Chapter Binomial Heaps. We will study another heap structure called, the binomial heap. The binomial heap allows for efficient union, which can not be done. Property. Height of complete binary tree with n nodes is ⎣log2 n⎦. . Theorem. In an explicit binary heap with n nodes, the operations INSERT. In computer science, a binomial heap is a heap similar to a binary heap but also supports quick .. "A data structure for manipulating priority queues" (PDF).

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Binomial Heaps. This chapter and Chapter 20 present data structures known as mergeable heaps, which support the following five operations. MAKE-HEAP(). binomial trees of orders 0, 1, 2, , k – 1 have 20, 21, 22, , 2k-1 nodes, then then number of nodes in an order-k binomial tree is. 20 + 21 + + 2k-1 + 1 = 2k . Binomial Heaps. CLRS, Chapters 6, Thanks to Kevin Wayne for the slides. 2. Priority Queues. (heaps). Supports the following operations. ▫ Insert element x.

Example of a binomial heap containing 13 nodes with distinct keys. The heap consists of three binomial trees with orders 0, 2, and 3. Implementation[ edit ] Because no operation requires random access to the root nodes of the binomial trees, the roots of the binomial trees can be stored in a linked list , ordered by increasing order of the tree. Merge[ edit ] To merge two binomial trees of the same order, first compare the root key. The result is a tree of order 3.

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.

Structure of a binomial heap A binomial heap is implemented as a set of binomial trees that satisfy the binomial heap properties: Each binomial tree in a heap obeys the minimum-heap property: the key of a node is greater than or equal to the key of its parent.

There can only be either one or zero binomial trees for each order, including zero order. The first property ensures that the root of each binomial tree contains the smallest key in the tree, which applies to the entire heap. In fact, the number and orders of these trees are uniquely determined by the number of elements n: each binomial tree corresponds to digit one in the binary representation of number n.

For example number 13 is in binary, FIGURE and thus a binomial heap with 13 elements will consist of three binomial trees of orders 3, 2, and 0 Implementation Because no operation requires random access to the root nodes of the binomial trees, the roots of the binomial trees can be stored in a linked list, ordered by increasing order of the tree. To merge two binomial trees of the same order, firstcompare the root key. The result is a tree Merge As mentioned above, the simplest and most important operation is the merging of two binomial trees of the same order within two binomial heaps.

Due to the structure of binomial trees, they can be merged trivially.

As their root node is the smallest element within the tree, by comparing the two keys, the smaller of them is the minimum key, and becomes the new root node.

Then the other tree become a subtree of the combined tree. This operation is basic to the complete merging of two binomial heaps. The lists of roots of both heaps are traversed simultaneously, similarly as in the merge algorithm. If only one of the heaps contains a tree of order j, this tree is moved to the merged heap.

In the course of the algorithm, we need to examine at most three trees of any order two from the two heaps we merge and one composed of two smaller trees.

Because each binomial tree in a binomial heap corresponds to a bit in the binary representation of its size, there is an analogy between the merging of two heaps and the binary addition of the sizes of the two heaps, from right-to-left. Whenever a carry occurs during addition, this corresponds to a merging of two binomial trees during the merge.

Each tree has order at most log n and therefore the running time is O log n. Due to the merge, insert takes O log n time, however it has an amortized time of O 1 i. Find minimum To find the minimum element of the heap, find the minimum among the roots of the binomial trees.