It contains a collection of binomial trees. Before diving further into the binomial heap, let us first try to understand what a binomial tree is. The recursive definition is given below. The binomial tree of order 0 i. Figure 1 illustrates the recursive definition. The binomial tree of order 1 is two binomial of trees of order 0 linked together.

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Binomial Heap is an extension of Binary Heap that provides faster union or merge operation together with other operations provided by Binary Heap. 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 as leftmost child or other.

A Binomial Tree of order k has following properties. And there can be at most one Binomial Tree of any degree. It is a collection of 3 Binomial Trees of orders 0, 2 and 3 from left to right. It is a collection of 2 Binomial Trees of orders 2 and 3 from left to right. 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 3 set bits in the binary representation of n , hence 3 Binomial Trees. We can also relate the degree of these Binomial Trees with positions of set bits. Operations of Binomial Heap: The main operation in Binomial Heap is 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. This implementation requires O Logn time. It can be optimized to O 1 by maintaining a pointer to minimum key root. 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. We stop when we either reach a node whose parent has a smaller key or we hit the root node.

Time complexity of decreaseKey is O Logn. In the following diagram, figure b shows the result after merging. 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 following 4 cases when we traverse the list of roots.

In following 3 cases orders of x and next-x are same. 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 extractMin and delete.

The idea is to represent Binomial Trees as the leftmost child and right-sibling representation, i.

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## Binomial Heap

Because the number of children for each node is variable, it does not work well for each node to have separate links to each of its children, as would be common in a binary tree ; instead, it is possible to implement this tree using links from each node to its highest-order child in the tree, and to its sibling of the next smaller order than it. These sibling pointers can be interpreted as the next pointers in a linked list of the children of each node, but with the opposite order from the linked list of roots: from largest to smallest order, rather than vice versa. This representation allows two trees of the same order to be linked together, making a tree of the next larger order, in constant time. The result is a tree of order 3. The operation of merging two heaps is used as a subroutine in most other operations.

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## Binomial Heaps (With C++ Implementation)

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