# Fibonacci Heap | Set 1 (Introduction)

• Difficulty Level : Medium
• Last Updated : 15 Dec, 2022

Heaps are mainly used for implementing priority queue. We have discussed the below heaps in previous posts.

In terms of Time Complexity, Fibonacci Heap beats both Binary and Binomial Heap.

Below are amortized time complexities of the Fibonacci Heap

```1) Find Min:      Θ(1)     [Same as  Binary but not Binomial since binomial has o(log n)]
2) Delete Min:    O(Log n) [Θ(Log n) in both Binary and Binomial]
3) Insert:        Θ(1)     [Θ(Log n) in Binary and Θ(1) in Binomial]
4) Decrease-Key:  Θ(1)     [Θ(Log n) in both Binary and Binomial]
5) Merge:         Θ(1)     [Θ(m Log n) or Θ(m+n) in Binary and
Θ(Log n) in Binomial]```

Like Binomial Heap, Fibonacci Heap is a collection of trees with min-heap or max-heap properties. In Fibonacci Heap, trees can have any shape even if all trees can be single nodes (This is unlike Binomial Heap where every tree has to be a Binomial Tree).

Below is an example Fibonacci Heap taken from here

Fibonacci Heap maintains a pointer to the minimum value (which is the root of a tree). All tree roots are connected using a circular doubly linked list, so all of them can be accessed using a single ‘min’ pointer.

The main idea is to execute operations in a “lazy” way. For example merge operation simply links two heaps, insert operation simply adds a new tree with a single node. The operation extract minimum is the most complicated operation. It does delay the work of consolidating trees. This makes delete also complicated as delete first decreases the key to minus infinite, then calls extract minimum.

Below are some interesting facts about Fibonacci Heap

1. The reduced time complexity of Decrease-Key has importance in Dijkstra and Prim algorithms. With Binary Heap, the time complexity of these algorithms is O(VLogV + ELogV). If Fibonacci Heap is used, then time complexity is improved to O(VLogV + E)
2. Although Fibonacci Heap looks promising time complexity-wise, it has been found slow in practice as hidden constants are high (Source Wiki).
3. Fibonacci heaps is mainly called so because Fibonacci numbers are used in the running time analysis. Also, every node in Fibonacci Heap has a degree at most O(log n) and the size of a subtree rooted in a node of degree k is at least Fk+2, where Fk is the kth Fibonacci number.

We will soon be discussing Fibonacci Heap operations in detail.