Data structures in typescript 22 fibonacci heap introduction

okay, let's dive into fibonacci heaps in typescript. this is a more advanced data structure, so we'll break it down step-by-step with explanations and code examples.

**introduction: what is a fibonacci heap?**

a fibonacci heap is a data structure that implements a heap priority queue. it's particularly efficient for operations involving *decrease-key* and *merge* (union) compared to binary heaps or binomial heaps.

**key features and properties:**

1. **heap property:** similar to other heap structures, a fibonacci heap maintains the *min-heap property* (or max-heap property, depending on the implementation). this means that the key of a node is less than or equal to (in a min-heap) the key of its children.

2. **structure:** a fibonacci heap is a collection of *trees*. unlike binomial heaps (which are made of binomial trees), fibonacci heaps don't have any strict structural requirements. the trees can vary significantly in shape and size.

3. **root list:** the roots of all trees in the heap are maintained in a circular, doubly linked list called the *root list*. this list is important for efficient merging.

4. **marking nodes:** each node has a "mark" bit. this mark bit is used to optimize the `decreasekey` operation. a marked node has lost one child since it was made a child of another node. this is crucial for the cascading cut process.

5. **potential function:** the analysis of fibonacci heaps relies heavily on a *potential function*, which is used to amortize the cost of operations. the potential function is defined as:

`φ(h) = t(h) + 2m(h)`

where:
* `t(h)` is the number of trees in the root list of heap `h`.
* `m(h)` is the number of marked nodes in heap `h`.

6. **degree:** the degree of a node is the number of children it has. the degree of the fibonacci heap is the max degree of any node in the rootlist.

**why fibonacci heaps?**

fibonacci heaps excel when you need to perform a large number of `decrease-key` operations ( ...

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Data Structures
TypeScript
Fibonacci Heap
Introduction
Algorithm
Data Management
Performance Analysis
Memory Efficiency
Graph Theory
Priority Queue
Complexity Analysis
Implementation
Optimization
Node Structure
Heap Operations