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90-723: Data Structuresand Algorithms for

Information ProcessingCopright ! "999# Carnegie $ellon% All &ights &eser'ed%

(ecture "0: Searching

Data Structures and

Algorithms for InformationProcessing

(ecture "0: Searching II

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)utline

* )ne more )+A scheme , )rdered

ashing ./ough Schoolo

prolem1

* Analsis of hashing algorithms

* Some practical considerations

* &adi searching

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)pen 's% Chained ashing

* o ig should the tale e4* )pen addressing can e incon'enient hen

the numer of insertions and deletions isunpredictale - o'erflo%

* Simple solution to o'erflo: &esi5e .doule1tale# rehashing e'erthing into the ne tale

* 6se nuth8s approach and doule hashing to

a'oid clustering%

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ariant: )rdered ashing

* In linear proing# e stop search hen

e find an empt cell or a record ith a

e e;ual to the search e

* In ordered hashing e stop hen e

find a e less than or e;ual to the

search e .tough schoolo hashing1

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/ough Schoolo hashing

* "3 chairs in the classroom

*

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Class in the morning

* Inserts

Don prefers 3 =umps 2

>ill prefers ? =umps @

Al prefers 3 =umps

Boe prefers 3 =umps @

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0 " 2 3 D)

@ ? 7

9 "0 ""

"2

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0 " 2 3 D)

@ ? >I(( 7

9 "0 ""

"2

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0 " 2 3 D)

Al can8t sit

hereEE

@ ? >I(( 7

9 "0 ""

"2

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0 " 2 3 D)

@ ? >I(( 7

9 Al "0 ""

"2

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0 " 2 3 D)

Boe ics

out Don

@ ? >I(( 7

9 Al "0 ""

"2

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0 " 2 3 Boe

@ ? Donicsout >ill

7

9 Al "0 ""

"2

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90-723: Data Structures

and Algorithms forInformation Processing

Copright ! "999# Carnegie $ellon% All &ights &eser'ed%


0 " 2 3 Boe

@ ? Don 7

9 Al and>ill argueand Al gets

iced out

"0 ""

"2

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0 " 2 A( 3 Boe

@ ? Don 7

9 >ill "0 ""

"2

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Searching the classroom

* Search for Don# >ill# Al# and Boe

* Search for en ho prefers 3 and =umps"

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ariant: )rdered ashing

* /his reduces the time of unsuccessful

search to aout the same as successful

search

* 6seful for applications here e epect

to ha'e a large numer of unsuccessful

searches

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/ime Analsis

* )pen address hashing methods store

records in a tale of si5e $% $ G

* /he performance of the operations

depends on the load factor

alpha H +$

* or chained hashing# alpha ma e

greater than "%

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(inear Proing

* )pen address hashing ith linear proingre;uires# on a'erage:

"+2 . " J "+."-alpha1K21 operations for

an unsuccessful search"+2 . " J "+."-alpha11 operations for a

successful search

*

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Doule ashing

* )pen address hashing ith doulehashing re;uires# on a'erage:

"+."-alpha1 operations for an

unsuccessful search-log."-alpha1+alpha operations for a

successful search

*

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Chained ashing

* Chained hashing re;uires# on a'erage:

"Jalpha operations for an unsuccessful

search

"Jalpha+2 operations for a successful

search

*

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/ime Analsis

* /hese formulas re;uire significantmathematical analsis# hich e on8tgo into%

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alpha (inear

proing)penaddressing

Chainedhashing

0%? "%?0 "%39 "%2?

0%A "%7? "%?3 "%30

0%7 2%"7 "%72 "%3?

0%D 3%00 2%0" "%@0

0%9 ?%?0 2%?A "%@?

"%0 F F "%?0

A'erage umer of ProesSuccessful Search

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&adi Searching

* or man applications# es can ethought of as numers

* Searching methods that tae ad'antage

of digital properties of these es arecalled radix searches

* &adi searches treat es as numers inase $ .the radix 1 and or ithindi'idual digits

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&adi Searching

* Pro'ide reasonale orst-caseperformance ithout complication ofalanced trees%

* Pro'ide a to handle 'ariale lengthes%

* >iased data can lead to degenerate datastructures ith ad performance%

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/he Simplest &adi Search

* Digital Search Trees F lie >S/s ut

ranch according to the e8s its%

* e comparison replaced function

that accesses the e8s net it%

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Digital Search

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Digital Search

* &e;uires ).log 1 comparisons ona'erage

* &e;uires comparisons in the orst

case for a tree uilt ith random -ites

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Digital Search

* Prolem: At each node e mae a fulle comparison F this ma eepensi'e# e%g% 'er long es

* Solution: store es onl at the lea'es#use radi epansion to do intermediatee comparisons

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&adi /ries

* 6sed for &etrie'al LsicM

* Internal nodes used for ranching#eternal nodes used for final e

comparison# and to store data

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&adi /rie

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&adi /ries

* (eft sutree has all es hich ha'e 0

for the leading it# right sutree has all

es hich ha'e " for the leading it* An insert or search re;uires ).log 1 it

comparisons in the a'erage case# and

it comparisons in the orst case

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&adi /ries

* Prolem: lots of etra nodes for es that

differ onl in lo order its .See & and S

nodes in eample ao'e1

* /his is addressed Patricia trees# hich allo NlooaheadO to the net rele'ant it

* Practical Algorithm /o &etrie'e Information

Coded In Alphanumeric .Patricia)

* In the slides that follo the entire alphaet

ould e included in the indees

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++ Insert ord .see Dro5de and Simon , needs or1

iH0

pHroot

Qhile not inserted if .LiM HH R081 set end-of-ord marer in p to true

else if .p%ptrsLLiMM HH null1 create leaf containing and put its

address in p%ptrsLLiMM

else if .refernce p%ptrsLLiMM refers to a leaf1

T( H e in leaf p%ptrsLLiMM

do create a non-leaf and put its address in p%ptrsLLiMM

p H the ne non-leaf

iJJ

hile .LiM HH T(LiM1 create a leaf containing and put its address in p%ptrsLL--iMM

if .end-of-ord reached1 set end-of-ord marer in p to true

else create leaf containing T( and put address in p%ptrsLT(LiMM

else p H p%ptrsLLiJJMM

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Insert NA&AO

U A < I P &

A&A

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U A < I P &

A&A

U A < I P &

PU A < I P &

P

Insert NA&

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U A < I P &

U A < I P &

U A < I P &

A&AA&

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A (

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A (

ADA$

)

V

V

I

&

<

D A

()VV

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d Al ith f ( t "0 S hi

A (0 @

I<? ?

D0

&0

A0

0

ADA$

()VV

Documents

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