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# Decrease Key and Delete Node Operations on a Fibonacci Heap

#### In this tutorial, you will learn how decrease key and delete node operations work. Also, you will find working examples of these operations on a fibonacci heap in C.

In a fibonacci heap, decrease-key and delete-node are important operations. These operations are discussed below.

## Decreasing a Key

In decreasing a key operation, the value of a key is decreased to a lower value.

Following functions are used for decreasing the key.

### Decrease-Key

1. Select the node to be decreased, x, and change its value to the new value k.
2. If the parent of xy, is not null and the key of parent is greater than that of the k then call `C``ut(x)` and `C``ascading-``C``ut(y)` subsequently.
3. If the key of x is smaller than the key of min, then mark x as min.

### Cut

1. Remove x from the current position and add it to the root list.
2. If x is marked, then mark it as false.

1. If the parent of y is not null then follow the following steps.
2. If y is unmarked, then mark y.
3. Else, call `Cut(y)` and `Cascading-Cut(parent of y)`.

## Decrease Key Example

The above operations can be understood in the examples below.

### Example: Decreasing 46 to 15.

1. Decrease the value 46 to 15.
2. Cut part: Since `24 ≠ nill` and `15 < its parent`, cut it and add it to the root list. Cascading-Cut part: mark 24.

### Example: Decreasing 35 to 5

1. Decrease the value 35 to 5.
2. Cut part: Since `26 ≠ nill` and `5<its parent`, cut it and add it to the root list.
3. Cascading-Cut part: Since 26 is marked, the flow goes to `Cut` and `Cascading-Cut`.
Cut(26): Cut 26 and add it to the root list and mark it as false.

Since the 24 is also marked, again call `Cut(24)` and `Cascading-Cut(7)`. These operations result in the tree below.

4. Since `5 < 7`, mark 5 as min.

## Deleting a Node

This process makes use of decrease-key and extract-min operations. The following steps are followed for deleting a node.

1. Let k be the node to be deleted.
2. Apply decrease-key operation to decrease the value of k to the lowest possible value (i.e. -∞).
3. Apply extract-min operation to remove this node.

## C Examples

``````// Operations on a Fibonacci heap in C

#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <math.h>

typedef struct _NODE
{
int key;
int degree;
struct _NODE *left_sibling;
struct _NODE *right_sibling;
struct _NODE *parent;
struct _NODE *child;
bool mark;
bool visited;
} NODE;

typedef struct fibanocci_heap
{
int n;
NODE *min;
int phi;
int degree;
} FIB_HEAP;

FIB_HEAP *make_fib_heap();
void insertion(FIB_HEAP *H, NODE *new, int val);
NODE *extract_min(FIB_HEAP *H);
void consolidate(FIB_HEAP *H);
void fib_heap_link(FIB_HEAP *H, NODE *y, NODE *x);
NODE *find_min_node(FIB_HEAP *H);
void decrease_key(FIB_HEAP *H, NODE *node, int key);
void cut(FIB_HEAP *H, NODE *node_to_be_decrease, NODE *parent_node);
void Delete_Node(FIB_HEAP *H, int dec_key);

FIB_HEAP *make_fib_heap(){
FIB_HEAP *H;
H = (FIB_HEAP *)malloc(sizeof(FIB_HEAP));
H->n = 0;
H->min = NULL;
H->phi = 0;
H->degree = 0;
return H;
}
void new_print_heap(NODE *n){
NODE *x;
for (x = n;; x = x->right_sibling)
{

if (x->child == NULL)
{
printf("node with no child (%d) n", x->key);
}
else
{

printf("NODE(%d) with child (%d)n", x->key, x->child->key);
new_print_heap(x->child);
}
if (x->right_sibling == n)
{
break;
}
}
}

void insertion(FIB_HEAP *H, NODE *new, int val){
new = (NODE *)malloc(sizeof(NODE));
new->key = val;
new->degree = 0;
new->mark = false;
new->parent = NULL;
new->child = NULL;
new->visited = false;
new->left_sibling = new;
new->right_sibling = new;
if (H->min == NULL)
{
H->min = new;
}
else
{
H->min->left_sibling->right_sibling = new;
new->right_sibling = H->min;
new->left_sibling = H->min->left_sibling;
H->min->left_sibling = new;
if (new->key < H->min->key)
{
H->min = new;
}
}
(H->n)++;
}

NODE *find_min_node(FIB_HEAP *H){
if (H == NULL)
{
printf(" n Fibonacci heap not yet created n");
return NULL;
}
else
return H->min;
}

FIB_HEAP *unionHeap(FIB_HEAP *H1, FIB_HEAP *H2){
FIB_HEAP *Hnew;
Hnew = make_fib_heap();
Hnew->min = H1->min;

NODE *temp1, *temp2;
temp1 = Hnew->min->right_sibling;
temp2 = H2->min->left_sibling;

Hnew->min->right_sibling->left_sibling = H2->min->left_sibling;
Hnew->min->right_sibling = H2->min;
H2->min->left_sibling = Hnew->min;
temp2->right_sibling = temp1;

if ((H1->min == NULL) || (H2->min != NULL && H2->min->key < H1->min->key))
Hnew->min = H2->min;
Hnew->n = H1->n + H2->n;
return Hnew;
}

int cal_degree(int n){
int count = 0;
while (n > 0)
{
n = n / 2;
count++;
}
return count;
}
void consolidate(FIB_HEAP *H){
int degree, i, d;
degree = cal_degree(H->n);
NODE *A[degree], *x, *y, *z;
for (i = 0; i <= degree; i++)
{
A[i] = NULL;
}
x = H->min;
do
{
d = x->degree;
while (A[d] != NULL)
{
y = A[d];
if (x->key > y->key)
{
NODE *exchange_help;
exchange_help = x;
x = y;
y = exchange_help;
}
if (y == H->min)
H->min = x;
if (y->right_sibling == x)
H->min = x;
A[d] = NULL;
d++;
}
A[d] = x;
x = x->right_sibling;
} while (x != H->min);

H->min = NULL;
for (i = 0; i < degree; i++)
{
if (A[i] != NULL)
{
A[i]->left_sibling = A[i];
A[i]->right_sibling = A[i];
if (H->min == NULL)
{
H->min = A[i];
}
else
{
H->min->left_sibling->right_sibling = A[i];
A[i]->right_sibling = H->min;
A[i]->left_sibling = H->min->left_sibling;
H->min->left_sibling = A[i];
if (A[i]->key < H->min->key)
{
H->min = A[i];
}
}
if (H->min == NULL)
{
H->min = A[i];
}
else if (A[i]->key < H->min->key)
{
H->min = A[i];
}
}
}
}

void fib_heap_link(FIB_HEAP *H, NODE *y, NODE *x){
y->right_sibling->left_sibling = y->left_sibling;
y->left_sibling->right_sibling = y->right_sibling;

if (x->right_sibling == x)
H->min = x;

y->left_sibling = y;
y->right_sibling = y;
y->parent = x;

if (x->child == NULL)
{
x->child = y;
}
y->right_sibling = x->child;
y->left_sibling = x->child->left_sibling;
x->child->left_sibling->right_sibling = y;
x->child->left_sibling = y;
if ((y->key) < (x->child->key))
x->child = y;

(x->degree)++;
}
NODE *extract_min(FIB_HEAP *H){

if (H->min == NULL)
printf("n The heap is empty");
else
{
NODE *temp = H->min;
NODE *pntr;
pntr = temp;
NODE *x = NULL;
if (temp->child != NULL)
{

x = temp->child;
do
{
pntr = x->right_sibling;
(H->min->left_sibling)->right_sibling = x;
x->right_sibling = H->min;
x->left_sibling = H->min->left_sibling;
H->min->left_sibling = x;
if (x->key < H->min->key)
H->min = x;
x->parent = NULL;
x = pntr;
} while (pntr != temp->child);
}

(temp->left_sibling)->right_sibling = temp->right_sibling;
(temp->right_sibling)->left_sibling = temp->left_sibling;
H->min = temp->right_sibling;

if (temp == temp->right_sibling && temp->child == NULL)
H->min = NULL;
else
{
H->min = temp->right_sibling;
consolidate(H);
}
H->n = H->n - 1;
return temp;
}
return H->min;
}

void cut(FIB_HEAP *H, NODE *node_to_be_decrease, NODE *parent_node){
NODE *temp_parent_check;

if (node_to_be_decrease == node_to_be_decrease->right_sibling)
parent_node->child = NULL;

node_to_be_decrease->left_sibling->right_sibling = node_to_be_decrease->right_sibling;
node_to_be_decrease->right_sibling->left_sibling = node_to_be_decrease->left_sibling;
if (node_to_be_decrease == parent_node->child)
parent_node->child = node_to_be_decrease->right_sibling;
(parent_node->degree)--;

node_to_be_decrease->left_sibling = node_to_be_decrease;
node_to_be_decrease->right_sibling = node_to_be_decrease;
H->min->left_sibling->right_sibling = node_to_be_decrease;
node_to_be_decrease->right_sibling = H->min;
node_to_be_decrease->left_sibling = H->min->left_sibling;
H->min->left_sibling = node_to_be_decrease;

node_to_be_decrease->parent = NULL;
node_to_be_decrease->mark = false;
}

NODE *aux;
aux = parent_node->parent;
if (aux != NULL)
{
if (parent_node->mark == false)
{
parent_node->mark = true;
}
else
{
cut(H, parent_node, aux);
}
}
}

void decrease_key(FIB_HEAP *H, NODE *node_to_be_decrease, int new_key){
NODE *parent_node;
if (H == NULL)
{
printf("n FIbonacci heap not created ");
return;
}
if (node_to_be_decrease == NULL)
{
printf("Node is not in the heap");
}

else
{
if (node_to_be_decrease->key < new_key)
{
printf("n Invalid new key for decrease key operation n ");
}
else
{
node_to_be_decrease->key = new_key;
parent_node = node_to_be_decrease->parent;
if ((parent_node != NULL) && (node_to_be_decrease->key < parent_node->key))
{
printf("n cut called");
cut(H, node_to_be_decrease, parent_node);
}
if (node_to_be_decrease->key < H->min->key)
{
H->min = node_to_be_decrease;
}
}
}
}

void *find_node(FIB_HEAP *H, NODE *n, int key, int new_key){
NODE *find_use = n;
NODE *f = NULL;
find_use->visited = true;
if (find_use->key == key)
{
find_use->visited = false;
f = find_use;
decrease_key(H, f, new_key);
}
if (find_use->child != NULL)
{
find_node(H, find_use->child, key, new_key);
}
if ((find_use->right_sibling->visited != true))
{
find_node(H, find_use->right_sibling, key, new_key);
}

find_use->visited = false;
}

FIB_HEAP *insertion_procedure(){
FIB_HEAP *temp;
int no_of_nodes, ele, i;
NODE *new_node;
temp = (FIB_HEAP *)malloc(sizeof(FIB_HEAP));
temp = NULL;
if (temp == NULL)
{
temp = make_fib_heap();
}
printf(" n enter number of nodes to be insert = ");
scanf("%d", &no_of_nodes);
for (i = 1; i <= no_of_nodes; i++)
{
printf("n node %d and its key value = ", i);
scanf("%d", &ele);
insertion(temp, new_node, ele);
}
return temp;
}
void Delete_Node(FIB_HEAP *H, int dec_key){
NODE *p = NULL;
find_node(H, H->min, dec_key, -5000);
p = extract_min(H);
if (p != NULL)
printf("n Node deleted");
else
printf("n Node not deleted:some error");
}

int main(int argc, char **argv){
NODE *new_node, *min_node, *extracted_min, *node_to_be_decrease, *find_use;
FIB_HEAP *heap, *h1, *h2;
int operation_no, new_key, dec_key, ele, i, no_of_nodes;
heap = (FIB_HEAP *)malloc(sizeof(FIB_HEAP));
heap = NULL;
while (1)
{

printf(" n choose below operations n 1. Create Fibonacci heap n 2. Insert nodes into fibonacci heap n 3. Find min n 4. Union n 5. Extract min n 6. Decrease key n 7.Delete node n 8. print heap n 9. exit n enter operation_no = ");
scanf("%d", &operation_no);

switch (operation_no)
{
case 1:
heap = make_fib_heap();
break;

case 2:
if (heap == NULL)
{
heap = make_fib_heap();
}
printf(" enter number of nodes to be insert = ");
scanf("%d", &no_of_nodes);
for (i = 1; i <= no_of_nodes; i++)
{
printf("n node %d and its key value = ", i);
scanf("%d", &ele);
insertion(heap, new_node, ele);
}
break;

case 3:
min_node = find_min_node(heap);
if (min_node == NULL)
printf("No minimum value");
else
printf("n min value = %d", min_node->key);
break;

case 4:
if (heap == NULL)
{
printf("n no FIbonacci heap is created please create fibonacci heap n ");
break;
}
h1 = insertion_procedure();
heap = unionHeap(heap, h1);
printf("Unified Heap:n");
new_print_heap(heap->min);
break;

case 5:
if (heap == NULL)
printf("Fibonacci heap is empty");
else
{
extracted_min = extract_min(heap);
printf("n min value = %d", extracted_min->key);
printf("n Updated heap: n");
new_print_heap(heap->min);
}
break;

case 6:
if (heap == NULL)
printf("Fibonacci heap is empty");
else
{
printf(" n node to be decreased = ");
scanf("%d", &dec_key);
printf(" n enter the new key = ");
scanf("%d", &new_key);
find_use = heap->min;
find_node(heap, find_use, dec_key, new_key);
printf("n Key decreased- Corresponding heap:n");
new_print_heap(heap->min);
}
break;
case 7:
if (heap == NULL)
printf("Fibonacci heap is empty");
else
{
printf(" n Enter node key to be deleted = ");
scanf("%d", &dec_key);
Delete_Node(heap, dec_key);
printf("n Node Deleted- Corresponding heap:n");
new_print_heap(heap->min);
break;
}
case 8:
new_print_heap(heap->min);
break;

case 9:
free(new_node);
free(heap);
exit(0);

default:
printf("Invalid choice ");
}
}
}``````

## Complexities

 Decrease Key O(1) Delete Node O(log n)

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