Implementation of Binomial Heap | Set – 2 (delete() and decreseKey())
In previous post i.e. Set 1 we have discussed that implements these below functions:
- insert(H, k): Inserts a key ‘k’ to Binomial Heap ‘H’. This operation first creates a Binomial Heap with single key ‘k’, then calls union on H and the new Binomial heap.
- getMin(H): A simple way to getMin() is to traverse the list of root of Binomial Trees and return the minimum key. This implementation requires O(Logn) time. It can be optimized to O(1) by maintaining a pointer to minimum key root.
- extractMin(H): This operation also uses union(). We first call getMin() to find the minimum key Binomial Tree, then we remove the node and create a new Binomial Heap by connecting all subtrees of the removed minimum node. Finally we call union() on H and the newly created Binomial Heap. This operation requires O(Logn) time.
Examples:
12------------10--------------------20
/ \ / | \
15 50 70 50 40
| / | |
30 80 85 65
|
100
A Binomial Heap with 13 nodes. It is a collection of 3
Binomial Trees of orders 0, 2 and 3 from left to right.
10--------------------20
/ \ / | \
15 50 70 50 40
| / | |
30 80 85 65
|
100
In this post, below functions are implemented.
- delete(H): Like Binary Heap, delete operation first reduces the key to minus infinite, then calls extractMin().
- decreaseKey(H): decreaseKey() is also similar to Binary Heap. We compare the decreases key with it parent and if parent’s key is more, we swap keys and recur for parent. We stop when we either reach a node whose parent has smaller key or we hit the root node. Time complexity of decreaseKey() is O(Logn)
// C++ program for implementation of// Binomial Heap and Operations on it#include <bits/stdc++.h>using namespace std; // Structure of Nodestruct Node{ int val, degree; Node *parent, *child, *sibling;}; // Making root global to avoid one extra// parameter in all functions.Node *root = NULL; // link two heaps by making h1 a child// of h2.int binomialLink(Node *h1, Node *h2){ h1->parent = h2; h1->sibling = h2->child; h2->child = h1; h2->degree = h2->degree + 1;} // create a NodeNode *createNode(int n){ Node *new_node = new Node; new_node->val = n; new_node->parent = NULL; new_node->sibling = NULL; new_node->child = NULL; new_node->degree = 0; return new_node;} // This function merge two Binomial TreesNode *mergeBHeaps(Node *h1, Node *h2){ if (h1 == NULL) return h2; if (h2 == NULL) return h1; // define a Node Node *res = NULL; // check degree of both Node i.e. // which is greater or smaller if (h1->degree <= h2->degree) res = h1; else if (h1->degree > h2->degree) res = h2; // traverse till if any of heap gets empty while (h1 != NULL && h2 != NULL) { // if degree of h1 is smaller, increment h1 if (h1->degree < h2->degree) h1 = h1->sibling; // Link h1 with h2 in case of equal degree else if (h1->degree == h2->degree) { Node *sib = h1->sibling; h1->sibling = h2; h1 = sib; } // if h2 is greater else { Node *sib = h2->sibling; h2->sibling = h1; h2 = sib; } } return res;} // This function perform union operation on two// binomial heap i.e. h1 & h2Node *unionBHeaps(Node *h1, Node *h2){ if (h1 == NULL && h2 == NULL) return NULL; Node *res = mergeBHeaps(h1, h2); // Traverse the merged list and set // values according to the degree of // Nodes Node *prev = NULL, *curr = res, *next = curr->sibling; while (next != NULL) { if ((curr->degree != next->degree) || ((next->sibling != NULL) && (next->sibling)->degree == curr->degree)) { prev = curr; curr = next; } else { if (curr->val <= next->val) { curr->sibling = next->sibling; binomialLink(next, curr); } else { if (prev == NULL) res = next; else prev->sibling = next; binomialLink(curr, next); curr = next; } } next = curr->sibling; } return res;} // Function to insert a Nodevoid binomialHeapInsert(int x){ // Create a new node and do union of // this node with root root = unionBHeaps(root, createNode(x));} // Function to display the Nodesvoid display(Node *h){ while (h) { cout << h->val << " "; display(h->child); h = h->sibling; }} // Function to reverse a list// using recursion.int revertList(Node *h){ if (h->sibling != NULL) { revertList(h->sibling); (h->sibling)->sibling = h; } else root = h;} // Function to extract minimum valueNode *extractMinBHeap(Node *h){ if (h == NULL) return NULL; Node *min_node_prev = NULL; Node *min_node = h; // Find minimum value int min = h->val; Node *curr = h; while (curr->sibling != NULL) { if ((curr->sibling)->val < min) { min = (curr->sibling)->val; min_node_prev = curr; min_node = curr->sibling; } curr = curr->sibling; } // If there is a single Node if (min_node_prev == NULL && min_node->sibling == NULL) h = NULL; else if (min_node_prev == NULL) h = min_node->sibling; // Remove min node from list else min_node_prev->sibling = min_node->sibling; // Set root (which is global) as children // list of min node if (min_node->child != NULL) { revertList(min_node->child); (min_node->child)->sibling = NULL; } // Do union of root h and children return unionBHeaps(h, root);} // Function to search for an elementNode *findNode(Node *h, int val){ if (h == NULL) return NULL; // check if key is equal to the root's data if (h->val == val) return h; // Recur for child Node *res = findNode(h->child, val); if (res != NULL) return res; return findNode(h->sibling, val);} // Function to decrease the value of old_val// to new_valvoid decreaseKeyBHeap(Node *H, int old_val, int new_val){ // First check element present or not Node *node = findNode(H, old_val); // return if Node is not present if (node == NULL) return; // Reduce the value to the minimum node->val = new_val; Node *parent = node->parent; // Update the heap according to reduced value while (parent != NULL && node->val < parent->val) { swap(node->val, parent->val); node = parent; parent = parent->parent; }} // Function to delete an elementNode *binomialHeapDelete(Node *h, int val){ // Check if heap is empty or not if (h == NULL) return NULL; // Reduce the value of element to minimum decreaseKeyBHeap(h, val, INT_MIN); // Delete the minimum element from heap return extractMinBHeap(h);} // Driver codeint main(){ // Note that root is global binomialHeapInsert(10); binomialHeapInsert(20); binomialHeapInsert(30); binomialHeapInsert(40); binomialHeapInsert(50); cout << "The heap is:\n"; display(root); // Delete a particular element from heap root = binomialHeapDelete(root, 10); cout << "\nAfter deleing 10, the heap is:\n"; display(root); return 0;} |
Output:
The heap is: 50 10 30 40 20 After deleing 10, the heap is: 20 30 40 50
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