priorityQueue-ADTLPO.li Stores a collection of pairs ( item , - - PowerPoint PPT Presentation

Priority Queue ! Heaps

priorityQueue-ADTLPO.li

Stores a collection of pairs ( item , priority)

• Priorities

are from

some

• rdered set

( For simplicity , we

from

0, 1,2 ,

with

highest priority

• Main operations

• insert ( item , priority)

adds item with priority priority

• mind

removes 4.returns)

item with least priority

• update( item , priority)

changes priority of

item to priority

RootedBmaryTreeter

• proof

node has

• r 2 children .
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same death .

• We want

a data

structure to implement

efficient

PQS

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use

a particular

kind of tree

• fordered-binarytrees.ir

isits each node of the tree

• nce
• visits every

node at depth i before any node at

depth it1 *

• visits

every

depth

• d descendent of

left G)

before any depth

• d descendent of right G) .

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bottom - up

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• proof

node has

• r 2 children .
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• n f.

has the

same death .

Complete Binary Tree

children

level - order

F-

traversal

A complete binary tree of height

h

is

;

a binary tree of height hi 2 with 2d nodes at depth d

0 ← dah

• 3. level order traversal visits every internal node before any leaf

**

• 4. every internal

node is proper4 except perhaps the last

which may have just a left child .

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Binary Heap Data Structure

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shape invariant

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some

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• This is the basic D.S. for implementing

PQS

(Binary

• How do

• perations

the invariants are maintained ?

• Consider Insertion:

If

14

to the heap ,

where should it

go ?

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there

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maintain the shape invariant

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re - establish the order invariant by

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from the heap:

1.

remove the root

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so as to maintain the shape invariant .

4.

restore the order invariant by calling

percolate

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return temp return temp

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check if

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a big value,

denoted a

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Suppose you have

can be done in time 04 log n) , with

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does it

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for ( i

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call percolate

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last internal node

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heap

if

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nodes does not

matter

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we

visit children

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It follows that it is easy to do

heap)

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early

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n percolate

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each 040g n) .

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mark

a

distinct edge for

ererypossibleswo.pl

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bounded by

max

possible)

←

I

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sln)

be the max

number of swaps

carried

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by

make- heap

• n a

set

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size

n .

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/

a

call to percolate

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• n

percolate

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t

a node at depth d

is called

there are

h-d

• n each

node at

2d nodes

each depth d

at depth d

from

0 to ht

she

2dL

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a

A- 0

t [' ( h -th

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set

i=h-d

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done by

make - heap is

bounded by

a

constant times the

number of swaps

so

is 0(hogn)T

0 (n) .

forrated

)

Updatingprioritie.si

update

• priority( item , j ) .
• We replace

• rder invariant :

if ja k

percolate

• up from the

modified

if

kaj

do

percolate - down from

the modified

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µ

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ADMD

Th takes ollogn) time

• bit howdowe-t.nl

the right

node to change ??

Restoring -

• T
• do this, we

need an auxiliary data structure .

End

=

Correctness of

swapping

in

percolate down

• b

A.c

the:c:D!:

III.

µ T

• e

a > e

and

Ed ,

[

A

D

we

swap

• c. e

Now :

know

bt.es c

and

be

• b

a#e

is

OK , except

µ

dfhq.ge

possibly

below

a

• which

[

A

D

we still

have to

look at

Correctness

• f swapping in

percolate

• up
• " suppose

up

°b

we know CED , ese , because we

Had

previously swapped

µ

Doe

• we know that be a

[

A D. if cab ,

we

swap e.b

Now :

we now know that

• a

cab se

{↳

and

cc bed

µ

dfh.ge

and

a

(

D

D

• rder

is ok , except

T3

possibly

with ancestors of

C