binomial heaps -...
TRANSCRIPT
![Page 1: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/1.jpg)
Binomial HeapsBryan M. Franklin
1Friday, October 10, 2008
![Page 2: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/2.jpg)
TradeoffsWorst Case
Operation Binary Heap Binomial HeapMake-Heap Θ(1) Θ(1)
Insert Θ(lg n) O(lg n)Minimum Θ(1) O(lg n)
Extract-Min Θ(lg n) Θ(lg n)Union Θ(n) Θ(lg n)
Decrease-Key Θ(lg n) Θ(lg n)Delete Θ(lg n) Θ(lg n)
2Friday, October 10, 2008
![Page 3: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/3.jpg)
Binomial Trees
Definition:
B0: A binomial tree with a single node.
Bk: Two Bk-1 binomial trees connected such that one tree has the other as its leftmost child.
3Friday, October 10, 2008
![Page 4: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/4.jpg)
Binomial Trees
B0 B1 B2 B3 B4
4Friday, October 10, 2008
![Page 5: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/5.jpg)
Properties of binomial tree Bk:
• 2k nodes
• Height of tree is k
• Exactly nodes at depth i
• Root has degree k
• Children of the root from left to right are binomial trees Bk-1, Bk-2, Bk-3, ... , B0
!k
i
"
Binomial Tree Properties
5Friday, October 10, 2008
![Page 6: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/6.jpg)
Binomial Tree Properties
B4
2k Nodes
k=424=16 nodes
1
54
11
3
109
15
2
87
14
6
1312
16
6Friday, October 10, 2008
![Page 7: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/7.jpg)
Binomial Tree Properties
B4
Height of k
k=4height is 4
4
01
0
2
01
0
3
01
0
2
01
0
7Friday, October 10, 2008
![Page 8: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/8.jpg)
Binomial Tree Properties
B4
Exactly nodes at depth i
1
43
6
2
54
4
1
32
3
1
21
1
!k
i
"
!40
"= 1
!41
"= 4
!42
"= 6
!43
"= 4
!44
"= 1
8Friday, October 10, 2008
![Page 9: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/9.jpg)
Binomial Tree Properties
B3
Root has degree k
4
01
0
2
01
0
3
01
0
2
01
0
k=4Root’s degree is 4
B2
B1
B0Root’s children:B3, B2, B1, B0
9Friday, October 10, 2008
![Page 10: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/10.jpg)
RepresentationLeft-child, Right-sibling (ch. 10)
Parent
Left-child
Right-sibling
10Friday, October 10, 2008
![Page 11: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/11.jpg)
Representation Example
B4
11Friday, October 10, 2008
![Page 12: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/12.jpg)
Binomial Heap
A binomial heap is a set, H, of binomial trees, such that:
1) Each tree in H satisfies the min-heap property.
2) The root of every tree in H has a unique degree.
12Friday, October 10, 2008
![Page 13: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/13.jpg)
Binomial Heaps
6
4410
17
29
3148
50
8
2223
24
30
3245
55
15
3328
41
3
37
18
7
25
12head[H1]
head[H2]
13Friday, October 10, 2008
![Page 14: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/14.jpg)
Heap Operations
• Make-Heap( )
• Minimum(H)
• Union(H1, H2)
• Insert(H, x)• Extract-Min(H)
• Decrease-Key(H, x, k)
• Delete(H, x)
14Friday, October 10, 2008
![Page 15: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/15.jpg)
Make-Binomial-Heap
Make-Binomial-Heap()
1 H ! new Heap2 head[H ]! NIL3 return H
15Friday, October 10, 2008
![Page 16: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/16.jpg)
Binomial-Heap-Minimum
Binomial-Heap-Minimum(H)
1 y ! NIL2 x! head[H ]3 min!"4 while x #= NIL5 do if key[x] < min6 then min! key[x]7 y ! x8 x! sibling [x]9 return y
16Friday, October 10, 2008
![Page 17: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/17.jpg)
Binomial-Heap-Minimum
15
3328
41
7
25
12head[H1]
x
min=∞
y
NIL
17Friday, October 10, 2008
![Page 18: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/18.jpg)
Binomial-Heap-Minimum
15
3328
41
7
25
12head[H1]
x
min=12
y
NIL
18Friday, October 10, 2008
![Page 19: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/19.jpg)
Binomial-Heap-Minimum
15
3328
41
7
25
12head[H1]
x
min=7
y
NIL
19Friday, October 10, 2008
![Page 20: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/20.jpg)
Binomial-Heap-Minimum
15
3328
41
7
25
12head[H1]
x
min=7
y
NIL
20Friday, October 10, 2008
![Page 21: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/21.jpg)
Binomial-Link
Binomial-Link(y, z)
1 p[y]! z2 sibling [y]! child [z]3 child [z]! y4 degree[z]! degree[z] + 1
21Friday, October 10, 2008
![Page 22: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/22.jpg)
Binomial-Link
y z
22Friday, October 10, 2008
![Page 23: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/23.jpg)
Binomial-Link
y z
p[y]
23Friday, October 10, 2008
![Page 24: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/24.jpg)
Binomial-Link
y z
sibling[y]
24Friday, October 10, 2008
![Page 25: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/25.jpg)
Binomial-Link
y z
child[z]
25Friday, October 10, 2008
![Page 26: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/26.jpg)
Binomial-Heap-UnionBinomial-Heap-Union(H1, H2)
1 H !Make-Binomial-Heap()2 head [H ]! Binomial-Heap-Merge(H1, H2)3 free H1 and H2, but not the lists they point to4 if head [H ] = NIL5 then return H6 prev -x ! NIL7 x! head [H ]8 next -x ! sibling [x]9 while next -x "= NIL
10 do if degree[x] "= degree[next -x ] or(sibling [next -x ] "= NIL and degree[sibling [next -x ]] = degree[x])
11 then prev -x ! x12 x! next -x13 else if key [x] # key [next -x ]14 then sibling [x]! sibling [next -x ]15 Binomial-Link(next -x , x)16 else if prev -x = NIL17 then head [H ]! next -x18 else sibling [prev -x ]! next -x19 Binomial-Link(x, next -x )20 x! next -x21 next -x ! sibling [x]22 return H
26Friday, October 10, 2008
![Page 27: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/27.jpg)
Binomial-Heap-Union
6
4410
17
29
3148
50
8
2223
24
30
3245
55
15
3328
41
3
37
18
7
25
12head[H1]
head[H2]
27Friday, October 10, 2008
![Page 28: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/28.jpg)
6
4410
17
29
3148
50
8
2223
24
30
3245
55
Binomial-Heap-Union
3
37
7
25
15
3328
41
12head[H] 18
x next-x
28Friday, October 10, 2008
![Page 29: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/29.jpg)
6
4410
17
29
3148
50
8
2223
24
30
3245
55
Binomial-Heap-Union
3
37
7
25
15
3328
41
head[H] 12
18
x next-x
29Friday, October 10, 2008
![Page 30: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/30.jpg)
6
4410
17
29
3148
50
8
2223
24
30
3245
55
Binomial-Heap-Union
3
37
7
25
15
3328
41
head[H] 12
18
x next-xprev-x
30Friday, October 10, 2008
![Page 31: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/31.jpg)
6
4410
17
29
3148
50
8
2223
24
30
3245
55
Binomial-Heap-Union
15
3328
41
head[H] 12
18
x next-xprev-x3
377
25
31Friday, October 10, 2008
![Page 32: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/32.jpg)
6
4410
17
29
3148
50
8
2223
24
30
3245
55
Binomial-Heap-Union
15
3328
41
head[H] 12
18
x next-xprev-x3
377
25
32Friday, October 10, 2008
![Page 33: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/33.jpg)
Binomial-Heap-InsertBinomial-Heap-Insert(H, x)
1 H ! " Make-Binomial-Heap()2 p[x] " NIL3 child [x] " NIL4 sibling [x] " NIL5 degree[x] " 06 head [H !] " x7 H " Binomial-Heap-Union(H, H !)
33Friday, October 10, 2008
![Page 34: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/34.jpg)
Extract-Min
Binomial-Heap-Extract-Min(H)
1 find root in H with minimum key value, remove as x2 H ! " Make-Binomial-Heap()3 fill heap H ! by linking x’s children in reverse order4 H " Binomial-Heap-Union(H, H !)5 return x
34Friday, October 10, 2008
![Page 35: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/35.jpg)
1
2512
18
16
2326
42
6
2914
38
8
1711
27
head[H] 37
41
10
1328
77
Extract-Min
35Friday, October 10, 2008
![Page 36: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/36.jpg)
1
2512
18
16
2326
42
6
2914
38
8
1711
27
head[H] 37
41
10
1328
77
x
Extract-Min
36Friday, October 10, 2008
![Page 37: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/37.jpg)
2512
18
16
2326
42
6
2914
38
8
1711
27
head[H] 37
41
10
1328
77
1
x
Extract-Min
37Friday, October 10, 2008
![Page 38: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/38.jpg)
25 12
18
16
2326
42
6
2914
38
8
1711
27
37
41
head[H] 10
1328
77
head[H’]
Extract-Min
1
x
38Friday, October 10, 2008
![Page 39: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/39.jpg)
25 12
18
16
2326
42
6
2914
38
8
1711
27
37
41
head[H]10
1328
77
Extract-Min
1
x
39Friday, October 10, 2008
![Page 40: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/40.jpg)
Decrease-KeyBinomial-Heap-Decrease-Key(H, x, k)
1 if k > key [x]2 then error ”new key is larger than current key”3 key [x]! k4 y ! x5 z ! p[y]6 while z "= NIL and key [y] < key [z]7 do exchange key [y]# key [z]8 ! exchange satellite data as well9 y ! z
10 z ! p[y]
40Friday, October 10, 2008
![Page 41: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/41.jpg)
Decrease-Key
25 12
18
16
2326
42
6
2914
38
8
1711
27
37
41
head[H]10
1328
77x
41Friday, October 10, 2008
![Page 42: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/42.jpg)
Decrease-Key
25 12
18
16
237
42
6
2914
38
8
1711
27
37
41
head[H]10
1328
77x
42Friday, October 10, 2008
![Page 43: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/43.jpg)
Decrease-Key
25 12
18
16
237
42
6
2914
38
8
1711
27
37
41
head[H]10
1328
77y
z
43Friday, October 10, 2008
![Page 44: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/44.jpg)
Decrease-Key
25 12
18
7
2316
42
6
2914
38
8
1711
27
37
41
head[H]10
1328
77y
z
44Friday, October 10, 2008
![Page 45: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/45.jpg)
Decrease-Key
25 12
18
7
2316
42
6
2914
38
8
1711
27
37
41
head[H]10
1328
77
zy
45Friday, October 10, 2008
![Page 46: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/46.jpg)
Decrease-Key
25 12
18
10
2316
42
6
2914
38
8
1711
27
37
41
head[H]7
1328
77
zy
46Friday, October 10, 2008
![Page 47: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/47.jpg)
Decrease-Key
25 12
18
10
2316
42
6
2914
38
8
1711
27
37
41
head[H]7
1328
77
y
z
47Friday, October 10, 2008
![Page 48: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/48.jpg)
Decrease-Key
25 12
18
10
2316
42
6
2914
38
8
1711
27
37
41
head[H]7
1328
77
y
z
48Friday, October 10, 2008
![Page 49: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/49.jpg)
Delete
Binomial-Heap-Delete(H, x)
1 Binomial-Heap-Decrease-Key(H, x,!")2 Binomial-Heap-Extract-Min()
49Friday, October 10, 2008
![Page 50: Binomial Heaps - pages.mtu.edupages.mtu.edu/~aebnenas/teaching/fall2008/cs5321/lectures/Bryan...Binomial Heaps Bryan M. Franklin bmfrankl@mtu.edu Friday, October 10, 2008 1](https://reader030.vdocuments.site/reader030/viewer/2022040713/5e18dd1633c76c119c7aad7d/html5/thumbnails/50.jpg)
The End
50Friday, October 10, 2008