[PPT] - W orst Case Ecien t Data Structures Gerth Stlting Bro dal PowerPoint Presentation

SLIDE 1 W

rst

Case Ecien t Data Structures Gerth Stlting Bro dal BRICS Departmen t

f

Computer Science Univ ersit y

f

Aarh us and MaxPlanc kInstitut f

ur

Informatik Saarbr

uc

k en

SLIDE 2 Ov erview

Priorit

y queues

Comparison

based data structures

Lo

w er b

unds

for comparison based data structures

RAM

data structures

P

arallel data structures

P

artial p ersisten t data structures G S Bro dal W

rst

case ecien t data structures

SLIDE 3 Priorit y Queues Main tain a set

f

n elemen ts from a totally

rdered

univ erse sa y in tegers under the

p

erations

FindMinQ
Inser

tQ e

DeleteMinQ
DeleteQ

e

MeldQ
Q
DecreaseKeyQ

e e

Assume

the lo cation

f

e is kno wn G S Bro dal W

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case ecien t data structures

SLIDE 4 Priorit y Queues Man y priorit y queues are based

n

heap

rdered

trees ie eac h no de stores an elemen t and the elemen t stored at a no de is

the

elemen t stored at the no des parren t

The

priorit y queues heaps

f

Willi ams are based

n
ne

heap

rdered

binary tree

Inser

t and DeleteMin can b e p er formed in O log n time Willi ams

G

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case ecien t data structures

SLIDE 5 Linking Heap Ordered T rees More recen t data structures are based

n

linking heap

rdered

trees

f

the same size

Ex

Binomial Queues are based

n

O log n heap

rdered

trees

Inser

t Theorem Binomial queues supp

rt

Inser t and Meld in amortized constan t time and DeleteMin in amortized O log n time V uillemin

G

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case ecien t data structures

SLIDE 6 Constan t Time Meldable Priorit y Queues Theorem Inser t and Meld can b e supp

rted

in w

rst

case constan t time and DeleteMin in w

rst

case O log n time

One

heap

rdered

tree

One

linking p er Inser t and Meld

r
sons
f

eac h rank less than the paren ts rank

ne

arbitrary rank ed son

The

ro

t

has rank

1

1 1 1 2 1 1 1 2 3

Bro

dal

G

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case ecien t data structures

SLIDE 7 Constan t Time DecreaseKey Theorem Fib

nacci

heaps supp

rt

DecreaseKey in amortized constan t time and DeleteMin in amortized O log n time F redman T arjan

Theorem

Relaxed heaps supp

rt

DecreaseKey in w

rst

case constan t time and DeleteMin in w

rst

case O log n time Meld requires log n time Driscoll Gab

w

Shrairman T arjan

Theorem

DecreaseKey and Meld can b e supp

rted

in w

rst

case constan t time and DeleteMin in w

rst

case O log n time Bro dal

O
heap
rdered

trees

Relaxed

heaps

A

n um b er

f

in v arian ts to solv e the tec hnical dep endencies

G

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case ecien t data structures

SLIDE 8 Comparison Based Priorit y Queues Driscoll Gab

w

F redman Shrairman Williams V uillemin Bro dal T arjan T arjan Bro dal

Heaps

Binomial Fib

nacci

Relaxed Queues Heaps Heaps FindMin O O O O O O Inser t Olog n O O O O O Meld On O O O Olog n O DeleteMin Olog n Olog n Olog n Olog n Olog n Olog n DecreaseKey Olog n Olog n Olog n O O O Amortized b

unds

G S Bro dal W

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case ecien t data structures

SLIDE 9 Comparison Based Priorit y Queues Lo w er Bounds Theorem Comparison based sorting requires n log n comparisons

Inser

t

r

DeleteMin require log n comparisons Theorem If Inser t and Delete mak e O t comparisons then FindMin requires n

O

t comparisons

a

doubly link ed list is an

ptimal

priorit y queue implemen tation Theorem If Meld mak es

n

comparisons and FindMin O n

comparisons
then

Delete and DeleteMin require log n comparisons Bro dal Chaudh uri Radhakrishnan

G

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case ecien t data structures

SLIDE 10 RAM Priorit y Queues Mo del A unit cost Random Access Mac hine with w

rd

size w

W
rd
p

erations

shifting

bitwise b

lean
p

erations Elemen ts In tegers in the range

w
Op

erations

FindMinQ
Inser

tQ e

DeleteQ

e

PredQ

e Pred f

g
Theorem

The ab

v

e

p

erations can b e p erformed in O log w

time

v an Emde Boas

an

O log log n priorit y queue for w

log

O

n

G S Bro dal W

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case ecien t data structures

SLIDE 11 RAM Priorit y Queues v an Emde Boas Thorup Andersson Bro dal

FindMin

O

O
O
O
O
Inser

t O log w

O

log log n O

p

log n O f n O log log n Delete O log w

O

log log n O

p

log n O f n O log log n Pred O log w

O
p

log n O

log

n f n

O
log

n log log n

log

log n

f

n

p

log n Amortized b

unds

G S Bro dal W

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case ecien t data structures

SLIDE 12 Outline

f

RAM Priorit y Queue

van Emde Boas

f

n elemen ts t in to a w

rd
log

n f n lev els f n lev els P ac k ed searc h trees

f

degree

f

n with buers

f

dela y ed Inser t and Delete

p

erations supp

rting

Inser t and Delete in w

rst

case O f n time

Tw
lev

el data structure v an Emde Boas and pac k ed

P

ac k ed searc h trees Andersson

Buer

trees for external memory

Arge
List

merging in O

w
rds

Alb ers Hagerup

Standard

deamortization tec hniques Bro dal

G

S Bro dal W

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case ecien t data structures

SLIDE 13 Adopting P arallelism to Priorit y Queues Question Is it p

ssible

to

btain

comparison based priorit y queues supp

rting
p

erations in

log

n time b y using a nonconstan t n um b er

f

pro cessors

Answ

er Y es O n pro cessors can supp

rt

Inser t and DeleteMin in constan t time

P
P
P
P
P
P
F
lklore

G S Bro dal W

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case ecien t data structures

SLIDE 14 Outline

f

P arallel Priorit y Queue

P
P
P
P
Pro

cessor P i main tains

r
trees
f

size

i
P

arallel linking and unlinking

f

trees Theorem O log n pro cessors can supp

rt

Inser t Meld and DeleteMin in constan t time An extension

f

the data structure supp

rts

Delete and DecreaseKey in constan t time to

Bro

dal

G

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case ecien t data structures

SLIDE 15 P arallel Priorit y Queues Pinotti Pinotti Ranade F

lklore

Pucci Das Crupi et al Bro dal

FindMin

O

O
O

log log n O

O
Inser

t O

O

log log n O log log n O

O
DeleteMin

O

O

log log n O log log n O

O
Meld

O log log n O

Delete

O log log n O

DecreaseKey

O log log n O

pro

cessors n log n log log n log n log log n log n log n Mo del Arra y EREW PRAM CREW PRAM Arra y CREW PRAM

r

Arra y G S Bro dal W

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case ecien t data structures

SLIDE 16 P arallel DecreaseKey

p

erations Essen tial to algorithms lik e Dijkstras algorithm for the singlesource shortest path problem are the

p

erations

DeleteMinQ
DecreaseKeyQ

e

e
e

k

e
k
Theorem

There exists an EREW PRAM data structure supp

rting

the ab

v

e

p

erations in constan t time pro vided e

e
e
k
Bro

dal T r a Zaroliagis

Previous

parallel data structures

nly

supp

rted

parallel Inser t and DeleteMin

p

erations G S Bro dal W

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case ecien t data structures

SLIDE 17 The SingleSource Shortest P ath Problem Theorem The singlesource shortest path problem can b e solv ed b y Dijkstras algorithm in sequen tial O n log n

m

time b y using Fib

nacci

heaps F redman T arjan

Driscoll

Han Gab

w

Bro dal P an Shrairman P aige T r a Reif T arjan Krusk al Zaroliagis

Time

O log

n

O n log n O n log log n O n O n W

rk

O n

log

n O n log n

m

O n

O

n

O

m log n Mo del EREW EREW CR CW CR CW CREW G S Bro dal W

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case ecien t data structures

SLIDE 18 P artial P ersistence A data structure is said to b e partial p ersisten t if

ld

v ersions are remem b ered and can b e accessed

nly

the latest v ersion can b e mo died Some naiv e approac hes

Store

a cop y

f

eac h v ersion

f

the data structure

space

and time

v

erhead p er up date

p

eration is O n

Only

store the sequence

f

c hanges done to the data structure

up

date steps in constan t time and space but accesses require O n time G S Bro dal W

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case ecien t data structures

SLIDE 19 P artial P ersitence T ec hniques Driscoll Sarnak Sleator Dietz T arjan Bro dal Raman

Dietz
Data

structures p

in

ter based p

in

ter based p

in

ter based arra ys Indegree O

O
log

O

n
Access

steps O

O
O
O

log log n Up date steps O

O
O
O

log log n amortized

tec

hnique requires the RAM

exp

ected amortized G S Bro dal W

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case ecien t data structures

SLIDE 20 Summary Comparison based priorit y queues W ADS SOD A NJC

Meld

can b e supp

rted

in w

rst

case constan t time

Meld

and DecreaseKey can b e supp

rted

in w

rst

case constan t time

A

lo w er b

und

tradeo b et w een up dating a priorit y queue and the query time RAM priorit y queues ST A CS

Delete

can b e supp

rted

in w

rst

case O log log n time

Pred

can b e supp

rted

in

log

n time while ha ving O log log n up date time P arallel priorit y queues SW A T IPPS

O

log n pro cessors can supp

rt

all

p

erations in constan t time

P

arallel DecreaseKey in constan t time P artial p ersisten t data structures NJC

Bounded

indegree data structures can b e made partially p ersisten t in w

rst

case constan t time G S Bro dal W

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case ecien t data structures

SLIDE 21 Appro ximate Dictionary Queries Dist H u v

is

the Hamming distance b et w een t w

binary

strings u and v

f

equal length u

v
Dist

H u v

Dictionary

W

fw
w
w

n g w i

f

g m for i

n
Query

string

f

g m

Question

Is there an y w i

W

suc h that Dist H

w

i

d
d
Y

ao Y ao Bro dal G

asieniec
Query

time O m log log n O m Space O nm log m O nm Prepro cessing time O nm log m O nm G S Bro dal W

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case ecien t data structures