Binomial Heap

The main application of Binary Heap is as implement a priority queue. Binomial Heap is an extension of Binary Heap that provides faster union or merge operation with other operations provided by Binary Heap. 

A Binomial Heap is a collection of Binomial Trees
What is a Binomial Tree?

A Binomial Tree of order 0 has 1 node. A Binomial Tree of order k can be constructed by taking two binomial trees of order k-1 and making one the leftmost child of the other. 

A Binomial Tree of order k the has following properties. 

• It has exactly 2^k nodes. 
• It has depth as k. 
• There are exactly kaiC_i nodes at depth i for i = 0, 1, . . . , k. 
• The root has degree k and children of the root are themselves Binomial Trees with order k-1, k-2,.. 0 from left to right. 

k = 0 (Single Node)

 o

k = 1 (2 nodes) 
[We take two k = 0 order Binomial Trees, and
make one as a child of other]
  o
 /  
o     

k = 2 (4 nodes)
[We take two k = 1 order Binomial Trees, and
make one as a child of other]
     o
   /   \
  o     o
 /       
o        

k = 3 (8 nodes)
[We take two k = 2 order Binomial Trees, and
make one as a child of other]
        o   
     /  | \ 
    o   o  o
   / \  | 
  o   o o   
 /              
o         

The following diagram is referred to form the 2nd Edition of the CLRS book. 

Binomial Heap:

A Binomial Heap is a set of Binomial Trees where each Binomial Tree follows the Min Heap property. And there can be at most one Binomial Tree of any degree. 
Examples Binomial Heap:

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

A Binomial Heap with 12 nodes. It is a collection of 2 
Binomial Trees of orders 2 and 3 from left to right. 

Python program to implement Binomial heap:

Python3

import math

class Node:
    def __init__(self, value):
        self.value = value
        self.parent = None
        self.children = []
        self.degree = 0
        self.marked = False

class BinomialHeap:
    def __init__(self):
        self.trees = []
        self.min_node = None
        self.count = 0

    def is_empty(self):
        return self.min_node is None

    def insert(self, value):
        node = Node(value)
        self.merge(BinomialHeap(node))

    def get_min(self):
        return self.min_node.value

    def extract_min(self):
        min_node = self.min_node
        self.trees.remove(min_node)
        self.merge(BinomialHeap(*min_node.children))
        self._find_min()
        self.count -= 1
        return min_node.value

    def merge(self, other_heap):
        self.trees.extend(other_heap.trees)
        self.count += other_heap.count
        self._find_min()

    def _find_min(self):
        self.min_node = None
        for tree in self.trees:
            if self.min_node is None or tree.value < self.min_node.value:
                self.min_node = tree

    def decrease_key(self, node, new_value):
        if new_value > node.value:
            raise ValueError("New value is greater than current value")
        node.value = new_value
        self._bubble_up(node)

    def delete(self, node):
        self.decrease_key(node, float('-inf'))
        self.extract_min()

    def _bubble_up(self, node):
        parent = node.parent
        while parent is not None and node.value < parent.value:
            node.value, parent.value = parent.value, node.value
            node, parent = parent, node

    def _link(self, tree1, tree2):
        if tree1.value > tree2.value:
            tree1, tree2 = tree2, tree1
        tree2.parent = tree1
        tree1.children.append(tree2)
        tree1.degree += 1

    def _consolidate(self):
        max_degree = int(math.log(self.count, 2))
        degree_to_tree = [None] * (max_degree + 1)

        while self.trees:
            current = self.trees.pop(0)
            degree = current.degree
            while degree_to_tree[degree] is not None:
                other = degree_to_tree[degree]
                degree_to_tree[degree] = None
                if current.value < other.value:
                    self._link(current, other)
                else:
                    self._link(other, current)
                degree += 1
            degree_to_tree[degree] = current

        self.min_node = None
        self.trees = [tree for tree in degree_to_tree if tree is not None]

    def __len__(self):
        return self.count

Javascript

// Javascript program for the above approach

class Node {
  constructor(value) {
    this.value = value;
    this.parent = null;
    this.children = [];
    this.degree = 0;
    this.marked = false;
  }
}

class BinomialHeap {
  constructor() {
    this.trees = [];
    this.min_node = null;
    this.count = 0;
  }

  is_empty() {
    return this.min_node === null;
  }

  insert(value) {
    let node = new Node(value);
    this.merge(new BinomialHeap(node));
  }

  get_min() {
    return this.min_node.value;
  }

  extract_min() {
    let min_node = this.min_node;
    this.trees.splice(this.trees.indexOf(min_node), 1);
    this.merge(new BinomialHeap(...min_node.children));
    this._find_min();
    this.count -= 1;
    return min_node.value;
  }

  merge(other_heap) {
    this.trees = [...this.trees, ...other_heap.trees];
    this.count += other_heap.count;
    this._find_min();
  }

  _find_min() {
    this.min_node = null;
    for (let tree of this.trees) {
      if (this.min_node === null || tree.value < this.min_node.value) {
        this.min_node = tree;
      }
    }
  }

  decrease_key(node, new_value) {
    if (new_value > node.value) {
      throw new Error("New value is greater than current value");
    }
    node.value = new_value;
    this._bubble_up(node);
  }

  delete(node) {
    this.decrease_key(node, -Infinity);
    this.extract_min();
  }

  _bubble_up(node) {
    let parent = node.parent;
    while (parent !== null && node.value < parent.value) {
      [node.value, parent.value] = [parent.value, node.value];
      [node, parent] = [parent, node];
    }
  }

  _link(tree1, tree2) {
    if (tree1.value > tree2.value) {
      [tree1, tree2] = [tree2, tree1];
    }
    tree2.parent = tree1;
    tree1.children.push(tree2);
    tree1.degree += 1;
  }

  _consolidate() {
    let max_degree = Math.floor(Math.log2(this.count)) + 1;
    let degree_to_tree = new Array(max_degree + 1).fill(null);

    while (this.trees.length) {
      let current = this.trees.shift();
      let degree = current.degree;
      while (degree_to_tree[degree] !== null) {
        let other = degree_to_tree[degree];
        degree_to_tree[degree] = null;
        if (current.value < other.value) {
          this._link(current, other);
        } else {
          this._link(other, current);
        }
        degree += 1;
      }
      degree_to_tree[degree] = current;
    }

    this.min_node = null;
    this.trees = degree_to_tree.filter((tree) => tree !== null);
  }

  get length() {
    return this.count;
  }
}

// This code is contributed by sdeadityasharma

Binary Representation of a number and Binomial Heaps
A Binomial Heap with n nodes has the number of Binomial Trees equal to the number of set bits in the binary representation of n. For example, let n be 13, there are 3 set bits in the binary representation of n (00001101), hence 3 Binomial Trees. We can also relate the degree of these Binomial Trees with positions of set bits. With this relation, we can conclude that there are O(Logn) Binomial Trees in a Binomial Heap with ‘n’ nodes. 
Operations of Binomial Heap:
The main operation in Binomial Heap is a union(), all other operations mainly use this operation. The union() operation is to combine two Binomial Heaps into one. Let us first discuss other operations, we will discuss union later.

1. insert(H, k): Inserts a key ‘k’ to Binomial Heap ‘H’. This operation first creates a Binomial Heap with a single key ‘k’, then calls union on H and the new Binomial heap. 
2. getting(H): A simple way to get in() is to traverse the list of the roots 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 the minimum key root. 
3. extracting(H): This operation also uses a 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. 
4. delete(H): Like Binary Heap, the delete operation first reduces the key to minus infinite, then calls extracting(). 
5. decrease key(H): decrease key() is also similar to Binary Heap. We compare the decreased key with its parent and if the parent’s key is more, we swap keys and recur for the parent. We stop when we either reach a node whose parent has a smaller key or we hit the root node. The time complexity of the decrease key() is O(Logn). 
   Union operation in Binomial Heap:
   Given two Binomial Heaps H1 and H2, union(H1, H2) creates a single Binomial Heap. 
6. The first step is to simply merge the two Heaps in non-decreasing order of degrees. In the following diagram, figure(b) shows the result after merging. 
7. After the simple merge, we need to make sure that there is at most one Binomial Tree of any order. To do this, we need to combine Binomial Trees of the same order. We traverse the list of merged roots, we keep track of three-pointers, prev, x, and next-x. There can be the following 4 cases when we traverse the list of roots. 
   —–Case 1: Orders of x and next-x are not the same, we simply move ahead. 
   In the following 3 cases, orders of x and next-x are the same. 
   —–Case 2: If the order of next-next-x is also the same, move ahead. 
   —–Case 3: If the key of x is smaller than or equal to the key of next-x, then make next-x a child of x by linking it with x. 
   —–Case 4: If the key of x is greater, then make x the child of next. 
   The following diagram is taken from the 2nd Edition of the CLRS book. 
   

Time Complexity Analysis:

Operations	Binary Heap	Binomial Heap	Fibonacci Heap
Procedure	Worst-case	Worst-case	Amortized
Making Heap	Θ(1)	Θ(1)	Θ(1)
Inserting a node	Θ(log(n))	O(log(n))	Θ(1)
Finding Minimum key	Θ(1)	O(log(n))	O(1)
Extract-Minimum key	Θ(log(n))	Θ(log(n))	O(log(n))
Union or merging	Θ(n)	O(log(n))	Θ(1)
Decreasing a Key	Θ(log(n))	Θ(log(n))	Θ(1)
Deleting a node	Θ(log(n))	Θ(log(n))	O(log(n))

How to represent Binomial Heap?
A Binomial Heap is a set of Binomial Trees. A Binomial Tree must be represented in a way that allows sequential access to all siblings, starting from the leftmost sibling (We need this in and extracting() and delete()). The idea is to represent Binomial Trees as the leftmost child and right-sibling representation, i.e., every node stores two pointers, one to the leftmost child and the other to the right sibling.  

Implementation of Binomial Heap