Binomial Heaps

p2.

Binomial trees:B0 has a single node

Bk:

Ex

B1 B2 B3

i

Bk-1

Bk-1

p3.

Lemma1For the binomial tree Bk,

1. there are 2k nodes,2. the height if the tree is k,3. there are exactly nodes at depth i for i= 0, 1, …., k, 4. the root has degree k, which is greater than that of

any other node; moreover if the children of the root are numbered from left to right by k-1, k-2, …, 0, child I is the root of a subtree Bi

Proof:1) By induction, 2k-1 + 2k-1 = 2k

2) By induction, 1 + (k-1) = k3) , by induction

ki

ki

1k1-i

1ki

p4.

Corollary:The maximum degree of any node in an n-node binomial tree is lg(n)

Binomial heaps:H: a set of binomial trees satisfying the following:

1. Each binomial tree in H is heap-ordered:the key of a node is greater than or equal to the key of its parent

2. There is at most one binomial tree in H whose root has a given degree

Note: By 2. an n-node binomial heap H consists of at most binomial trees

1n lg

p5.

Representation of binomial heaps

(a) head[H] 10 1

1711

29148

18

12 25

38

27

6

p6.

/100/

180/ /

121

25

/ /

/12

/63/

270/ /

111

170/

380/ /

82

141

290/ /

(b) head[H]

pkey

degree

child

sibling

p7.

Operations on binomial heaps Creating a new binomial heap

Finding the minimum key Uniting 2 binomial heaps

time:θ(1) NIL,Hhead

}y return

} }

x y key[x] min

{ then min x[k] if

{ do NIL x while

min head[H] x NIL y

{Minimum(H)HeapBinomial

Time: O( lg n)

p8.

Binomial-Heap-Merge

}1 degree[z] degree[z]

child[z] child[z] sibling[y]

z p[y] {

z) Link(y,Binomial

y

Bk-1

Bk-1

Y, z : Bk-1 trees

p9.

12 7 15

28 33

41

25 37

3 6

4410298

1731482223

32 2445

55

18

30

50

(a) head[H1] head[H2]

37

3 6

4410298

1731482223

32 2445

55

18

30

50

(b) head[H] 12 7 15

28 33

41

25

x next-xsorted degree

Output of

Binomial-Heap-Merge

Case 3

p10.

37

3 6

4410298

1731482223

32 2445

55

18

30

50

(c) head[H] 12 7 15

28 33

41

25

x next-x

Case 2

sibling[next-x]

37

3 6

4410298

1731482223

32 2445

55

18

30

50

(d) head[H] 12 7 15

28 33

41

25

x next-x

Case 4

prev-x

p11.

37

3 6

4410298

1731482223

32 2445

55

18

30

50

(d) head[H] 12

7

15

28 33

4125

x next-x

Case 3

prev-x

37

3 6

4410298

1731482223

32 2445

55

18

30

50

(d) head[H] 12

715

28 33

41

25

x next-x

Case 1

prev-x

p12.

.......

{

then ) ]degree[ ]]ling[degree[sib and NIL ]sibling[ (

or ) ]degree[ ]degree[ ( if {

)do NIL ( while

sibling[H] head[H]

NIL H return then NILhead[H] if

to pointthey liststhe not but H and Hfree )H ,Merge(H-Heap-Binomial head[H]

) Heap(-Binomial-Make H {

)H ,Union(HHeapBinomial

21

21

21

next-xxxprev-x

xnext-xnext-x

next-xx

next-x

next-xxprev-x

Case1

p13.

}y return

} ]sibling[

}

) , Link(-Binomial ]sibling[

else head[H]

then ) NIL ( if {else }

) , Link(-Binomial ]sibling[ ]sibling[

{ then ) ]key[ ]key[ ( ifelse }

xnext-x

next-xxnext-xx

next-xprev-x

next-xprev-x

xnext-xnext-xx

next-xx

......

Case 3

Case 4

p14.

….a b c dprev-x x

next-xsibling[next-x]

Bk Bl

….….(a)

Case 1

a b c d

prev-x x next-x

Bk Bl

….

a b c dprev-x x

next-xsibling[next-x]

Bk Bk

….….(b)

Case 2

a b c d

prev-x x next-x

Bk Bk

….

Bk Bk

p15.

a b c dprev-x x

next-xsibling[next-x]

Bk Bk

….….(c)

Case 3

a bc

d

prev-x x next-x

Bk

Bk

….

Bl Bl

a b c dprev-x x

next-xsibling[next-x]

Bk Bk

….….(d)

Case 4

Bk

][][ xnextkeyxkey

][][ xnextkeyxkey

Bk+1

a cb

d

prev-x x

Bk

Bk

….

Bl

Bk+1

p16.

Insert a node

Extracting the node with minimum key

})'

'

'

H H, Union(-Heap-Binomial H x ]head[H NIL sibling[x]

NIL child[x] NIL p[x]

) Heap(-Binomial-Make H {

x) Insert(H,HeapBinomial

}

)x return

H H, Union(-Heap-Binomial H listresulting

the of headthe to point to ]head[H set and children sx' of list linkedthe of orderthe reverse

) Heap(-Binomial-Make H H of list root

the inkey minthe withrootthe remove and find {

Min(H)-ExtractHeapBinomial

'

'

'

z

y xx

p17.

13

10 1

2512166

1823262914

17 3811

27

41

8

42

(a) head[H] 37

28

77

13

10 1

2512166

1823262914

17 3811

27

418

42

(b) head[H] 37

28

77

x

p18.

13

10 25

41

(c) head[H] 37

28

77

6

2914

17 3811

27

8

16

2326

42

12

18

head[H’]

25 6

2914

17 3811

27

8

12

18

41

37

(d)head[H]

16

2326

42

13

10

28

77

p19.

Decreasing a key

Deleting a key

}}

p[y] z z y

fields other and key[z] key[y]exchange {

do ) key[z] key[y] and NIL z ( whilep[y] z x y

k key[x] " key[x] k" error then key[x] k if

{k) x, Key(H,-DecreaseHeapBinomial

} ) H Min(-Extract-Heap-Binomial

) -x, H, Key(-Decrease-Heap-Binomial {

x) Delete(H,HeapBinomial

p20.

25 6

2914

17 3811

27

8

12

18

41

37

(a)head[H]

16

237

42

13

10

28

77

z

y

25 6

2914

17 3811

27

8

12

18

41

37

(b)head[H]

7

2316

42

13

10

28

77

z

y

p21.

25 6

2914

17 3811

27

8

12

18

41

37

(c)head[H]

10

2316

42

13

7

28

77

z

y