Algorithm Analysis 2

7/25/2019 Algorithm Analysis 2

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CSCE 3110Data Structures &Algorithm Analysis

Rada Mihalcea

http://www.cs.unt.edu/~rada/CSCE3110

Algorithm Analsis !!

Reading: Weiss, chap. 2
http://www.cs.unt.edu/~rada/CSCE31101http://www.cs.unt.edu/~rada/CSCE31101


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Last ime

Steps in pro"lem sol#ing

Algorithm analsis

Space comple$it

%ime comple$it

&seudo'code


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Algorithm Analysis

(ast time:E$perimental approach ) pro"lems

(ow le#el analsis ) count operations

A"stract e#en *urtherCharacteri+e an algorithm as a *unction o* the,pro"lem si+e-

E.g.!nput data arra pro"lem si+e is length o*arra

!nput data matri$ pro"lem si+e is $ M


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Asymptotic !otation

2oal: to simpli* analsis " getting rid o*unneeded in*ormation lie ,rounding-

14000400151400040006e want to sa in a *ormal wa 3n75 n7

%he ,8ig'9h- otation:

gi#en *unctionsf(n)and gn4 we sa thatf(n)is 9g(n) i* and onl i* there are positi#econstants cand n0such thatf(n) c g(n)*ornn0


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"raphic #llustration

*n 7n;*n is 9n

%he order o* *nis n

g(n) =n

c g(n) =4n

n

(n)= 2n +6


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$ore e%amples

6hat a"out *n >n7 ? !s it 9n?

@ind a c such that >n7 = cn *or an n n0

B0n3

; 70n ; > is 9n3

6ould "e correct to sa is 9n3;n

ot use*ul4 as n3e$ceeds " *ar n4 *or large #alues

6ould "e correct to sa is 9nB

9D4 "ut gn should "e as closed as possi"le to *n

3logn ; log log n 9 ?

Simple Rule: Drop lower order

terms and constant factors


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roperties o' (ig)*h

!* *n is 9gn then a*n is 9gn *or an a.

!* *n is 9gn and hn is 9gn then *n;hn is 9gn;gn

!* *n is 9gn and hn is 9gn then *nhn is 9gngn

!* *n is 9gn and gn is 9hn then *n is 9hn

!* *n is a polnomial o* degree d 4 then *n is 9n dn$ 9an4 *or an *i$ed $ 0 and a 1

An algorithm o* order n to a certain power is "etter than an algorithm o* order a 1 tothe power o* n

log n$is 9log n4 *o$ $ 0 ) how?

log $n is 9n *or $ 0 and 0An algorithm o* order log n to a certain power is "etter than an algorithm o* n raisedto a power .


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Asymptotic analysis )terminology

Special classes o* algorithms:logarithmic: O(log n)

linear: O(n)

quadratic: O(n2

)polynomial: O(nk), k 1

exponential: O(an), n > 1

&olnomial #s. e$ponential ?

(ogarithmic #s. polnomial ?


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Some !um+ers

log n n n log n n2

n3

2n

0 1 0 1 1 21 2 2 4 8 4

2 4 8 16 64 16

3 8 24 64 512 256

4 16 64 256 4096 65536

5 32 160 1024 32768 4294967296


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-elaties/ o' (ig)*h

,Relati#es- o* the 8ig'9h

*n: 8ig 9mega ) asmptotic loer"ound

*n: 8ig %heta) asmptotic tight"ound8ig'9mega ) thin o* it as the in#erse o* 9n

gn is *n i* *n is 9gn

8ig'%heta ) com"ine "oth 8ig'9h and 8ig'9mega

*n isgn i* *n is 9gn and gn is *n

Mae the di**erence:

3n;3 is 9n and is n

3n;3 is 9n7 "ut is not n7


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$ore relaties/

(ittle'oh) *n is ogn i* *or an c0 thereis n0 such that *n = cgn *or n n0.

(ittle'omega

(ittle'theta

7n;3 is on7

7n ; 3 is on ?


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E%ample

Remem"er the algorithm *or computing pre*i$ a#erages

' compute an arra A starting with an arra F

' e#er element AGiH is the a#erage o* all elements FGIH with I = i

Remem"er some pseudo'code J Solution 1

Algorithmpre'i%Aerages14Input4 An n)element array o' num+ers.

Output4 An n )element array A o' num+ers such that A5i6 is the aerageo' elements 506, ... , 5i6.

Let A +e an array o' nnum+ers.

fori0 ton) 1 do

a 0

for7 0 toido

a a 8 5j6

A5i6 a9i8 1

returnarray A

Analyze this


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E%ample cont:;

Algorithmpre'i%Aerages24

#nput4 An n)element array o' num+ers.

*utput4 An n )element array A o' num+ers such that

A5i6 is the aerage o' elements 506, ... , 5i6.Let A +e an array o' nnum+ers.

s0

fori 0 tondo

s s 8 5i6

A5i6 s9i8 1

returnarray A


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(ac< to the original =uestion

6hich solution would ou choose?9n7 #s. 9n

Some math J

properties o* logarithms:log"$ log"$ ; log"

log"$/ log"$ ' log"

log"$a alog"$

log"a log$a/log$"

properties o* e$ponentials:a";c a"a ca"c a"c

a"/ac a"'c

" a loga"

"c a cKloga"


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#mportant Series

Sum o* sLuares:

Sum o* e$ponents:

2eometric series:

Special case when A 7 70; 71; 77; J ; 7 7;1' 1

Nlargefor36

)12)(1( 3

1

2 NNNNi

N

i

++

=

=

-1kandNlargefor|1|

1

1

+

+

=

k

Ni

kN

i

k

1

11

0

=

+

=

A

AA

NN

i

i

=

+==+++=N

i

NNiNNS1

2/)1(21)(


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Analy>ing recursie algorithms

*unction *oo param A4 param 8

statement 1N

statement 7N

i* termination condition returnN

*ooA4 8N

O


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Soling recursie e=uations +yrepeate; su+stitution

T(n) =T(n/2)+ c substitute for T(n/=T(n/4) + c+ c substitute for T(n/4)= T(n/8) + c + c + c

= T(n/23) + 3c in more compact form= = T(n/2k) + kc inductive leap

T(n) = T(n/2logn

) + clogn choose k = logn=T(n/n) + clogn=T(!) + clo"n = b + clo"n = #(lo"n)


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Soling recursie e=uations +ytelescoping

%n %n/7 ; c initial eLuation%n/7 %n/> ; c so this holds

%n/> %n/P ; c and this J

%n/P %n/1 %7 ; c e#entuall J

%7 %1 ; c and this J

%n %1 ; clogn sum eLuations4 canceling theterms appearing on "oth sides

%n Qlogn


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ro+lem

Running time *or *inding a num"er in a sortedarra

G"inar searchH

&seudo'code

Running time analsis


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AD

A% A"stract ata %pes

A logical #iew o* the data o"Iects togetherwith speci*ications o* the operations reLuired

to create and manipulate them.

escri"e an algorithm ) pseudo'code

escri"e a data structure ) A%


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What is a data type?

A set of objets! eah alled an instane of the data ty"e#

$o%e objets are s&ffiiently i%"ortant to be "ro'ided

ith a s"eial na%e#

A set of o"erations# "erations an be realized 'iao"erators! f&ntions! "roed&res! %ethods! and s"eial

synta* (de"ending on the i%"le%enting lang&age)

+ah objet %&st ha'e so%e re"resentation (not

neessarily knon to the &ser of the data ty"e)

+ah o"eration %&st ha'e so%e i%"le%entation (also not

neessarily knon to the &ser of the data ty"e)


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What is a representation?

A s"eifi enoding of an instane

,his enoding .$, be knon to i%"le%entors

of the data ty"e b&t N++ N, be knon to

&sers of the data ty"e

,er%inology "we implement data types using

data structures


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Two varieties of data types

"a&e data ty"esin hih the re"resentation isnot knon to the &ser#

,rans"arent data ty"esin hih the re"resentationis "rofitably knon to the &ser- i#e# the enodingis diretly aessible and/or %odifiable by the&ser#

6hich one ou thin is "etter?6hat are the means pro#ided " C;; *orcreating opaLue data tpes?


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Why are opaque data types etter?

e"resentation an be hanged itho&t affeting&ser

ores the "rogra% designer to onsider the

o"erations %ore aref&lly+na"s&lates the o"erations

Allos less restriti'e designs hih are easier toe*tend and %odify

esign alays done ith the e*"etation that thedata ty"e ill be "laed in a library of ty"esa'ailable to all#


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!ow to design a data type

Step # Specification

ake a list of the o"erations (j&st their na%es)

yo& think yo& ill need# e'ie and refine the

list#

eide on any onstants hih %ay be re&ired#

esribe the "ara%eters of the o"erations in detail#

esribe the se%antis of the o"erations (hatthey do) as "reisely as "ossible#


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Step $# Application

e'elo" a real or i%aginary a""liation to test the

s"eifiation#

issing or ino%"lete o"erations are fo&nd as a

side-effet of trying to &se the s"eifiation#


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Step %# &mplementation

eide on a s&itable re"resentation#

4%"le%ent the o"erations#

,est! deb&g! and re'ise#


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E%ample ) AD #nteger

ame o* A% !nteger

9peration escription C/C;;

Create e*ines an identi*ier with an

unde*ined #alue int id1N

Assign Assigns the #alue o* one integer id1 id7N

identi*ier or #alue to another integer

identi*ier

isELual Returns true i* the #alues associated id1 id7Nwith two integer identi*iers are the

same


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E%ample ? AD #nteger

(ess%han Returns true i* an identi*ier integer is

less than the #alue o* the second id1=id7

integer identi*ier

egati#e Returns the negati#e o* the integer #alue 'id1

Sum Returns the sum o* two integer #alues id1;id7

9peration Signatures

Create: identi*ier !nteger

Assign: !nteger !denti*ier

!sELual: !nteger4!nteger 8oolean(ess%han: !nteger4!nteger 8oolean

egati#e: !nteger !nteger

Sum: !nteger4!nteger !nteger


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$ore e%amples

6ell see more e$amples throughout thecourse

Stac

ueue%ree

And more

Documents

Algorithm Analysis 2