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Chapter 4Chapter 4Analysis ToolsAnalysis Tools

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Which is faster – selection sort or insertion sort?

Potential method for evaluation:Implement each as a method and then Time each method to see which is faster

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What are the most important criteria that influence our algorithm implementation choices?

What do each of these criteria directly affect?

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Experimental StudiesExperimental Studies

Write a program Write a program implementing the algorithmimplementing the algorithmRun the program with Run the program with inputs of varying size and inputs of varying size and compositioncompositionUse a method like Use a method like System.currentTimeMillis()System.currentTimeMillis() to to get an accurate measure get an accurate measure of the actual running timeof the actual running timePlot the resultsPlot the results

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Limitations of ExperimentsLimitations of Experiments

It is necessary to implement the It is necessary to implement the algorithm, which may be difficultalgorithm, which may be difficultResults may not be indicative of the Results may not be indicative of the running time on other inputs not included running time on other inputs not included in the experiment. in the experiment. In order to compare two algorithms, the In order to compare two algorithms, the same hardware and software same hardware and software environments must be usedenvironments must be used

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Theoretical AnalysisTheoretical Analysis

Uses a high-level description of the Uses a high-level description of the algorithm instead of an implementationalgorithm instead of an implementation

Characterizes running time as a Characterizes running time as a function of the input size, function of the input size, nn..

Takes into account all possible inputsTakes into account all possible inputs

Allows us to evaluate the speed of an Allows us to evaluate the speed of an algorithm independent of the algorithm independent of the hardware/software environmenthardware/software environment

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Big-O

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Two important rules:Make g(n) as small as possibleg(n) never contains unnecessary terms

Asymptotic AnalysisThe goal of asymptotic analysis is to determine the complexity order of an algorithm

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Big-Oh RulesBig-Oh Rules

If is If is ff((nn)) a polynomial of degree a polynomial of degree dd, then , then ff((nn)) is is OO((nndd)), i.e.,, i.e.,

1.1. Drop lower-order termsDrop lower-order terms

2.2. Drop constant factorsDrop constant factors

Use the smallest possible class of functionsUse the smallest possible class of functions Say “Say “22nn is is OO((nn))”” instead of “instead of “22nn is is OO((nn22))””

Use the simplest expression of the classUse the simplest expression of the class Say “Say “33nn 55 is is OO((nn))”” instead of “instead of “33nn 55 is is OO(3(3nn))””

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Figure 9.6Figure 9.6An insertion sort partitions the array into two regions

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Figure 9.7Figure 9.7An insertion sort of an array of five integers.

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Figure 9.4Figure 9.4A selection sort of an array of five integers

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Order of Complexity•Exponential•Polynomial•Log•Linear•Constant

Exponential > Polynomial > Log > Linear > Constant

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Figure 9.3aFigure 9.3aA comparison of growth-rate functions: a) in tabular form

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Figure 9.3bFigure 9.3bA comparison of growth-rate functions: b) in graphical form

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Worst case: largest value for any problem of size n

Best case: smallest value for any problem of size n

Average case: (weighted) average of all problems of size n

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Figure 9.5Figure 9.5The first two passes of a bubble sort of an array of five integers: a) pass 1; b) pass 2

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Figure 9.8Figure 9.8A mergesort with an auxiliary temporary array

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Figure 9.9Figure 9.9A mergesort of an array of six integers

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Figure 9.10Figure 9.10A worst-case instance of the merge step in mergesort

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Figure 9.11Figure 9.11Levels of recursive calls to mergesort given an array of eight items

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Figure 9.12Figure 9.12A partition about a pivot

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Figure 9.13Figure 9.13kSmall versus quicksort

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Figure 9.14Figure 9.14Invariant for the partition algorithm

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Figure 9.15Figure 9.15Initial state of the array

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Figure 9.16Figure 9.16Moving theArray[firstUnknown] into S1 by swapping it with theArray[lastS1+1] and by incrementing both lastS1 and firstUnknown

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Figure 9.17Figure 9.17Moving theArray[firstUnknown] into S2 by incrementing firstUnknown

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Figure 9.18aFigure 9.18aDeveloping the first partition of an array when the pivot is the first item

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Figure 9.18bFigure 9.18bDeveloping the first partition of an array when the pivot is the first item

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Figure 9.19Figure 9.19A worst-case partitioning with quicksort

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Figure 9.20Figure 9.20A average-case partitioning with quicksort

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Figure 9.21Figure 9.21A radix sort of eight integers

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Figure 9.22Figure 9.22Approximate growth rates of time required for eight sorting algorithms