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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.

## 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

- Select the node to be decreased,
`x`, and change its value to the new value`k`. - If the parent of
`x`,`y`, 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. - If the key of
`x`is smaller than the key of min, then mark`x`as min.

### Cut

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

### Cascading-Cut

- If the parent of
`y`is not null then follow the following steps. - If
`y`is unmarked, then mark`y`. - 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.

- Decrease the value 46 to 15.

**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

- Decrease the value 35 to 5.

- Cut part: Since
`26 ≠ nill`

and`5<its parent`

, cut it and add it to the root list.

- 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.**Cascading-Cut(24)**:

Since the 24 is also marked, again call`Cut(24)`

and`Cascading-Cut(7)`

. These operations result in the tree below. - 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.

- Let
`k`be the node to be deleted. - Apply decrease-key operation to decrease the value of
`k`to the lowest possible value (i.e. -∞). - 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 cascading_cut(FIB_HEAP *H, 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;
fib_heap_link(H, y, 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;
}
void cascading_cut(FIB_HEAP *H, NODE *parent_node){
NODE *aux;
aux = parent_node->parent;
if (aux != NULL)
{
if (parent_node->mark == false)
{
parent_node->mark = true;
}
else
{
cut(H, parent_node, aux);
cascading_cut(H, 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);
printf("n cascading cut called");
cascading_cut(H, 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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