1. Chapter 3: The Efficiency of Algorithms
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2. Objectives In this chapter, you will learn about:
Attributes of algorithms
Measuring efficiency
Analysis of algorithms
When things get out of hand
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3. Introduction Desirable characteristics in an algorithm Correctness
Ease of understanding
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
How easy is it to understand or alter an algorithm?
Important for program maintenance
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5. Attributes of Algorithms (continued)
Elegance
How clever or sophisticated is an algorithm?
Sometimes elegance and ease of understanding work at cross-purposes
Efficiency
How much time and/or space does an algorithm require when executed?
Perhaps the most important desirable attribute
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6. Attributes of Algorithms (continued)
Example: …+100=?
sum=0
x=1
While x is less than or equal to 100 do step 4 and 5
sum=sum+x
x=x+1
Print the value of x
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7. Attributes of Algorithms (continued)
Example: …+100=?
1+100=101
2+99=101 …
50+51=101
(100/2)*101=5050
Formula:
Sum=(n+1)*n/2
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8. Measuring Efficiency Analysis of algorithms
Study of the efficiency of various algorithms
Efficiency measured as function relating size of input to time or space used
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
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9. 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
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10. Sequential Search Algorithm
Figure 3.1
Sequential Search Algorithm
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11. 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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12. 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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13. Sequential Search (continued)
Space efficiency
Uses essentially no more memory storage than original input requires
Very space-efficient
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14. Order of Magnitude: Order n
As n grows large, order of magnitude dominates running time, minimizing effect of coefficients and lower-order terms
All functions that have a linear shape are considered equivalent
Order of magnitude n
Written (n)
Functions vary as a constant times n
e.x., n, 2n, 10n and ½n have the same order (n)
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15. Work = cn for Various Values of c
Figure 3.4
Work = cn for Various Values of c
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16. Practice Problem n Best Case Worst Case Average Case 10 1 5 50 25 100
1,000
500
10,000
5,000
100,000
50,000
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17. 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
Uses the Find Largest algorithm
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18. Selection Sort Algorithm
Figure 3.6
Selection Sort Algorithm
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19. Unsorted section | Sorted section
Loop#3
3. Not empty
4. 5, 3, 2| 7, 8
5. 2, 3, 5| 7, 8
6. 2, 3| 5, 7, 8
Loop#4
3. Not empty
4. 2, 3| 5, 7, 8
5. 2, 3| 5, 7, 8
6. 2| 3, 5, 7, 8
Loop#5
Not empty
4. 2| 3, 5, 7, 8
5. 2| 3, 5, 7, 8
6. |2, 3, 5, 7, 8
Loop#6
3. Empty
End loop
N=5 List is 5, 7, 2, 8, 3
5, 7, 2, 8, 3|
Loop#1
3. Not empty
5, 7, 2, 3, 8|
5, 7, 2, 3| 8
Loop#2
3. Not empty
4. 5, 7, 2, 3| 8
5. 5, 3, 2, 7| 8
6. 5, 3, 2| 7, 8
Unsorted section | Sorted section
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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 and worst cases
Space efficiency
Space for the input sequence, plus a constant number of local variables
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22. Selection Sort (continued)X Y 3 5 X Y 5 5
X = Y X Y 5 5
Y = X
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23. Selection Sort (continued)X Y 3 3 5
T = X -- Copy X to T T X Y 5 5 3
X = Y -- Copy Y to X T X Y 5 3 3
Y = T -- Copy T to X
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24. Order of Magnitude – Order n2
All functions with highest-order term cn2 have similar shape
An algorithm that does cn2 work for any constant c is order of magnitude n2, or (n2)
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25. Order of Magnitude – Order n2 (continued)
Anything that is (n2) will eventually have larger values than anything that is (n), no matter what the constants are
An algorithm that runs in time (n) will outperform one that runs in (n2)
e.x. (½)n n, 5n2 + 1 and 10 n2 run in the same (n2)
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26. Work = cn2 for Various Values of c
Figure 3.10
Work = cn2 for Various Values of c
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27. Figure 3.11 A Comparison of n and n2
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28. Figure 3.11 A Comparison of n and n2
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29. Analysis of Algorithms
Multiple algorithms for one task may be compared for efficiency and other desirable attributes
Data cleanup problem
Search problem
Pattern matching
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30. Data Cleanup Algorithms
Given a collection of numbers, find and remove all zeros
Possible algorithms
Shuffle-left
Copy-over
Converging-pointers
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31. The Shuffle-Left Algorithm
Scan list from left to right
When a zero is found, shift all values to its right one slot to the left
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32. The Shuffle-Left Algorithm for Data Cleanup
Figure 3.14
The Shuffle-Left Algorithm for Data Cleanup
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33. An Example Legit=5 Legit=4 24 36 45 24 45 36 24 45 36 Legit=4 24 36 4524 36 45 24 45 36 24 45 36
Legit=4 24 36 45 24 45 36
Legit=3 24 45 36 24 45 36
An Example
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34. The Shuffle-Left Algorithm (continued)
Time efficiency
Count examinations of list values and shifts
Best case
No shifts, n examinations
(n)
Worst case
Shift at each pass, n passes
n2 shifts plus n examinations
(n2)
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35. The Shuffle-Left Algorithm (continued)
Space efficiency
n slots for n values, plus a few local variables
(n)
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36. The Copy-Over Algorithm
Use a second list
Copy over each nonzero element in turn
Time efficiency
Count examinations and copies
Best case
All zeros
n examinations and 0 copies
(n)
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37. The Copy-Over Algorithm for Data Cleanup
Figure 3.15
The Copy-Over Algorithm for Data Cleanup
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38. The Copy-Over Algorithm (continued)
Time efficiency (continued)
Worst case
No zeros
n examinations and n copies
(n)
Space efficiency
2n slots for n values, plus a few extraneous variables
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39. The Copy-Over Algorithm (continued)
Time/space tradeoff
Algorithms that solve the same problem offer a tradeoff:
One algorithm uses more time and less memory
Its alternative uses less time and more memory
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40. The Converging-Pointers Algorithm
Swap zero values from left with values from right until pointers converge in the middle
Time efficiency
Count examinations and swaps
Best case
n examinations, no swaps
(n)
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41. The Converging-Pointers Algorithm for Data Cleanup
Figure 3.16
The Converging-Pointers Algorithm for Data Cleanup
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42. The Converging-Pointers Algorithm (continued)
Time efficiency (continued)
Worst case
n examinations, n swaps
(n)
Space efficiency
n slots for the values, plus a few extra variables
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43. Analysis of Three Data Cleanup Algorithms
Figure 3.17
Analysis of Three Data Cleanup Algorithms
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44. Binary Search 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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45. Binary Search Algorithm (list must be sorted)
Figure 3.18
Binary Search Algorithm (list must be sorted)
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46. Binary Search (continued)
Efficiency
Best case
1 comparison
(1)
Worst case
lg n comparisons
lg n: The number of times n may be divided by two before reaching 1
(lg n)
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47. Binary Search (continued)
Tradeoff
Sequential search
Slower, but works on unordered data
Binary search
Faster (much faster), but data must be sorted first
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48. A Comparison of n and lg n
Figure 3.21
A Comparison of n and lg n
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49. Pattern Matching Analysis involves two measures of input size
m: length of pattern string
n: length of text string
Unit of work
Comparison of a pattern character with a text character
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50. Pattern Matching (continued)
Efficiency
Best case
Pattern does not match at all
n - m + 1 comparisons
(n)
Worst case
Pattern almost matches at each point
(m -1)(n - m + 1) comparisons
(m x n)
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51. Order-of-Magnitude Time Efficiency Summary
Figure 3.22
Order-of-Magnitude Time Efficiency Summary
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52. When Things Get Out of Hand
Polynomially bound algorithms
Work done is no worse than a constant multiple of n2
Intractable algorithms
Run in worse than polynomial time
Examples
Hamiltonian circuit
Bin-packing
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53. When Things Get Out of Hand (continued)
Exponential algorithm
(2n)
More work than any polynomial in n
Approximation algorithms
Run in polynomial time but do not give optimal solutions
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54. Comparisons of lg n, n, n2 , and 2n
Figure 3.25
Comparisons of lg n, n, n2 , and 2n
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55. A Comparison of Four Orders of Magnitude
Figure 3.27
A Comparison of Four Orders of Magnitude
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56. Summary of Level 1 Level 1 (Chapters 2 and 3) explored algorithms
Pseudocode
Sequential, conditional, and iterative operations
Algorithmic solutions to three practical problems
Chapter 3
Desirable properties for algorithms
Time and space efficiencies of a number of algorithms
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57. Summary Desirable attributes in algorithms:
Correctness
Ease of understanding
Elegance
Efficiency
Efficiency – an algorithm’s careful use of resources – is extremely important
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58. Summary
To compare the efficiency of two algorithms that do the same task
Consider the number of steps each algorithm requires
Efficiency focuses on order of magnitude
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