as n increases. You want to write an algorithm with an order of magnitude that grow slowly as n increases.
For example, which one of the following algorithm has the shortest execution time?
John’s algorithm: (1)
Kim’s algorithm: (n)
Mathew’s algorithm: (n2)
Mark’s algorithm: (lg n) (note: logarithm of n to the base 2)
Answers:
When the data set size n is 4, the total execution time of
John’s algorithm is 1
Kim’s algorithm is 4
Mathew’s algorithm is 16
Mark’s algorithm is 2
When the data set size n is 16, the total execution time of
Kim’s algorithm is 16
Mathew’s algorithm is 256
Mark’s algorithm is 4
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12. Sequential Search
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13. Sequential Search (continued)
Comparison of the NAME being searched for against a name in the list
Central unit of work
Used for efficiency analysis
For lists with n entries
Best case
NAME is the first name in the list
1 comparison
(1)
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14. Sequential Search (continued)
For lists with n entries
Worst case
NAME is the last name in the list
NAME is not in the list
n comparisons
(n)
Average case
Roughly n/2 comparisons
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15. Sequential Search (continued)
Space efficiency
Uses essentially no more memory storage than original input requires
Very space efficient
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16. Selection Sort Sorting Selection sort algorithm
Take a sequence of n values and rearrange them into order
Selection sort algorithm
Repeatedly searches for the largest value in a section of the data
Moves that value into its correct position in a sorted section of the list
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17. Shows the data list after each selection sort swap (exchange) on the following data set:
1st swap:
2nd swap:
3rd swap:
4th swap:
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18. Quiz Use the Sort Animator to answer the question: How many times has Step 8 been executed? _____
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19. Work = cn for Various Values of c
Figure 3.4
Work = cn for Various Values of c
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20. Selection Sort (continued)
Count comparisons of largest so far against other values
Find Largest, given m values, does m-1 comparisons
Selection sort calls Find Largest n times,
Each time with a smaller list of values
Cost = n-1 + (n-2) + … = n(n-1)/2
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21. Selection Sort (continued)
Time efficiency
Comparisons: n(n-1)/2
Exchanges: n (swapping largest into place)
Overall: (n2), best, average and worst cases
Space efficiency
Space for the input sequence, plus a constant number of local variables
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22. Quiz
(T/F)The best case of the time efficiency of sequential search algorithm is (n2).
(T/F)The average case of the time efficiency of sequential search algorithm is (n).
(T/F)The worst case of the time efficiency of sequential search algorithm is (n2).
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23. Quiz
(T/F)The best case of the time efficiency of selection sort algorithm is (n2).
(T/F)The aerage case of the time efficiency of selection sort algorithm is (n).
(T/F)The worst case of the time efficiency of selection sort algorithm is (n2).
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24. Quiz 1.(T/F) Order of magnitude n is written (n).
2.(T/F) Order of magnitude is used to measure the space efficiency of algorithms.
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25. Quiz
__(T/F) If each comparison takes about 1 second, then the total time for a sequential search algorithm to search an item in a list with 100 entries at the best case is (1) seconds.
__(T/F) If each comparison takes about 1 second, then the total time for a sequential search algorithm to search an item in a list with 100 entries at the worst case is (1) seconds.
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26. __(T/F) If each comparison takes about 1 second, then the total time for a sequential search algorithm to search an item in a list with 100 entries at the average case is (1) seconds.
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27. __(T/F) If each comparison takes about 1 second, then the total time for a selection sort algorithm to search an item in a list with 10 entries at the best case is (10) seconds.
__(T/F) If each comparison takes about 1 second, then the total time for a selection sort algorithm to search an item in a list with 100 entries at the average case is (1) seconds.
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28. __(T/F) If each comparison takes about 1 second, then the total time for a selection sort algorithm to search an item in a list with 100 entries at the worst case is (1) seconds.
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29. Binary Search Algorithm
Given ordered data
Search for NAME by comparing to middle element
If not a match, restrict search to either lower or upper half only
Each pass eliminates half the data
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30. Binary Search Algorithm (list must be sorted)
Figure 3.18
Binary Search Algorithm (list must be sorted)
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31. Binary Search Algorithm (continued)
Efficiency
Best case
1 comparison
(1)
Worst case
lg n comparisons
lg n: The number of times n can be divided by two before reaching 1
(lg n)
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32. Binary Search Exercise
Use the binary search algorithm to decide whether 35 is in the following list:
3, 6, 7, 10, 35, 37, 39
What numbers will be compared to 35?
Ans: 10 37 35
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33. Binary Search Quiz
Use the binary search algorithm to decide whether 2 is in the following list:
3, 6, 7, 10, 35, 37, 39
What numbers will be compared to 2?
Ans:
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34. Answers of #28 on page 123. #28a. 100 10 ------ = 500 50 #28b =
#28b
100* =
500 *
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