Why we do Amortized Analysis for Fibonacci Heap?

Question

In Fibonacci heap for all operations analysis are Amortized in nature. Why cant we have normal analysis as in case of Binomial Heap.

Answer 1

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.