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This means it might contain formatting issues, incorrect code, conceptual problems, or other severe issues.

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• Implement queue operations for '''Fibonacci heaps'''. Where H is heap, x node with data value, k integer. *Operations: ** MakeHeap() - create new empty Fibonacci heap ** Insert(H,x) - insert new element x into heap H ** Union(H1, H2) - union heap H1 and heap H2 ** Minimum(H) - return minimum value from heap H ** ExtractMin(H) - (or DeleteMin(H)) - return minimum value from heap H and remove it from heap ** DecreaseKey(H,x,k) - decrease value of element x in heap H to value k ** Delete(H,x) - remove element x from heap H

## C++

```template <class V>
class FibonacciHeap;

template <class V> struct node {
private:
node<V>* prev;
node<V>* next;
node<V>* child;
node<V>* parent;
V value;
int degree;
bool marked;
public:
friend class FibonacciHeap<V>;
node<V>* getPrev() {return prev;}
node<V>* getNext() {return next;}
node<V>* getChild() {return child;}
node<V>* getParent() {return parent;}
V getValue() {return value;}
bool isMarked() {return marked;}

bool hasChildren() {return child;}
bool hasParent() {return parent;}
};

template <class V> class FibonacciHeap {
protected:
node<V>* heap;
public:

FibonacciHeap() {
heap=_empty();
}
virtual ~FibonacciHeap() {
if(heap) {
_deleteAll(heap);
}
}
node<V>* insert(V value) {
node<V>* ret=_singleton(value);
heap=_merge(heap,ret);
return ret;
}
void merge(FibonacciHeap& other) {
heap=_merge(heap,other.heap);
other.heap=_empty();
}

bool isEmpty() {
return heap==NULL;
}

V getMinimum() {
return heap->value;
}

V removeMinimum() {
node<V>* old=heap;
heap=_removeMinimum(heap);
V ret=old->value;
delete old;
return ret;
}

void decreaseKey(node<V>* n,V value) {
heap=_decreaseKey(heap,n,value);
}

node<V>* find(V value) {
return _find(heap,value);
}
private:
node<V>* _empty() {
return NULL;
}

node<V>* _singleton(V value) {
node<V>* n=new node<V>;
n->value=value;
n->prev=n->next=n;
n->degree=0;
n->marked=false;
n->child=NULL;
n->parent=NULL;
return n;
}

node<V>* _merge(node<V>* a,node<V>* b) {
if(a==NULL)return b;
if(b==NULL)return a;
if(a->value>b->value) {
node<V>* temp=a;
a=b;
b=temp;
}
node<V>* an=a->next;
node<V>* bp=b->prev;
a->next=b;
b->prev=a;
an->prev=bp;
bp->next=an;
return a;
}

void _deleteAll(node<V>* n) {
if(n!=NULL) {
node<V>* c=n;
do {
node<V>* d=c;
c=c->next;
_deleteAll(d->child);
delete d;
} while(c!=n);
}
}

child->prev=child->next=child;
child->parent=parent;
parent->degree++;
parent->child=_merge(parent->child,child);
}

void _unMarkAndUnParentAll(node<V>* n) {
if(n==NULL)return;
node<V>* c=n;
do {
c->marked=false;
c->parent=NULL;
c=c->next;
}while(c!=n);
}

node<V>* _removeMinimum(node<V>* n) {
_unMarkAndUnParentAll(n->child);
if(n->next==n) {
n=n->child;
} else {
n->next->prev=n->prev;
n->prev->next=n->next;
n=_merge(n->next,n->child);
}
if(n==NULL)return n;
node<V>* trees[64]={NULL};

while(true) {
if(trees[n->degree]!=NULL) {
node<V>* t=trees[n->degree];
if(t==n)break;
trees[n->degree]=NULL;
if(n->value<t->value) {
t->prev->next=t->next;
t->next->prev=t->prev;
} else {
t->prev->next=t->next;
t->next->prev=t->prev;
if(n->next==n) {
t->next=t->prev=t;
n=t;
} else {
n->prev->next=t;
n->next->prev=t;
t->next=n->next;
t->prev=n->prev;
n=t;
}
}
continue;
} else {
trees[n->degree]=n;
}
n=n->next;
}
node<V>* min=n;
do {
if(n->value<min->value)min=n;
n=n->next;
} while(n!=n);
return min;
}

node<V>* _cut(node<V>* heap,node<V>* n) {
if(n->next==n) {
n->parent->child=NULL;
} else {
n->next->prev=n->prev;
n->prev->next=n->next;
n->parent->child=n->next;
}
n->next=n->prev=n;
n->marked=false;
return _merge(heap,n);
}

node<V>* _decreaseKey(node<V>* heap,node<V>* n,V value) {
if(n->value<value)return heap;
n->value=value;
if(n->value<n->parent->value) {
heap=_cut(heap,n);
node<V>* parent=n->parent;
n->parent=NULL;
while(parent!=NULL && parent->marked) {
heap=_cut(heap,parent);
n=parent;
parent=n->parent;
n->parent=NULL;
}
if(parent!=NULL && parent->parent!=NULL)parent->marked=true;
}
return heap;
}

node<V>* _find(node<V>* heap,V value) {
node<V>* n=heap;
if(n==NULL)return NULL;
do {
if(n->value==value)return n;
node<V>* ret=_find(n->child,value);
if(ret)return ret;
n=n->next;
}while(n!=heap);
return NULL;
}
};
```

## Go

A package. Implementation follows Fredman and Tarjan's 1987 paper.

```package fib

import "fmt"

type Value interface {
LT(Value) bool
}

type Node struct {
value      Value
parent     *Node
child      *Node
prev, next *Node
rank       int
mark       bool
}

func (n Node) Value() Value { return n.value }

type Heap struct{ *Node }

func MakeHeap() *Heap { return &Heap{} }

func (h *Heap) Insert(v Value) *Node {
x := &Node{value: v}
if h.Node == nil {
x.next = x
x.prev = x
h.Node = x
} else {
meld1(h.Node, x)
if x.value.LT(h.value) {
h.Node = x
}
}
return x
}

func meld1(list, single *Node) {
list.prev.next = single
single.prev = list.prev
single.next = list
list.prev = single
}

func (h *Heap) Union(h2 *Heap) {
switch {
case h.Node == nil:
*h = *h2
case h2.Node != nil:
meld2(h.Node, h2.Node)
if h2.value.LT(h.value) {
*h = *h2
}
}
h2.Node = nil
}

func meld2(a, b *Node) {
a.prev.next = b
b.prev.next = a
a.prev, b.prev = b.prev, a.prev
}

func (h Heap) Minimum() (min Value, ok bool) {
if h.Node == nil {
return
}
return h.value, true
}

func (h *Heap) ExtractMin() (min Value, ok bool) {
if h.Node == nil {
return
}
min = h.value
roots := map[int]*Node{}
r.prev = r
r.next = r
for {
x, ok := roots[r.rank]
if !ok {
break
}
delete(roots, r.rank)
if x.value.LT(r.value) {
r, x = x, r
}
x.parent = r
x.mark = false
if r.child == nil {
x.next = x
x.prev = x
r.child = x
} else {
meld1(r.child, x)
}
r.rank++
}
roots[r.rank] = r
}
for r := h.next; r != h.Node; {
n := r.next
r = n
}
if c := h.child; c != nil {
c.parent = nil
r := c.next
for r != c {
n := r.next
r.parent = nil
r = n
}
}
if len(roots) == 0 {
h.Node = nil
return min, true
}
var mv *Node
var d int
for d, mv = range roots {
break
}
delete(roots, d)
mv.next = mv
mv.prev = mv
for _, r := range roots {
r.prev = mv
r.next = mv.next
mv.next.prev = r
mv.next = r
if r.value.LT(mv.value) {
mv = r
}
}
h.Node = mv
return min, true
}

func (h *Heap) DecreaseKey(n *Node, v Value) error {
if n.value.LT(v) {
return fmt.Errorf("DecreaseKey new value greater than existing value")
}
n.value = v
if n == h.Node {
return nil
}
p := n.parent
if p == nil {
if v.LT(h.value) {
h.Node = n
}
return nil
}
h.cutAndMeld(n)
return nil
}

func (h Heap) cut(x *Node) {
p := x.parent
p.rank--
if p.rank == 0 {
p.child = nil
} else {
p.child = x.next
x.prev.next = x.next
x.next.prev = x.prev
}
if p.parent == nil {
return
}
if !p.mark {
p.mark = true
return
}
h.cutAndMeld(p)
}

func (h Heap) cutAndMeld(x *Node) {
h.cut(x)
x.parent = nil
meld1(h.Node, x)
}

func (h *Heap) Delete(n *Node) {
p := n.parent
if p == nil {
if n == h.Node {
h.ExtractMin()
return
}
n.prev.next = n.next
n.next.prev = n.prev
} else {
h.cut(n)
}
c := n.child
if c == nil {
return
}
for {
c.parent = nil
c = c.next
if c == n.child {
break
}
}
meld2(h.Node, c)
}

func (h Heap) Vis() {
if h.Node == nil {
fmt.Println("<empty>")
return
}
var f func(*Node, string)
f = func(n *Node, pre string) {
pc := "│ "
for x := n; ; x = x.next {
if x.next != n {
fmt.Print(pre, "├─")
} else {
fmt.Print(pre, "└─")
pc = "  "
}
if x.child == nil {
fmt.Println("╴", x.value)
} else {
fmt.Println("┐", x.value)
f(x.child, pre+pc)
}
if x.next == n {
break
}
}
}
f(h.Node, "")
}
```

A demonstration:

```package main

import (
"fmt"

"fib"
)

type str string

func (s str) LT(t fib.Value) bool { return s < t.(str) }

func main() {
fmt.Println("MakeHeap:")
h := fib.MakeHeap()
h.Vis()

fmt.Println("\nInsert:")
h.Insert(str("cat"))
h.Vis()

fmt.Println("\nUnion:")
h2 := fib.MakeHeap()
h2.Insert(str("rat"))
h.Union(h2)
h.Vis()

fmt.Println("\nMinimum:")
m, _ := h.Minimum()
fmt.Println(m)

fmt.Println("\nExtractMin:")
// happens on delete min.
h.Insert(str("bat"))
x := h.Insert(str("meerkat")) // save x for decrease key and delete demos
m, _ = h.ExtractMin()
fmt.Printf("(extracted %v)\n", m)
h.Vis()

fmt.Println("\nDecreaseKey:")
h.DecreaseKey(x, str("gnat"))
h.Vis()
fmt.Println("\nDelete:")
// add yet a couple more items to show how F&T's original delete was
// lazier than CLRS's delete.
h.Insert(str("bobcat"))
h.Insert(str("bat"))
fmt.Printf("(deleting %v)\n", x.Value())
h.Delete(x)
h.Vis()
}
```

{{out}}

```
MakeHeap:
<empty>

Insert:
└─╴ cat

Union:
├─╴ cat
└─╴ rat

Minimum:
cat

ExtractMin:
(extracted bat)
├─┐ cat
│ └─╴ rat
└─╴ meerkat

DecreaseKey:
├─┐ cat
│ └─╴ rat
└─╴ gnat

Delete:
(deleting gnat)
├─╴ bat
├─╴ bobcat
└─┐ cat
└─╴ rat

```

## Kotlin

{{trans|Go}}

```// version 1.2.21

class Node<V : Comparable<V>>(var value: V) {
var parent: Node<V>? = null
var child:  Node<V>? = null
var prev:   Node<V>? = null
var next:   Node<V>? = null
var rank = 0
var mark = false

fun meld1(node: Node<V>) {
this.prev?.next = node
node.prev = this.prev
node.next = this
this.prev = node
}

fun meld2(node: Node<V>) {
this.prev?.next = node
node.prev?.next = this
val temp = this.prev
this.prev = node.prev
node.prev = temp
}
}

fun <V: Comparable<V>> makeHeap() = FibonacciHeap<V>()

class FibonacciHeap<V: Comparable<V>>(var node: Node<V>? = null) {

fun insert(v: V): Node<V> {
val x = Node(v)
if (this.node == null) {
x.next = x
x.prev = x
this.node = x
}
else {
this.node!!.meld1(x)
if (x.value < this.node!!.value) this.node = x
}
return x
}

fun union(other: FibonacciHeap<V>) {
if (this.node == null) {
this.node = other.node
}
else if (other.node != null) {
this.node!!.meld2(other.node!!)
if (other.node!!.value < this.node!!.value) this.node = other.node
}
other.node = null
}

fun minimum(): V? = this.node?.value

fun extractMin(): V? {
if (this.node == null) return null
val min = minimum()
val roots = mutableMapOf<Int, Node<V>>()

r.prev = r
r.next = r
var rr = r
while (true) {
var x = roots[rr.rank] ?: break
roots.remove(rr.rank)
if (x.value < rr.value) {
val t = rr
rr = x
x = t
}
x.parent = rr
x.mark = false
if (rr.child == null) {
x.next = x
x.prev = x
rr.child = x
}
else {
rr.child!!.meld1(x)
}
rr.rank++
}
roots[rr.rank] = rr
}

var r = this.node!!.next
while (r != this.node) {
val n = r!!.next
r = n
}
val c = this.node!!.child
if (c != null) {
c.parent = null
var rr = c.next!!
while (rr != c) {
val n = rr.next!!
rr.parent = null
rr = n
}
}
if (roots.isEmpty()) {
this.node = null
return min
}
val d = roots.keys.first()
var mv = roots[d]!!
roots.remove(d)
mv.next = mv
mv.prev = mv
for ((_, rr) in roots) {
rr.prev = mv
rr.next = mv.next
mv.next!!.prev = rr
mv.next = rr
if (rr.value < mv.value) mv = rr
}
this.node = mv
return min
}

fun decreaseKey(n: Node<V>, v: V) {
require (n.value > v) {
"In 'decreaseKey' new value greater than existing value"
}
n.value = v
if (n == this.node) return
val p = n.parent
if (p == null) {
if (v < this.node!!.value) this.node = n
return
}
cutAndMeld(n)
}

private fun cut(x: Node<V>) {
val p = x.parent
if (p == null) return
p.rank--
if (p.rank == 0) {
p.child = null
}
else {
p.child = x.next
x.prev?.next = x.next
x.next?.prev = x.prev
}
if (p.parent == null) return
if (!p.mark) {
p.mark = true
return
}
cutAndMeld(p)
}

private fun cutAndMeld(x: Node<V>) {
cut(x)
x.parent = null
this.node?.meld1(x)
}

fun delete(n: Node<V>) {
val p = n.parent
if (p == null) {
if (n == this.node) {
extractMin()
return
}
n.prev?.next = n.next
n.next?.prev = n.prev
}
else {
cut(n)
}
var c = n.child
if (c == null) return
while (true) {
c!!.parent = null
c = c.next
if (c == n.child) break
}
this.node?.meld2(c!!)
}

fun visualize() {
if (this.node == null) {
println("<empty>")
return
}

fun f(n: Node<V>, pre: String) {
var pc = "│ "
var x = n
while (true) {
if (x.next != n) {
print("\$pre├─")
}
else {
print("\$pre└─")
pc = "  "
}
if (x.child == null) {
println("╴ \${x.value}")
}
else {
println("┐ \${x.value}")
f(x.child!!, pre + pc)
}
if (x.next == n) break
x = x.next!!
}
}
f(this.node!!, "")
}
}

fun main(args: Array<String>) {
println("MakeHeap:")
val h = makeHeap<String>()
h.visualize()

println("\nInsert:")
h.insert("cat")
h.visualize()

println("\nUnion:")
val h2 = makeHeap<String>()
h2.insert("rat")
h.union(h2)
h.visualize()

println("\nMinimum:")
var m = h.minimum()
println(m)

println("\nExtractMin:")
// happens on delete min.
h.insert("bat")
val x = h.insert("meerkat")  // save x for decrease key and delete demos.
m = h.extractMin()
println("(extracted \$m)")
h.visualize()

println("\nDecreaseKey:")
h.decreaseKey(x, "gnat")
h.visualize()

println("\nDelete:")
// add a couple more items.
h.insert("bobcat")
h.insert("bat")
println("(deleting \${x.value})")
h.delete(x)
h.visualize()
}
```

{{out}}

```
MakeHeap:
<empty>

Insert:
└─╴ cat

Union:
├─╴ cat
└─╴ rat

Minimum:
cat

ExtractMin:
(extracted bat)
├─┐ cat
│ └─╴ rat
└─╴ meerkat

DecreaseKey:
├─┐ cat
│ └─╴ rat
└─╴ gnat

Delete:
(deleting gnat)
├─╴ bat
├─╴ bobcat
└─┐ cat
└─╴ rat

```

## Phix

{{trans|Go}}

```enum VALUE, PARENT, CHILD, PREV, NEXT, RANK, MARK, NODELEN=\$

function new_node()
return repeat(NULL,NODELEN)
end function

sequence nodes = {}
integer freelist = NULL

function new_slot()
integer res
if freelist!=NULL then
res = freelist
freelist = nodes[freelist]
nodes[freelist] = NULL
else
nodes = append(nodes,NULL)
res = length(nodes)
end if
return res
end function

procedure release_slot(integer n)
nodes[n] = freelist
freelist = n
end procedure

function MakeHeap()
return new_slot()
end function

procedure meld1(integer list, single)
nodes[nodes[list][PREV]][NEXT] = single
nodes[single][PREV] = nodes[list][PREV]
nodes[single][NEXT] = list
nodes[list][PREV] = single
end procedure

function Insert(integer h, object v)
integer n = 0
sequence x = new_node()
x[VALUE] = v
if nodes[h] == NULL then
x[NEXT] = h
x[PREV] = h
nodes[h] = x
else
n = new_slot()
nodes[n] = x
meld1(h, n)
if nodes[n][VALUE]<nodes[h][VALUE] then
h = n
end if
end if
return {h,n}
end function

procedure meld2(integer a, b)
nodes[nodes[a][PREV]][NEXT] = b
nodes[nodes[b][PREV]][NEXT] = a
{nodes[a][PREV], nodes[b][PREV]} = {nodes[b][PREV], nodes[a][PREV]}
end procedure

function Union(integer h, h2)
if nodes[h] == NULL then
h = h2
elsif nodes[h2] != NULL then
meld2(h, h2)
if nodes[h2][VALUE]<nodes[h][VALUE] then
h = h2
end if
else
release_slot(h2)
end if
return {h,NULL} -- (h2:=NULL implied)
end function

function Minimum(integer h)
if nodes[h] == NULL then
return {"<none>",false}
end if
return {nodes[h][VALUE], true}
end function

nodes[r][PREV] = r
nodes[r][NEXT] = r
while true do
integer node = getd_index(nodes[r][RANK],roots)
if node=NULL then exit end if
integer x = getd_by_index(node,roots)
deld(nodes[r][RANK],roots)
if nodes[x][VALUE]<nodes[r][VALUE] then
{r, x} = {x, r}
end if
nodes[x][PARENT] = r
nodes[x][MARK] = false
if nodes[r][CHILD] == NULL then
nodes[x][NEXT] = x
nodes[x][PREV] = x
nodes[r][CHILD] = x
else
meld1(nodes[r][CHILD], x)
end if
nodes[r][RANK] += 1
end while
setd(nodes[r][RANK],r,roots)
end procedure

function ExtractMin(integer h)
if nodes[h] == NULL then
return {h,"<none>",false}
end if
object minimum = nodes[h][VALUE]
integer roots = new_dict()
integer r = nodes[h][NEXT], n
while r != h do
n := nodes[r][NEXT]
r = n
end while
integer c = nodes[h][CHILD]
if c != NULL then
nodes[c][PARENT] = NULL
r := nodes[c][NEXT]
while r != c do
n := nodes[r][NEXT]
nodes[r][PARENT] = NULL
r = n
end while
end if
if dict_size(roots) == 0 then
destroy_dict(roots)
return {NULL, minimum, true}
end if
integer d = getd_partial_key(0,roots)
integer mv = getd(d,roots)
deld(d,roots)
nodes[mv][NEXT] = mv
nodes[mv][PREV] = mv
sequence rs = getd_all_keys(roots)
for i=1 to length(rs) do
r = getd(rs[i],roots)
nodes[r][PREV] = mv
nodes[r][NEXT] = nodes[mv][NEXT]
nodes[nodes[mv][NEXT]][PREV] = r
nodes[mv][NEXT] = r
if nodes[r][VALUE]<nodes[mv][VALUE] then
mv = r
end if
end for
h = mv
destroy_dict(roots)
return {h, minimum, true}
end function

procedure cut_and_meld(integer h, x, bool meld)
integer p := nodes[x][PARENT]
nodes[p][RANK] -= 1
if nodes[p][RANK] == 0 then
nodes[p][CHILD] = NULL
else
nodes[p][CHILD] = nodes[x][NEXT]
nodes[nodes[x][PREV]][NEXT] = nodes[x][NEXT]
nodes[nodes[x][NEXT]][PREV] = nodes[x][PREV]
end if
if nodes[p][PARENT] == NULL then
return
end if
if not nodes[p][MARK] then
nodes[p][MARK] = true
return
end if
cut_and_meld(h,p,true)
if meld then
nodes[x][PARENT] = NULL
meld1(h, x)
end if
end procedure

function DecreaseKey(integer h, n, object v)
if nodes[n][VALUE]<v then
crash("DecreaseKey new value greater than existing value")
end if
nodes[n][VALUE] = v
if n!=h then
integer p := nodes[n][PARENT]
if p == NULL then
if v<nodes[h][VALUE] then
h = n
end if
else
cut_and_meld(h,n,true)
end if
end if
return h
end function

function Delete(integer h, n)
integer p := nodes[n][PARENT]
if p == NULL then
if n == h then
{h} = ExtractMin(h)
return h
end if
nodes[nodes[n][PREV]][NEXT] = nodes[n][NEXT]
nodes[nodes[n][NEXT]][PREV] = nodes[n][PREV]
else
cut_and_meld(h,n,false)
end if
integer c := nodes[n][CHILD]
if c != NULL then
while true do
nodes[c][PARENT] = NULL
c = nodes[c][NEXT]
if c == nodes[n][CHILD] then
exit
end if
end while
meld2(h, c)
end if
return h
end function

constant W=platform()=WINDOWS,
Horizontal = iff(W?#C4:'-'),
Vertical   = iff(W?#B3:'|'),
sBtmLeft   = iff(W?#C0:'+'),
sLeftTee   = iff(W?#C3:'+'),
sTopRight  = iff(W?#BF:'+')

procedure vis(integer n, string pre)
string pc = Vertical&" "
sequence x = nodes[n]
while true do
integer next = x[NEXT]
if next!=n then
printf(1,pre&sLeftTee&Horizontal)
else
printf(1,pre&sBtmLeft&Horizontal)
pc = "  "
end if
if x[CHILD] == NULL then
printf(1,"%c %s\n",{Horizontal,sprint(x[VALUE])})
else
printf(1,"%c %s\n",{sTopRight,sprint(x[VALUE])})
vis(x[CHILD], pre&pc)
end if
if next=n then exit end if
x = nodes[next]
end while
end procedure

procedure Vis(integer h)
if nodes[h] == NULL then
printf(1,"<empty>\n")
return
end if
vis(h,"")
end procedure

printf(1,"MakeHeap:\n")
integer h := MakeHeap()
Vis(h)

printf(1,"\nInsert:\n")
{h} = Insert(h,"cat")
Vis(h)

printf(1,"\nUnion:\n")
integer h2 := MakeHeap()
{h2} = Insert(h2,"rat")
{h,h2} = Union(h,h2)     -- (h2:=NULL)
Vis(h)

printf(1,"\nMinimum:\n")
{object m, {}} = Minimum(h)
?m

printf(1,"\nExtractMin:\n")
-- happens on delete min.
{h} = Insert(h,"bat")
{h,integer x} = Insert(h,"meerkat") -- save x for decrease key and delete demos
{h,m,{}} = ExtractMin(h)
printf(1,"(extracted %s)\n", {sprint(m)})
Vis(h)

printf(1,"\nDecreaseKey:\n")
h = DecreaseKey(h, x, "gnat")
Vis(h)

printf(1,"\nDelete:\n")
-- add yet a couple more items to show how F&T's original delete was
-- lazier than CLRS's delete.
{h} = Insert(h,"bobcat")
{h} = Insert(h,"bat")
printf(1,"(deleting %s)\n", {sprint(nodes[x][VALUE])})
h = Delete(h,x)
Vis(h)
```

{{out}}

```
MakeHeap:
<empty>

Insert:
└── "cat"

Union:
├── "cat"
└── "rat"

Minimum:
"cat"

ExtractMin:
(extracted "bat")
├─┐ "cat"
│ └── "rat"
└── "meerkat"

DecreaseKey:
├─┐ "cat"
│ └── "rat"
└── "gnat"

Delete:
(deleting "gnat")
├── "bat"
├── "bobcat"
└─┐ "cat"
└── "rat"

```

## Python

```class FibonacciHeap:

# internal node class
class Node:
def __init__(self, data):
self.data = data
self.parent = self.child = self.left = self.right = None
self.degree = 0
self.mark = False

# function to iterate through a doubly linked list
flag = False
while True:
if node == stop and flag is True:
break
elif node == stop:
flag = True
yield node
node = node.right

# pointer to the head and minimum node in the root list
root_list, min_node = None, None

# maintain total node count in full fibonacci heap
total_nodes = 0

# return min node in O(1) time
def find_min(self):
return self.min_node

# extract (delete) the min node from the heap in O(log n) time
# amortized cost analysis can be found here (http://bit.ly/1ow1Clm)
def extract_min(self):
z = self.min_node
if z is not None:
if z.child is not None:
# attach child nodes to root list
children = [x for x in self.iterate(z.child)]
for i in xrange(0, len(children)):
self.merge_with_root_list(children[i])
children[i].parent = None
self.remove_from_root_list(z)
# set new min node in heap
if z == z.right:
self.min_node = self.root_list = None
else:
self.min_node = z.right
self.consolidate()
self.total_nodes -= 1
return z

# insert new node into the unordered root list in O(1) time
def insert(self, data):
n = self.Node(data)
n.left = n.right = n
self.merge_with_root_list(n)
if self.min_node is None or n.data < self.min_node.data:
self.min_node = n
self.total_nodes += 1

# modify the data of some node in the heap in O(1) time
def decrease_key(self, x, k):
if k > x.data:
return None
x.data = k
y = x.parent
if y is not None and x.data < y.data:
self.cut(x, y)
if x.data < self.min_node.data:
self.min_node = x

# merge two fibonacci heaps in O(1) time by concatenating the root lists
# the root of the new root list becomes equal to the first list and the second
# list is simply appended to the end (then the proper min node is determined)
def merge(self, h2):
H = FibonacciHeap()
H.root_list, H.min_node = self.root_list, self.min_node
# fix pointers when merging the two heaps
last = h2.root_list.left
h2.root_list.left = H.root_list.left
H.root_list.left.right = h2.root_list
H.root_list.left = last
H.root_list.left.right = H.root_list
# update min node if needed
if h2.min_node.data < H.min_node.data:
H.min_node = h2.min_node
# update total nodes
H.total_nodes = self.total_nodes + h2.total_nodes
return H

# if a child node becomes smaller than its parent node we
# cut this child node off and bring it up to the root list
def cut(self, x, y):
self.remove_from_child_list(y, x)
y.degree -= 1
self.merge_with_root_list(x)
x.parent = None
x.mark = False

# cascading cut of parent node to obtain good time bounds
z = y.parent
if z is not None:
if y.mark is False:
y.mark = True
else:
self.cut(y, z)

# combine root nodes of equal degree to consolidate the heap
# by creating a list of unordered binomial trees
def consolidate(self):
A = [None] * self.total_nodes
nodes = [w for w in self.iterate(self.root_list)]
for w in xrange(0, len(nodes)):
x = nodes[w]
d = x.degree
while A[d] != None:
y = A[d]
if x.data > y.data:
temp = x
x, y = y, temp
A[d] = None
d += 1
A[d] = x
# find new min node - no need to reconstruct new root list below
# because root list was iteratively changing as we were moving
# nodes around in the above loop
for i in xrange(0, len(A)):
if A[i] is not None:
if A[i].data < self.min_node.data:
self.min_node = A[i]

# actual linking of one node to another in the root list
# while also updating the child linked list
self.remove_from_root_list(y)
y.left = y.right = y
self.merge_with_child_list(x, y)
x.degree += 1
y.parent = x
y.mark = False

# merge a node with the doubly linked root list
def merge_with_root_list(self, node):
if self.root_list is None:
self.root_list = node
else:
node.right = self.root_list.right
node.left = self.root_list
self.root_list.right.left = node
self.root_list.right = node

# merge a node with the doubly linked child list of a root node
def merge_with_child_list(self, parent, node):
if parent.child is None:
parent.child = node
else:
node.right = parent.child.right
node.left = parent.child
parent.child.right.left = node
parent.child.right = node

# remove a node from the doubly linked root list
def remove_from_root_list(self, node):
if node == self.root_list:
self.root_list = node.right
node.left.right = node.right
node.right.left = node.left

# remove a node from the doubly linked child list
def remove_from_child_list(self, parent, node):
if parent.child == parent.child.right:
parent.child = None
elif parent.child == node:
parent.child = node.right
node.right.parent = parent
node.left.right = node.right
node.right.left = node.left
```