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# Difference between binary heap, binomial heap, and Fibonacci heap

Last Updated on June 9, 2023 by Mayank Dham

A heap is a specialized tree-based data structure that satisfies the heap property. It is commonly used to implement priority queues and efficient sorting algorithms like heapsort. The heap property states that for a max heap, the value of each node is greater than or equal to the values of its children, while for a min heap, the value of each node is less than or equal to the values of its children.

### Binary Heap:

Binary heap is a complete tree i.e. All the levels of the tree are completely filled except the leaf nodes or last level and have all keys on the left. Due to this property, Binary heaps are suitable to be stored in an array.
A binary heap is either a min-heap or a max heap.

In the min-heap, the value at the root node is smaller than all the nodes present in the binary heap, and in the max heap, the value at the root node is greater than all the nodes present in the binary heap.

### Binomial Heap:

Binomial heap is an extension of binary heaps. Binomial heaps provide faster operations.
A binomial heap is a collection of binomial trees.

So what is a binomial tree?

A binomial tree of order 0 has 1 node only. We can construct a binomial tree of k order by taking two binomial trees of order k-1 and making one tree as the leftmost child or other.

Some properties of order k binomial tree:

• The binomial tree has exactly 2k nodes.
• The binomial tree has a depth of k.
• The root of the binomial tree has degree k and children of the root node are also binomial trees.

A binomial heap is a set of binomial trees where each binomial tree follows the min-heap property.

### Fibonacci Heap:

Fibonacci heap is a collection of trees that follows the property of min heap or max heap. Fibonacci heaps can have any shape even if all trees can have a single node. In Fibonacci heaps, we’ll maintain a pointer to a minimum value i.e. root of the tree.
All tree roots are connected using a circular doubly linked. Therefore, we can access all of them using a single pointer.

Example:

This article tried to discuss the Difference between binary heap, binomial heap, and Fibonacci heap. Hope this blog helps you understand the concept. To practice problems on Heap you can check out MYCODE | Competitive Programming – Heaps.

## Difference Between Binomial Heap and Fibonacci Heap

Feature Binomial Heap Fibonacci Heap
Structure Collection of binomial trees Collection of min-heap-ordered trees
– Insertion O(log n) O(1) amortized
– Deletion O(log n) O(log n)
– Decrease-key O(log n) O(1) amortized
– Merging O(log n) O(1) amortized
Space Efficiency Requires more space due to binomial trees Requires more space due to additional properties
Key Advantage Efficient decrease-key operations Extremely fast insertion and merging operations
Applications Priority queues, graph algorithms Advanced algorithms, specifically decrease-key ops

## Difference Between the Fibonacci and Binomial Heap

Feature Fibonacci Heap Binomial Heap
Structure Collection of min-heap-ordered trees Collection of binomial trees
– Insertion O(1) amortized O(log n)
– Deletion O(log n) O(log n)
– Decrease-key O(1) amortized O(log n)
– Merging O(1) amortized O(log n)
Space Efficiency Requires more space due to additional properties Requires less space due to simpler structure
Key Advantage Extremely fast insertion and merging operations Efficient decrease-key operations
Applications Advanced algorithms, specifically decrease-key operations Priority queues, graph algorithms
Potential Trade-offs Increased space overhead, larger constant factors Slightly slower operations compared to Fibonacci heaps
Notable Algorithms Dijkstra’s algorithm, Prim’s algorithm Heapsort

Conclusion
Binary heaps, binomial heaps, and Fibonacci heaps are all variations of heap data structures, each with its own characteristics and performance trade-offs. Binary heaps are simple and commonly used, providing efficient insertion, deletion, and retrieval operations with logarithmic time complexity. Binomial heaps offer improved performance for certain operations, such as faster insertion and merging, while still maintaining efficient deletion and retrieval. Fibonacci heaps, the most advanced among them, excel in insertion and merging operations with constant amortized time complexity, but may have slightly slower deletion and retrieval operations. The choice of heap structure depends on the specific requirements of the application, with binary heaps being a solid choice for most scenarios, while binomial heaps and Fibonacci heaps offer further optimizations for specific use cases that demand more efficient operations.

Q1. What is the main difference between a binary heap, a binomial heap, and a Fibonacci heap?
The main difference lies in their underlying structure and performance characteristics. Binary heaps are implemented as complete binary trees and offer efficient operations with logarithmic time complexity. Binomial heaps are composed of binomial trees and provide improved performance for certain operations like merging and insertion. Fibonacci heaps, on the other hand, are made up of min-heap-ordered trees and have exceptionally fast insertion and merging operations with constant amortized time complexity.

Q2. Which heap is the most efficient for insertion and merging operations?
Fibonacci heaps are the most efficient for insertion and merging operations. They have constant amortized time complexity for these operations, meaning their performance remains constant even as the size of the heap grows. Binomial heaps also have efficient insertion and merging, but Fibonacci heaps offer further optimization.

Q3. Which heap is commonly used for implementing priority queues?
Binary heaps are commonly used for implementing priority queues due to their simplicity and efficient operations. They provide logarithmic time complexity for insertion, deletion, and retrieval of the maximum or minimum value. Binomial heaps and Fibonacci heaps can also be used for priority queues, with binomial heaps offering additional advantages such as efficient merging and Fibonacci heaps excelling in insertion and merging.

Q4. Are binomial heaps or Fibonacci heaps more space-efficient than binary heaps?
Both binomial heaps and Fibonacci heaps require more space than binary heaps due to their additional structural properties. Binomial heaps require space for multiple binomial trees, while Fibonacci heaps store multiple trees as well. However, the space overhead of binomial and Fibonacci heaps is generally reasonable and does not significantly impact their performance.

Q5. Which heap is recommended for applications that frequently require decrease-key operations?
Binomial heaps are often recommended for applications that frequently require decrease-key operations. They offer efficient decrease-key operations with logarithmic time complexity, making them suitable for scenarios where priorities of elements in the heap need to be modified frequently. Fibonacci heaps also support efficient decrease-key operations, but they come with additional complexity and may not always provide significant advantages unless decrease-key operations dominate the overall workload.