1.2 Comparing Algorithms2

8/12/2019 1.2 Comparing Algorithms2

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1.2 Comparing Algorithms


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What is an algorithm?

A set of unambiguous instructions for solving a

problem

Algorithms are used to transform inputs into

outputs

Computer programs would not exist without them

There are several ways to write an algorithm to

solve a problem e.g. bubble sort & insertion sort

Must produce correct output for all valid inputs

Amount of memory used and speed of computation

are important


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Computational Complexity

This depends on the space complexity and the timecomplexity

Time complexity indicates how fast it runs

Space complexity indicates how much memory space isneeded

The number of items to be searched or sorted impacts on

how long it takes to carry out the operation

Time complexity can be improved at the expense of spacecomplexity

See the presentationon examples of problems with

different time complexities
http://www.school-portal.co.uk/GroupDownloadFile.asp?GroupId=1010236&ResourceID=4106083http://www.school-portal.co.uk/GroupDownloadFile.asp?GroupId=1010236&ResourceID=4106083


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Complexity of a problem

Worst-case complexity

e.g. search for a value where every number in a list is

examined

Best-case complexity e.g. search for a value where only one comparison is

required

Average-case complexity

The complexity of a problem is taken to be the worst-

case complexity of the most efficient algorithm

which solves the problem.


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Units for measuring time

Time in seconds is not used because there are

too many factors that can vary e.g. computer

speed

Time efficiency is based on counting the

number of times the algorithms basic

operation is executed on inputs of size n


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Order of growth

Assesses by what factor execution time increases

when the size of the input is increased.

E.g. if the input size (n) doubles and the algorithm

takes four times longer to execute, the order of

growth is quadratic. Therefore, the algorithm is

O(n)


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Big O notation

Allows us to talk about the complexity of the

algorithm irrespective of the machine that the

algorithm is implemented on

YouTube - explains and demonstrates the main

asymptotic bounds associated with measuring

algorithm performance

O(g) is called big O of g it represents the class

of functions that grow no faster than g

e.g. O(n)
http://www.youtube.com/watch?v=6Ol2JbwoJp0http://www.youtube.com/watch?v=6Ol2JbwoJp0


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Travelling salesman problem

Given a list of cities and their pair wise

distances, the task is to find the least possibledistance that visits each city exactly once

No general method of solution is known, and

the problem is non-deterministic polynomial-

time hard.

Travelling salesman problem wiki

TSP website including game
http://en.wikipedia.org/wiki/Travelling_salesman_problemhttp://en.wikipedia.org/wiki/Travelling_salesman_problemhttp://www.tsp.gatech.edu/index.htmlhttp://www.tsp.gatech.edu/index.htmlhttp://en.wikipedia.org/wiki/Travelling_salesman_problemhttp://en.wikipedia.org/wiki/Travelling_salesman_problem


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Sorting algorithms

Many different methods that can be used to sort an

unordered list

As the size of the list increases the efficiency of an

algorithm becomes critical

When choosing an algorithm consider both the

time efficiency and the space efficiency.

Animations about the different sorting algorithms

Detailed look at some sorting algorithms
http://www.sorting-algorithms.com/http://www.sorting-algorithms.com/http://localhost/var/www/apps/conversion/tmp/scratch_4/Case_study_Comparing_algorithms.dochttp://www.iti.fh-flensburg.de/lang/algorithmen/sortieren/algoen.htmhttp://localhost/var/www/apps/conversion/tmp/scratch_4/Case_study_Comparing_algorithms.dochttp://www.iti.fh-flensburg.de/lang/algorithmen/sortieren/algoen.htmhttp://localhost/var/www/apps/conversion/tmp/scratch_4/Case_study_Comparing_algorithms.dochttp://www.sorting-algorithms.com/http://www.sorting-algorithms.com/


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Bubble sort

Videographical demonstration and analysis of its timecomplexity.

It works by going through the list swapping items with the itemsnext to them if they are the wrong way round. It will need to gothrough the list multiple times to sort it.

If there aren

items in the list then the algorithm will maken

-1passes through the list at most; on each pass it will make n-1comparisonsgiving a maximum of (n-1)(n-1) comparisonswhich gives a time complexity of O(n2). This means that as thelist size grows bigger the Bubble Sort becomes increasinglyinefficient.

The space complexity of the Bubble Sort algorithm is O(1)itonly ever needs one additional memory location (to store one ofthe two values being swapped) no matter how many items are inthe list.

Find out more
http://www.school-portal.co.uk/GroupDownloadFile.asp?GroupId=1010236&ResourceID=4488273http://www.theteacher99.btinternet.co.uk/theteacher/newalevel/cp1_4_1.htmhttp://www.theteacher99.btinternet.co.uk/theteacher/newalevel/cp1_4_1.htmhttp://www.school-portal.co.uk/GroupDownloadFile.asp?GroupId=1010236&ResourceID=4488273


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Quick sort YouTube - explains and demonstrates graphically

There are many versions of the Quick Sort algorithm.

In fact, it is one of the most frequently used algorithms for

sorting.

The time complexity of Quick Sort is O(n2), though under most

circumstances it is O(nlog n).

In practice, Quick Sort is faster than other O(nlog n) sorting

algorithms.

The space complexity is O(log n), which is considered to be

good (though it is less space efficient than Bubble Sort).

Quick Sort works by picking an item in the list as a pivot and

placing all items in the list on the correct side of the pivot. For

each of the two sublists a pivot is chosen and the process

continues in this manner until the list is sorted.

Find out more
http://www.youtube.com/watch?v=y_G9BkAm6B8http://www.theteacher99.btinternet.co.uk/theteacher/newalevel/cp4_4_3.htmhttp://www.theteacher99.btinternet.co.uk/theteacher/newalevel/cp4_4_3.htmhttp://www.youtube.com/watch?v=y_G9BkAm6B8


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Merge sort

YouTube - explains and demonstrates graphically Merge Sort is a fast sorting algorithm. Time complexity is O(nlog n). Space complexity is O(n)which is not bad but

does make it one of the least space efficient ofthe sorting algorithms.

Merge Sort works by dividing a list in half, sortingthe two sublists and then merging them together.

This process is applied to each of the sublists, thesublists of the sublists, and so on. Find out more
http://www.youtube.com/watch?v=GCae1WNvnZMhttp://www.cse.iitk.ac.in/users/dsrkg/cs210/applets/sortingII/mergeSort/mergeSort.htmlhttp://www.cse.iitk.ac.in/users/dsrkg/cs210/applets/sortingII/mergeSort/mergeSort.htmlhttp://www.youtube.com/watch?v=GCae1WNvnZM


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Class work

1. Study the slides in this presentation.

2. Study the presentationon examples ofproblems with different time complexities.

3. Start the sorting algorithm animationsandwatch how each one operates.

4. Watch the bubble sort demo video including

analysis of its time complexity.5. Answer questions 1-11 in the A2 Computing

book pages 21-24.
http://www.school-portal.co.uk/GroupDownloadFile.asp?GroupId=1010236&ResourceID=4106083http://www.sorting-algorithms.com/http://www.school-portal.co.uk/GroupDownloadFile.asp?GroupId=1010236&ResourceID=4488273http://www.school-portal.co.uk/GroupDownloadFile.asp?GroupId=1010236&ResourceID=4488273http://www.sorting-algorithms.com/http://www.school-portal.co.uk/GroupDownloadFile.asp?GroupId=1010236&ResourceID=4106083


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Homework

Answer the past paper questions:

1. Specimen paper COMP 3 Q12. June 2010 COMP 3 Q5

Hand-in Friday 16thMarch 2012


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AQA, Specimen paper question

1 The big O notation is often used to describe the efficiency of an algorithm.

(a) Place the following algorithms in order of efficiency, the most efficient first.

Algorithm A that is O(n)

Algorithm B that is O(a)

Algorithm C that is O(n) (1 mark)

(b) Describe a linear search and explain why it is O(n).

(4 marks)

(c) Describe a bubble sort and explain why it is O(n).

(4 marks)

Documents

1.2 Comparing Algorithms2