# Chapter 3 The Efficiency of Algorithms

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Chapter 3 The Efficiency of Algorithms
Algorithmic problem solving Chapter 3 The Efficiency of Algorithms

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

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

Invitation to Computer Science, 5th Edition
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

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

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

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

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

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

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

Invitation to Computer Science, 5th Edition
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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Invitation to Computer Science, 5th Edition
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

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

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

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

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

Invitation to Computer Science, 5th Edition
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

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

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

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

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

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

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

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

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