A binomial heap is typically represented as a collection of binomial trees with specific properties. The primary data structure used for this representation is a linked list of binomial trees. Each binomial tree in the linked list follows the binomial tree properties, and the entire linked list of binomial trees forms the binomial heap. Here's a high-level representation:
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Linked List of Binomial Trees:
- The binomial heap is represented as a linked list, where each node in the list corresponds to a binomial tree.
- The linked list is ordered by increasing order of binomial trees.
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Binomial Tree Structure:
- Each binomial tree in the linked list follows the structure of a binomial tree with a root node and its children.
- The order of a binomial tree determines the number of nodes it contains and its position in the linked list.
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Heap Order Property:
- The linked list satisfies the heap order property, ensuring that the root of each binomial tree has a key value less than or equal to the key values of its children.
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Minimum Pointer:
- The binomial heap often includes a pointer to the minimum element in the heap. This pointer allows for constant-time access to the minimum element.
Here's a simple textual representation to illustrate how a binomial heap might be represented:
Binomial Heap:
B_0 B_1 B_2
| | |
4 7 12
| |
15 24
|
27
In this example, the binomial heap consists of binomial trees B_0, B_1, and B_2, arranged in increasing order of their orders. Each binomial tree follows the properties of a binomial tree, with nodes containing key values. The minimum pointer would point to the root of the binomial tree with the smallest key value.
The actual implementation details might involve additional structures to facilitate efficient operations such as insertion, union, and extraction of the minimum element. Each binomial tree can be implemented as a separate data structure, and the linked list of binomial trees forms the overall structure of the binomial heap.
