### amortized-analysis

#### Why we do Amortized Analysis for Fibonacci Heap?

```In Fibonacci heap for all operations analysis are Amortized in nature. Why cant we have normal analysis as in case of Binomial Heap.
```
```In a binomial heap, each operation is guaranteed to run with a certain worst-case performance. An insertion will never take more than time O(log n), a merge will never take more than time O(log n + log m), etc. Therefore, when analyzing the efficiency of a binomial heap, it's common to use a more traditional algorithmic analysis.
Now, that said, there are several properties of binomial heaps that only become apparent when doing an amortized analysis. For example, what's the cost of doing n consecutive insertions into a binomial heap, assuming the heap is initially empty? You can show that, in this case, the amortized cost of an insertion is O(1), meaning that the total cost of doing n insertions is O(n). In that sense, using an amortized analysis on top of a traditional analysis reveals more insights about the data structure than might initially arise from a more conservative worst-case analysis.
In some sense, Fibonacci heaps are best analyzed in an amortized sense because even though the worst-case bounds on many of the operations really aren't that great (for example, a delete-min or decrease-key can take time Θ(n) in the worst case), across any series of operations the Fibonacci heap has excellent amortized performance. Even though an individual delete-min might take Θ(n) time, it's never possible for a series of m delete-mins to take more than Θ(m log n) time.
In another sense, though, Fibonacci heaps were specifically designed to be efficient in an amortized sense rather than a worst-case sense. They were initially invented to speed up Dijkstra's and Prim's algorithms, where all that mattered were the total cost of doing m decrease-keys and n deletes on an n-node heap, and since that was the design goal, the designers made no attempt to make the Fibonacci heap efficient in the worst case.```

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