What's the difference between a binary heap and a Fibonacci heap?

A binary heap is a simple data structure, while a Fibonacci heap is a more complex, optimised version of a heap.

A binary heap is a complete binary tree, which can be either a max-heap or a min-heap. In a max-heap, the parent node is always larger than or equal to its child nodes, while in a min-heap, the parent node is always smaller than or equal to its child nodes. Binary heaps are used in various algorithms, including Dijkstra's algorithm and the heap sort algorithm. They are simple to understand and implement, but they can be inefficient in some cases.

On the other hand, a Fibonacci heap is a more advanced data structure that was developed to improve the efficiency of certain operations in comparison to binary heaps. Fibonacci heaps are not binary trees, but rather a collection of trees with a more relaxed structure. This allows for more efficient merging of heaps, which can be a significant advantage in algorithms that frequently combine heaps, such as Prim's algorithm for finding the minimum spanning tree.

The main advantage of Fibonacci heaps over binary heaps is their efficiency in performing decrease-key and delete operations. In a binary heap, these operations can take O(log n) time, where n is the number of nodes in the heap. However, in a Fibonacci heap, the decrease-key operation can be done in constant amortised time, and the delete operation can be done in O(log n) amortised time. This makes Fibonacci heaps more efficient for algorithms that frequently perform these operations.

However, Fibonacci heaps are more complex to implement than binary heaps, and their constant factors are larger. This means that for small inputs, or for applications where decrease-key and delete operations are not common, a binary heap may be more efficient. Furthermore, the structure of a Fibonacci heap is more difficult to understand and analyse, which can make it a less attractive choice for some applications.