CSC335-Chapter10

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CSC-335CSC-335

ADTs & Data StructuresADTs & Data Structures

Instructor:Instructor:

ChristosChristosKolonisKolonis

(Chapter 10 ADT Ipleentation:(Chapter 10 ADT Ipleentation:

!ecursion" Al#orith Anal$sis" an%!ecursion" Al#orith Anal$sis" an%Stan%ar% Al#orithsStan%ar% Al#oriths

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Chapter ContentsChapter Contents

10'1 !ecursion10'1 !ecursion10' )*aples o+ !ecursion: To,ers o+ anoi.10' )*aples o+ !ecursion: To,ers o+ anoi.

/arsin#/arsin#

10'3 Ipleentin# !ecursion10'3 Ipleentin# !ecursion

10' Al#orith )++icienc$10' Al#orith )++icienc$

10'5 Stan%ar% Al#oriths in C10'5 Stan%ar% Al#oriths in C

10'2 /roin# Al#oriths Correct (4ptional10'2 /roin# Al#oriths Correct (4ptional

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Chapter 46ectiesChapter 46ecties

!eie, recursion $ loo7in# at e*aples!eie, recursion $ loo7in# at e*aples Sho, ho, recursion is ipleente% usin# aSho, ho, recursion is ipleente% usin# a

run-tie stac7run-tie stac7

8oo7 at the iportant topic o+ al#orith8oo7 at the iportant topic o+ al#orith

e++icienc$ an% ho, it is easure%e++icienc$ an% ho, it is easure% Descrie soe o+ the po,er+ul an% use+ulDescrie soe o+ the po,er+ul an% use+ul

stan%ar% C +unction teplates in ST8stan%ar% C +unction teplates in ST8

(4ptional Intro%uce rie+l$ the topic o+(4ptional Intro%uce rie+l$ the topic o+al#orith eri+icational#orith eri+ication

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10'1 !ecursion10'1 !ecursion

A +unction is %e+ine% recursiel$ i+ it has theA +unction is %e+ine% recursiel$ i+ it has the+ollo,in# t,o parts+ollo,in# t,o parts

An anchor or ase caseAn anchor or ase case The +unction is %e+ine% +or one or ore speci+icThe +unction is %e+ine% +or one or ore speci+ic

alues o+ the paraeter(salues o+ the paraeter(s An in%uctie or recursie caseAn in%uctie or recursie case

The +unction9s alue +or current paraeter(s isThe +unction9s alue +or current paraeter(s is%e+ine% in ters o+ preiousl$ %e+ine% +unction%e+ine% in ters o+ preiousl$ %e+ine% +unction

alues an%or paraeter(salues an%or paraeter(s

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!ecursie )*aple!ecursie )*aple

Consi%er a recursie po,er +unctionConsi%er a recursie po,er +unctiondouble power (double x, unsigned n)double power (double x, unsigned n){ if ( n == 0 ){ if ( n == 0 )

return 1.0; return 1.0;

else else

return x * power (x, n-1); } return x * power (x, n-1); } ;hich is the anchor

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!ecursie )*aple!ecursie )*aple

=ote the=ote theresults o+results o+a calla call

!ecursie!ecursie

callscalls !esolution!esolution

o+ theo+ thecallscalls

Fig 10.2

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AA >a%>a%?se o+ !ecursion?se o+ !ecursion

@ionacci nuers@ionacci nuers1" 1" " 3" 5" " 13" 1" 31" 1" " 3" 5" " 13" 1" 3++

11B 1" +B 1" +

((B 1 +B 1 +

nnB +B +

n -(n -( + +

n -1n -1

A recursie +unctionA recursie +unctiondouble Fib (unsigned n)double Fib (unsigned n)

{ if (n

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AA >a%>a%?se o+ !ecursion?se o+ !ecursion

;h$ is this ine++icient

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?ses o+ !ecursion?ses o+ !ecursion

>inar$ Search>inar$ Search

SeeSee source co%esource co%e

=ote results o+=ote results o+recursie callrecursie call
http://codesampleschapter10.htm/http://codesampleschapter10.htm/http://codesampleschapter10.htm/

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?ses o+ !ecursion?ses o+ !ecursion

/alin%roe chec7er/alin%roe chec7er A palin%roe has sae alue ,ith characters reerse%A palin%roe has sae alue ,ith characters reerse%

1331 racecar1331 racecar

!ecursie al#orith +or an inte#er!ecursie al#orith +or an inte#er I+I+ nu#$igit%s

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!ecursion )*aple:!ecursion )*aple:To,ers o+ anoiTo,ers o+ anoi

!ecursie al#orith especiall$ appropriate +or!ecursie al#orith especiall$ appropriate +orsolution $ recursionsolution $ recursion

Tas7Tas7 oe %is7s +ro le+t pe# to ri#ht pe#oe %is7s +ro le+t pe# to ri#ht pe#

;hen %is7 oe%" ust e place% on a pe#;hen %is7 oe%" ust e place% on a pe#

4nl$ one %is7 (top %is7 on a pe# oe% at a tie4nl$ one %is7 (top %is7 on a pe# oe% at a tie 8ar#er %is7 a$ neer e place% on a saller %is78ar#er %is7 a$ neer e place% on a saller %is7

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I%enti+$ ase case:I%enti+$ ase case:I+ there is one %is7 oe +ro A to CI+ there is one %is7 oe +ro A to C

In%uctie solution +or n E 1 %is7sIn%uctie solution +or n E 1 %is7s

oe topost n 1 %is7s +ro A to >" usin# Coe topost n 1 %is7s +ro A to >" usin# C

+or teporar$ stora#e+or teporar$ stora#e oe +inal %is7 reainin# on A to Coe +inal %is7 reainin# on A to C

oe the n 1 %is7 +ro > to C usin# A +oroe the n 1 %is7 +ro > to C usin# A +orteporar$ stora#eteporar$ stora#e

Fie, co%e +or solution"Fie, co%e +or solution" @i# 10'@i# 10'
http://codesampleschapter10.htm/http://codesampleschapter10.htm/http://codesampleschapter10.htm/

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=ote the #raphical steps to the solution=ote the #raphical steps to the solution

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!ecursion )*aple: /arsin#!ecursion )*aple: /arsin#

)*aples so +ar are %irect recursion)*aples so +ar are %irect recursion @unction calls itsel+ %irectl$@unction calls itsel+ %irectl$

In%irect recursion occurs ,henIn%irect recursion occurs ,hen

A +unction calls other +unctionsA +unction calls other +unctions

Soe chain o+ +unction calls eentuall$ resultsSoe chain o+ +unction calls eentuall$ resultsin a call to ori#inal +unction a#ainin a call to ori#inal +unction a#ain

An e*aple o+ this is the prole o+An e*aple o+ this is the prole o+

processin# arithetic e*pressionsprocessin# arithetic e*pressions

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/arser is part o+ the copiler/arser is part o+ the copiler

Input to a copiler isInput to a copiler ischaracterscharacters >ro7en up into eanin#+ul #roups>ro7en up into eanin#+ul #roups

I%enti+iers" resere% ,or%s" constants" operatorsI%enti+iers" resere% ,or%s" constants" operators

These units are calle% to7ensThese units are calle% to7ens

!eco#niGe% $ le*ical anal$Ger!eco#niGe% $ le*ical anal$Ger S$nta* rules applie%S$nta* rules applie%

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/arser #enerates a/arser #enerates aparse tree usin# theparse tree usin# theto7ens accor%in# toto7ens accor%in# torules elo,:rules elo,:

An e*pression:An e*pression:ter ter H ter ter H terter ter H ter ter H ter

A ter:A ter:+actor +actor H +actor +actor H +actor+actor +actor H +actor +actor H +actor

A +actor:A +actor:( e*pression H letter H %i#it( e*pression H letter H %i#it

Note the indirect

recursion

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10'3 Ipleentin# !ecursion10'3 Ipleentin# !ecursion

!ecall section J'" actiation recor% create%!ecall section J'" actiation recor% create%+or +unction call+or +unction call

Actiation recor%s place% on run-tie stac7Actiation recor%s place% on run-tie stac7

!ecursie calls #enerate stac7 o+ siilar!ecursie calls #enerate stac7 o+ siilaractiation recor%sactiation recor%s

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Ipleentin# !ecursionIpleentin# !ecursion

;hen ase case reache% an% successie;hen ase case reache% an% successiecalls resole%calls resole%

Actiation recor%s are poppe% o++ the stac7Actiation recor%s are poppe% o++ the stac7

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10' Al#orith )++icienc$10' Al#orith )++icienc$

o, %o ,e easure e++icienc$o, %o ,e easure e++icienc$ Space utiliGation aount o+ eor$ reuire%Space utiliGation aount o+ eor$ reuire% Tie reuire% to accoplish the tas7Tie reuire% to accoplish the tas7

Tie e++icienc$ %epen%s on :Tie e++icienc$ %epen%s on :

siGe o+ inputsiGe o+ input

spee% o+ achinespee% o+ achine

ualit$ o+ source co%eualit$ o+ source co%e

ualit$ o+ copilerualit$ o+ copiler These vary fro one!"atfor to another

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Al#orith )++icienc$Al#orith )++icienc$

;e;e cancancount the nuer o+ tiescount the nuer o+ tiesinstructions are e*ecute%instructions are e*ecute%

This #ies us a easure o+ e++icienc$ o+ anThis #ies us a easure o+ e++icienc$ o+ anal#orithal#orith

So ,e easure coputin# tie as:So ,e easure coputin# tie as:

T(nT(n B coputin# tie o+ an al#orith +or input o+ siGe nB coputin# tie o+ an al#orith +or input o+ siGe n B nuer o+ ties the instructions are e*ecute% B nuer o+ ties the instructions are e*ecute%

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1

)*aple: Calculatin# the ean)*aple: Calculatin# the ean

Tas7Tas7 L ties e*ecute%L ties e*ecute%

1'1' InitialiGe theInitialiGe the sumsumto 0to 0 11

'' InitialiGe in%e*InitialiGe in%e* iito 0to 0 11

3'3' ;hile;hile iiMM nn%o +ollo,in#%o +ollo,in# n1n1

'' a A%% *NiO to sua A%% *NiO to su nn

5'5' Increent Increent ii$ 1$ 1 nn

2'2' !eturn!eturn mean = sum/nmean = sum/n 11

Total 3n Total 3n

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Coputin# Tie 4r%er o+ a#nitu%eCoputin# Tie 4r%er o+ a#nitu%e

As nuer o+ inputs increasesAs nuer o+ inputs increases T(n B 3n #ro,s at a rate proportional toT(n B 3n #ro,s at a rate proportional to nn

Thus T(n has the Por%er o+ a#nitu%ePThus T(n has the Por%er o+ a#nitu%eP nn

TheThe computing timecomputing timeo+ an al#oritho+ an al#orith on inputon inputof size n,of size n,

T(n)T(n)sai% to haesai% to hae order of magnitude f(n)order of magnitude f(n)""

,ritten,ritten T(n) is O(f(n))T(n) is O(f(n)) i+ i+ there is some constant C such thatthere is some constant C such that

T(nT(n MMCC+(n +or all su++icientl$ lar#e alues o+ n+(n +or all su++icientl$ lar#e alues o+ n

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>i# 4h =otation>i# 4h =otation

Another ,a$ o+ sa$in# this:Another ,a$ o+ sa$in# this:Q TheThe complexitycomplexityo+ the al#orith iso+ the al#orith is4(+(n'4(+(n'

Q )*aple: @or the ean-Calculation)*aple: @or the ean-Calculation

Al#orith:Al#orith:

T(n isT(n is O(n)O(n)

Q =ote that constants an% ultiplicatie=ote that constants an% ultiplicatie+actors are i#nore%'+actors are i#nore%'

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+(n is usuall$ siple:+(n is usuall$ siple: n" nn" n((" n" n33" '''" '''

nn1" lo#1" lo#

((nn

n lo#n lo#((nnlo#lo#

((lo#lo#

((nn

=ote #raph=ote #raph

o+ coono+ cooncoputin# tiescoputin# ties

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Rraphs o+ coon coputin# tiesRraphs o+ coon coputin# ties

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Coon Coputin# Tie @unctionsCoon Coputin# Tie @unctions

lo#lo#lo#lo#nn lo#lo#nn nn n lo#n lo#nn nn nn33 nn

------ 00 11 00 11 11

0'000'00 11

1'001'00

1212

22

1212

1'51'5 33 22 5151 5252

'00'00 1212 22 5252 0202 2553225532

'3'3 55 33 120120 1010 3J23J2 2J22J2

'5'5 22 22 33 0202 2121 1'2J)11'2J)1

3'003'00 5252 00 2553225532 12JJJ1212JJJ12 1'15J)JJ1'15J)JJ

3'33'3 1010 1010 100100 105J2105J2 1'0J)01'0J)0 1')301')30

'3'3 00 105J2105J2 0J1500J150 1'1)11'1)1 1'15)11'15)1 2'J)315252'J)31525

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Coputin# in !eal TieCoputin# in !eal Tie

Suppose each instruction can e %one in 1Suppose each instruction can e %one in 1icrosecon%icrosecon%

@or n B 52 inputs ho, lon# +or arious +(n@or n B 52 inputs ho, lon# +or arious +(n

@unction@unction TieTie

lo#lo#lo#lo#nn 3 icrosecon%s3 icrosecon%s

8o#8o#nn icrosecon%s icrosecon%s

nn '5 illisecon%s'5 illisecon%s

n lo#n lo#nn illisecon%s illisecon%s

nn 25 illisecon%s25 illisecon%s

nn33 1J secon%s1J secon%s

nn 3'J)23'J)2 centuries!!centuries!!

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10'5 ST89s Al#oriths10'5 ST89s Al#oriths

ST8 has a collection o+ ore than 0ST8 has a collection o+ ore than 0#eneric#eneric al#orithsal#oriths'' not eer +unctions o+ ST89s container classesnot eer +unctions o+ ST89s container classes

%o not access containers %irectl$'%o not access containers %irectl$'

The$ are stan%-alone +unctionsThe$ are stan%-alone +unctions operate on %ata $ eans o+ iterators 'operate on %ata $ eans o+ iterators '

a7es it possile to ,or7 ,ith re#ular C-a7es it possile to ,or7 ,ith re#ular C-st$le arra$s as ,ell as containers'st$le arra$s as ,ell as containers'

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)*aple o+)*aple o+ sortsortAl#orithAl#orith

Coes in seeral %i++erent +orsCoes in seeral %i++erent +ors

4ne uses the4ne uses the

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10'2 /roin# Al#oriths Correct10'2 /roin# Al#oriths Correct

De%uctie proo+ o+ correctness a$ eDe%uctie proo+ o+ correctness a$ ereuire%reuire%

In sa+et$-critical s$stes ,here lies at ris7In sa+et$-critical s$stes ,here lies at ris7

ust speci+$ust speci+$

The P#ienP or precon%itionsThe P#ienP or precon%itions

The Pto sho,P or post con%itionsThe Pto sho,P or post con%itions

Pre and AlgorithmPre and Algorithm =>=>PostPost

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)*aple: !ecursie /o,er @unction)*aple: !ecursie /o,er @unction

@unction:@unction:double power (double x, unsigned n)double power (double x, unsigned n){ if ( n == 0 ){ if ( n == 0 )

return 1.0; return 1.0;

else else

return x * power (x, n-1); } return x * power (x, n-1); }

/recon%ition:/recon%ition: Input consists o+ a real nuerInput consists o+ a real nuer xxan% a nonne#atiean% a nonne#atie

inte#erinte#er nn

/ostcon%ition:/ostcon%ition:

)*ecution o+ +unction terinates)*ecution o+ +unction terinates ;hen it terinates alue returne% is *;hen it terinates alue returne% is *nn

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)*aple: !ecursie /o,er @unction)*aple: !ecursie /o,er @unction

?se atheatical in%uction on n?se atheatical in%uction on n Sho, postcon%ition +ollo,s i+Sho, postcon%ition +ollo,s i+ n = 0n = 0

Assue +orAssue +or n = n = " e*ecution terinates an%" e*ecution terinates an%returns correct aluereturns correct alue

;hen calle% ,ith;hen calle% ,ith n = " 1n = " 1" in%uctie case" in%uctie case return x * power (x, n ! 1)return x * power (x, n ! 1)is e*ecute%is e*ecute%

Falue o+Falue o+ n ! 1n ! 1isis

It +ollo,s that ,ithIt +ollo,s that ,ith n = " 1,n = " 1,returnsreturns x * xx * x))

,hich euals,hich euals xx)"1)"1

CSC335-Chapter10

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