Priority Queue ! Heaps

priorityQueue-ADTLPO.liStores a collection of pairs ( item , priority)

- Priorities are from some ordered set

( For simplicity , we use prioritiesfrom 0

, 1,2, . . . with 0"

highest priority" )

- Main operations :- insert ( item , priority)adds item with priority priority

. extract-

mindremoves 4.returns) item with least priority

- update ( item , priority)changes priority of item to priority

RootedBmaryTreeter- proof = every node has 0 or

2 children .

- perfect = proper BT. where everyt.nl has the same death .

• We want a data structure to implementefficient PQS

.

Eeg . 040g n) time for all operations .

. We (again) will use a particular kind of tree

. .

.

. . .ofordered-binarytrees.iris its each node of the tree once

• visits every node at depth i before anynode at depth it1 *

• visits every depth -d descendent of left G)before any depth

-d descendent of right G) .

F- - Isg. Is# III.• → → → sA1→AH#↳ t.to#.-shg-7-z

> → →

* in some texts,it is bottom - up ,

not top - down.

RootedBmaryTreeter- proof = every node has 0 or

2 children .

- perfect = proper BT. where every1on f. has the same death .

* has 2 ** in theComplete Binary Tree children level - orderF- traversal

A complete binary tree of height h is1

;a binary tree of height hi

2 with 2d nodes at depth d, for every 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.

. :*, F. 157,53F. IF, µ

✓

#bt.sk#osITNjho7Ibbh.JbbA µ sinkG) × 4) * ✓ ✓

Binary Heap Data Structure÷ "

- a complete binary tree]shape invariant

- with vertices labelled by keys fromsome ordered set ,ftp.iorities-st.keyldtkeiylpareutlvD-YIIIaut "for every node V

←

Eg . f- ①

③ ② / -⑥

{ to ⑤ '④ ④ I✓ ② to £1.0

⑨ ⑤ ×-This is the basic D.S. for implementing PQS(Binary min- cheap)

..

- How do we implement the operations so that

the invariants are maintained ?

- Consider Insertion:If we want to insert14 to the heap ,where should it go ?

-Notice: there no choice about how the shade changes :

T 1ha I✓ to do d. do To do1 AM. hobnob

HeapInTo insert an item with key K :1. add a new leaf v with keyG) = K , so as to

maintain the shape invariant

2. re - establish the order invariant byexecuting percolate-upG) .

percolate-upG) {while ( r is not root and keyLv ) a key

.

(parent lol) ) I

swap positions of v and parentG) in the tree

} ②} ¥ ←

48×

/ ③4 \€↳←8h10 }i.%# 2

Insert-2.thenI.thenzinto.io①

¥1 To ④ TO

④ ¥¥ ¥

I①,①I← ↳ ② 70s

A ④ ¥ ⇐ ⑤ D ' ⑧

④ ⑤④ ⑥To i

Heap Extract - Min .

-

Consider : ssmo%¥

⇒ §yjthese

heap

we must replace the root with the smaller of its children :

f-① 8%8 ?10 12

0k Not ok !

Heap Extract - Min .-

To remove the litem with the) smallest keyfrom the heap :

1. remove the root

2. replace the root with the"

last leaf ",

so as to maintain the shape invariant .

4. restore the order invariant by callingpercolate-down(root)

Percolate-down is more work than percolate-up ,

because it must look at both children

to see what to do ( and the childrenmay or may not exist)

Heapreml- remove the root- fix the shape invariant by moving

the

" bottom right"or" last " nude to the root .

X.tl #.

percolate -down 4){while ( r has a child c with keyG) a key as){

c ← child of r with the smallest keyamong the children

of r.

swap r and c in the tree

}I

Notice that :- V may have 0,1 , or 2 children

. if v has 2 children , we care about

the one with the smallest key .

Do extract -min 3 times:- AT

*

⇒i

④ ¥h

#④⑤ '④⑤

Doextraot-m

⇒ ⇒

.

I⑦'①

④

Complexity of Heap Insert ! Extract -min-Claim : Insert to Extract - min take time 0( log a)

for heaps of size n.

Recall: A perfect binary tree of height h has2h"-I nodes

.

II. By induction on h ( or"the structure of the tree.

B¥s : If h=o then we have It'- I = 1 nodes . V

III. : Consider some H70and assume the perfect

binarytree of height h has 2h"- I ii.des .

+ ,

II ! show the p .b.t .of height htt has 21h") - I nodes s .

The tree is:}A DI¥"

so it has 2h"- it 2h11 + I = 2. 2h11 = 21""I 1 modesty

sizeboundsoncompletebinarytrees.IEvery complete binary tree with height hand in nodes

satisfies : 2h ± in ⇐ zhtly

Smallest : Left:

I p.IT . of height h- I p.b.tv of height h .

# nodes = 21h-htt. It , = 2h

So,we have : 2h In

log.2h I loginh E loginh= Ollogn)

⇒ Heap insert 4 extract min take time 0( log a)-

hinkedImplemewtatimr.atµ,

D"last"

I 1¥X. *1€

last

N¥ " ¥t¥E

Array-BasedBinarytteapImplemeutatimllse.cothis embedding of a complete binary treeof site n in a size- n array :

⑥

←↳ ith node in

③/h④ ⑤ To level- order

④¥ ④ ④ ⑤ traversal

4 Itith array element

• Children of node i are nodes 2it I & 2 it 2

• Parent of node i is node KE14

1AX0*14*18%11www.

Array-BasedBinarytteapImplemeutatimllse.cothis embedding of a complete binary treeof site n in a size- n array :

⑥

←↳ ith node in

③/h④ ⑤ To level- order

④¥ ④ ④ ⑤ ⑤ traversal

4 Itith array element

• Children of node i are nodes 2it I ¢ 2 it 2

• Parent of node i is node kindly

I

was× Growing 4 Shrinking the tree is easy in the array embedding

partiaky-fttedArrayImpkmentationofBinarytteap.LI②¥←↳ =

'

④←

Insert④¥, ¥57

⑧%¥i%① 6 I ¥ 6

2

① 1-← to Id'

%

Insert for Array- based Heap-+

. Variables : ar# ,size

- Heap elements are in A fo) . .Afsite - I

insert4)I

÷÷÷÷:*:*:*:]:::±.while ( v > o and AG3 < Alps){Swap AG3 and A Cp]✓ ← p

p← 4-1) 12J

}

}site← site t t ;

Partially- fitted Array Implementation of Binary Heap. : Extract-min-I

← ¥ = ¥%HHXN'

9itV 11

⑥ l④ it # 6 It

←¥8¥ toa#⑥ € 5¥ "

①

0¥ ¥11 to

Extraot.minforArray-basedtte.ptextract - milktemp ← A Lo] A record value to returnsite ← size - I

AC03 ← A Lsieej Amove old last leaf to rooti← 0while ( ait t a site ) while i not a a leafchild← Zit 1 11 the left child of i

percolate if ( 2 it 2 a site AND AGit 23 L AL2 itD) {11 use the right child if it exists

down } *xieo! ÷÷::?.swap A Lehild] and Ali]

[ ← child

} else {

[ , 3return temp

return temp .}

A small space - for - time trade- off_inExtraot-m- Extract -min does many comparisons, eg this size) tocheck if i is a leaf .

- Suppose we ensure the array has site7 2. size

a¥ there is a big value, denoted a , that can bestored in the array but will never

be a key .and every array entry

that is not a key is a .

-Then,we can skip the explicit checks for

being a leaf .

Extract-mmrariantextract-m.NL i has a child

temp ← Alo] A record value to return that is out of

site ← size - Iorder

move # last leaf to

i← 0

while ( AL2it is a ACD ¥ Akita]< ALII){

peggwlatefg.tl?!II?nII.Isi:IIci,} it is the left chidSetseswap AL2 it is and Acis } it is the right childis 2it 2}

" {eturw temp}

makingatteapfromase.toSuppose you have n keys and want to make a heap with them .

.Clearly can be done in time 04 log n) , with n inserts .• Claim : the following alg . does it in time 0(a) .

make-heap ( T) SXT is a complete bit . with n keys .for ( i = tht -1 down to a){call percolate- down on node i

}3

Howdoesmake-heapwork.7.hn/2J-lis the last internal node

- the algorithm does a percolate- down at eachinternal node

, working bottom - up .

( percolate- down makes a tree into a heapif the only node violating the orderproperty is the root )

0•

I✓\]N4 5TG

15 16 " '8 "9 "

Lnfhj - I = Leith - I = I

;%D'D

tceheapExam, ④ .

- .-#④✓ 4 ④

2

¥1 ⑤ 708z 4

* ¥ #aD= 10 ; Ln I2J- I

= 4

Notice: The exact order of visiting nodes does notmatter - as long as we visit childrenbefore parents .

[It follows that it is easy to do a recursive make- 'heap)

Make-heaplomplexitf.CIearly Oln log n) : n percolate- down calls , each 040gn) .

. How can we see it is actually 0(n) ?Intuition : mark a distinct edge forererypossibleswo.plTime taken is bounded by max . # of swaps possible)

⑧

←I \

*

Timelomplexityoftlakeheapletsln) be the max number of swaps carriedout by make- heap on a set of size n.

We can bound Scn) by :

541 I 2dL - d)d-- o b [The max # of swaps forA / a call to percolate- down on

percolate - down ) t a node at depth dis called

,at most

,there are

is h-don each node at 2d nodeseach depth d at depth dfrom 0 to ht

.

she 2dL-d) aA-0 11

=I

= 21h - o) t2Yh- 1) t . . . t ['

( h -th-D) t 2h" (h- Chi))set i=h-d

,so D= h - i and while d ranges over

0,1 , - . . , h- I , i will range over he;o , h- I , . . . h- (in - 1)"h 1

Now

she iii.ili) = Etta.ie?IEiilbenYI)=nIIi.snhEoIit2n-( £o = It fit f. + It . . . = It It IttIIy )

Lomplexityotmakeheapblorkdone by make - heap is bounded by a

constant times the number of swaps

so is 0(hogn)T 0 (n) .

forrated)

Updatingprioritie.siSuppose a heap contains an item with priority K ,and we execute update-priority ( item , j ) .

- We replace K with j in the heap , and thenrestore the order invariant :

if ja k , do percolate-up from themodified node

if kaj , do percolate -down fromthe modified node .

X. 13>I µ^ ANAD ADMD

Th takes ollogn) time - bit howdowe-t.nlis the right node to change ??Restoring-

-To do this,we need an auxiliary data structure .

End=

Correctness of swapping in percolate down-

ob . Suppose we are percolating down cA.c the:c:D!: III.µ Toe . If a > e and Ed ,[ A D

we swap c. e

Now : . we know bt.es c

and be. es dob

a#e . so order is OK , except

µ dfhq.ge possibly below a - which[ A D we still have to look at .

Correctness of swapping in percolate - up-

" suppose we are percolating up a°b

we know CED ,ese , because weHad previously swapped c with dore

µ Doe - we know that bea

[ A D.if cab , we swap e.b

Now :. we now know that

•a cab se

{↳ and cc bed

µ dfh.ge and a < bsa

( D D ' so order is ok, except

T3 possibly with ancestors ofC

,which we still must check.