Teach A level Compu/ng: Algorithms and Data …...Week 3: Sor*ng and Searching 14/03/2017 1 Teach A level Compu/ng: Algorithms and Data Structures Eliot Williams @MrEliotWilliams Course

Week3:Sor*ngandSearching 14/03/2017

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Teach A level Compu/ng: Algorithms and Data

Structures

EliotWilliams

@MrEliotWilliams

Course Outline

1 Representa+onsofdatastructures:Arrays,tuples,Stacks,Queues,Lists

2 RecursiveAlgorithms

3 SearchingandSor*ng-EWwillbelate!

4 HashingandDic*onaries,GraphsandTrees

5 Depthandbreadthfirstsearching;treetraversals


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Searching, sor/ng and algorithm analysis

Algorithm Analysis

• Analgorithmisaself-containedsequenceofac*onstobeperformed

• Wecananalysisanalgorithmsefficiencyusingthefollowingmeasures

•  The*meittakestofinish•  Howmuchmemoryitneeds


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Measuring /me

•  FirsTme.py

• Runtheprogramhowlongdothesimpleexpressionstake• Changethenumbersandopera*onsinthefirstthreestatements

• Uncommentthefurtherstatements(eachblockata*me)

Measuring /me

•  FirsTme.py

• Differentcomputerswillproducedifferent*mes•  Timetakentoexecuteacommandisnotequalonthesamecomputer•  Timetakentoexecuteasequenceofcommandsgrowswiththelengthofthesequence


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We don’t need absolute /me measures • Wewanttounderstandfundamentalproper*esofouralgorithms,notthingsspecifictoapar*cularinputormachine.

• Wethereforeonlycareabouthowthe*meincreases•  Maybethe*mestaysthesame•  Maybedoublingthesize,doublesthe*me•  Maybedoublingthesize,morethandoublesthe*me

AnyComputer

Problem(sizeN) Time:T1

Problem(size2*N) Time:T2

Big O Nota/on

• BigONota*onisaformalwayofsta*nghowthe*metakenbyanalgorithmrelatestoaninputsize

•  Trysecond*me.py•  andsecond*men.py

• O(1)inputsizedoesnoteffectthe*me• O(n)inputgrowslinearlywithinputsize


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Big O Nota/on

•  BigONota*onisaformalwayofsta*nghowthe*metakenbyanalgorithmrelatestoaninputsize–weonlyneedtoconsiderthelargestterm

•  O(1)inputsizedoesnoteffectthe*me•  O(logn)logarithmicgrowth•  O(n)*megrowslinearlywithinputsize•  O(n2)*meisdirectlypropor*onaltothesquareoftheinputsize•  O(ny)Exponen*algrowth•  O(2n)polynomialgrowth–intractableproblems….

O(n)

O(logn)

O(n2)

O(1)


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Search: The Problem •  Findatargetvalueinthelist•  Isitthere?•  Ifso,atwhatindex?

•  Target=41,foundatindex4•  Target=27,notfound

01234567891011

173152194134761128924461

Linear Search

•  Idea:lookateachentryinturn•  Steps•  Startatindex=0•  IsArray[Index]equaltothetarget?•  Ifyes,stop;otherwiseincrementtheindex•  Stopwhenindexisonelessthanthelength

0123456789

1731521941347611289234Target

Index

0


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Exercise 1.1 and 1.2

Linear Search – Algorithm • Algorithminpseudocode• ArrayisA

index = 0 !while index < length of array! if A[index] equals target! return index! index = index + 1!return -1 to show not found !

for i is 0 to length of array-1! if A[i] equals target! return i!return -1 to show not found !


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Binary Search Searchingasortedlist

Searching a Sorted List

• Ques*on:whyarebooksinthelibrarykeptinorder?


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Searching a Sorted List

• Ques*on:whyarebooksinthelibrarykeptinorder?

•  Inanorderedarray,wedonothavetolookateveryitem•  “Beforethisone”•  “Aperthisone”•  …quicklyfindthecorrectloca*on

• Whatisthebestalgorithmforlooking?

Binary Search – Sorted Lists • Whichhalfisitin?Lookinthemiddle.

0123456789

1117192831344152769241Target

Index

4

7

6


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Binary Search – Algorithm • Keyidea:inwhichpartofthearrayarewelooking?

lep right

Binary Search – 3 Cases

• Case1:Xequalstargetvalue• Case2:X<targetvalue

• Case3:X>targetvalue

lep rightmidX

rightlep

lepX

right


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Binary Search – Algorithm left = 0!right = length of array!while right > left:! mid = average of left and right! if A[mid] equals target! found it at 'mid'! if A[mid] < target! search between mid+1 & right! otherwise ! search between left & mid!return not found!

Binary Search – Python def BSearch(A, target):! left = 0! right = len(A)! while right > left:! mid = (left + right) // 2! if A[mid] == target:! return mid! elif A[mid] < target:! left = mid+1! else:! right = mid! return -1!


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Binary Search – Complexity

• Numberofsteps=numberofbinarydigittoindex• O(logN)

1117192831344152

000

001

010

011

100

101

110

111

Binary Search (Recursive) •  Searchasortedarray–isEinthearray?

• Basecase:•  emptyarrayorEfound

• Recursivecase:•  CompareEwiththemiddleelement•  IfEsmaller,searchthelephalf•  IfElarger,searchtherighthalf


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Exercise 1.3 • Completetheitera5veandrecursivebinarysearchproceduresinbinarysearch.py

Sor/ng


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Sor/ng: The Problem

• Arrangearrayinorder•  Sameentries;inorder–swapentries

• Proper*es•  Speed,space,stable,

0123456789

17315219413476112892

11171928313441527692

Discussion

•  Sortapackoftoptrumps• Describehowyoudoit


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Bubble Sort – Insight

•  Librarianfindstwobooksoutoforder•  Swapthemover!• Repeatedly

0123456789

17315219413476112892

Bubble Sort – Descrip/on

• Passthroughthearray(star*ngonthelep)•  Swapanyentriesthatareoutoforder• Repeatun*lnoswapsneeded

0123456789

17315219413476112892

Quiz:showarrayaperfirstpass


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Bubble Sort – Algorithm •  Sor*ngArrayA•  Assumeindices0tolength-1

while swaps happen! index = 1! while index < length! if A[index-1] > A[index]! swap A[index-1] and A[index]! index = index + 1!

Exercise 2.1 Bubble Sort

• Completethetabletoshowthesuccessivepassesofabubblesort


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Demo sor*ngDemo.py

Bubble Sort – Proper/es

•  Stable•  Inefficient• O(N2)•  Doublelength–*meincreases4-fold

hvp://www.sor*ng-algorithms.com/bubble-sort


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Inser/on Sort – Insight

•  Imaginepartofthearrayisordered•  Insertthenextitemintothecorrectplace

17315219413476112892

orderednotyetordered

17315219413411287692


Inser/on Sort – Descrip/on

•  Startwithoneentry–ordered•  Takeeachentryinturn•  Insertintoorderedpartbyswappingwithlowervalues•  Stopwhenallentriesinserted

17315219413476112892



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Inser/on Sort – Algorithm •  Sor*ngArrayA•  Assumeindices0tolength-1

index = 1 !while index < length of array! ix = index! while A[ix] < A[ix-1] and ix > 0! swap A[ix] and A[ix-1]! ix = ix – 1! index = index + 1!

•  A[0:index]ordered•  Samevalues

Innerloop:insertintoorderedlist

Exercises 3.x


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Quicksort – Insight

• Howcouldweshareoutsor*ngbetweentwopeople?•  ChooseavalueV•  Givefirstpersonallvalues<V•  Givesecondpersonallvalues>V

• Whenthereisonlyasingleentry–itissorted

Quicksort Example 173152194134761128

281719113134765241

all<31 all>=31

11171928

1928

41345276

3441


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Quicksort Descrip/on

• Chooseapivotvalue• Par**onthearray•  Valueslessthanthepivottothelep•  Thepivot•  Valuesgreaterthanthepivottotheright

• Repeatprocessoneachpar**on• …un*lpar**onhasnomorethanonevalue

• Workdoneinpar++on

Tasks

• Quickfirststep.py–goingthroughonlyonce•  quickS1

•  Separateintotwonewlistsandputbacktogetherwithpivot•  Checksbasicunderstanding


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Recursive Sor/ng • Concept•  Splitarrayintwohalves,sorteachhalf•  Combinetwosortedarrays•  Singleitemsorted(basecase)

•  Twoalgorithms• Mergesort•  Halvearray–mergetwosortedlists

• Quicksort•  Par**onarray–combineeasy

Merge Sort – Insight

• Howcouldweshareoutsor*ngbetweentwopeople?•  Halfitandsorteachhalf•  Mergethetwosortedlists

• Whenthereisonlyasingleentry–itissorted


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Merge Sort

1731521941347611

1731521941347611

1719315211344176

Decompose

1117193134415276

Merge

MAGICRecursion

Merging • Workdoneinthemerge

• Repeatedlyselectsmallest

17193152

11344176

171931 5211 344176


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Proper/es

•  Inser*onsort•  O(N2)–sameasbubblesort•  Stable

•  Quicksort

•  Moreefficient:O(NlogN)•  Notstable

• Mergesort•  O(NlogN)–sameasquicksortbutextraspace•  Stable

hvp://www.sor*ng-algorithms.com/inser*on-sort

hvp://www.sor*ng-algorithms.com/quick-sort

hvp://www.sor*ng-algorithms.com/merge-sort

Documents

Teach A level Compu/ng: Algorithms and Data …...Week 3: Sor*ng and Searching 14/03/2017 1 Teach A level Compu/ng: Algorithms and Data Structures Eliot Williams @MrEliotWilliams Course