1
$\begingroup$

The following excerpts are from the section Fibonacci Heap from the text Introduction to Algorithms by Cormen et. al


The potential function for the Fibonacci Heaps $H$ is defined as follows:

$$\Phi(H)=t(H)+2m(H)$$

where $t(H)$ is the number of trees in the root list of the heap $H$ and $m(H)$ is the number of marked nodes in the heap.


The question here asked why the potential function $\Phi(H)$ is chosen as above and the answer given is quite appropriate : "because it works". My question is "detailed intuition of how the potential function works"

Before diving into the Fibonacci Heap operations the authors try to convince us about the essence of Fibonacci Heaps as follows:

The key idea in the mergeable-heap operations on Fibonacci heaps is to delay work as long as possible.

There is a performance trade-off among implementations of the various operations.($\color{green}{\text{ I do not get why }}$)

If the number of trees in a Fibonacci heap is small, then during an $\text{Extract-Min}$ operation we can quickly determine which of the remaining nodes becomes the new minimum node ($\color{blue}{\text{why?}}$).

However, as we saw with binomial heaps, we pay a price for ensuring that the number of trees is small: it can take up to $\Omega (\lg n)$ time to insert a node into a binomial heap or to unite two binomial heaps. As we shall see, we do not attempt to consolidate trees in a Fibonacci heap when we insert a new node or unite two heaps. We save the consolidation for the $\text{Extract-Min}$ operation, which is when we really need to find the new minimum node.


Now the problem which I am facing with the text is that they dive into proving the amortized cost mathematically using the potential method without going into the vivid intuition of the how or when the "credits" are stored as potential in the heap data structure and when it is actually used up. Moreover in most of the places what is used is "asymptotic" analysis instead of actual mathematical calculations, so it is not quite possible to conjecture whether the constant in $O(1)$ for the amortized cost ( $\widehat{c_i}$ ) is greater or less than the constant in $O(1)$ for the actual cost ($c_i$) for an operation.

[The answer here talks about the possible charging and discharging of the potential function (and also possibly briefly deals with the logic behind the highlighted portion in the above blockquote) but it is after all an answer to a different question. I wish to have a greater details of the same]


$$\begin{array}{|c|c|c|} \hline \text{Sl no.}&\text{Operation}&\widehat{c_i}&c_i&\text{Method of cal. of $\widehat{c_i}$}&\text{Cal. Steps}&\text{Intuition}\\ \hline 1&\text{Make-Fib-Heap}&O(1)&O(1)&\text{Asymptotic}&\Delta\Phi=0\text{ ; $\widehat{c_i}=c_i=O(1)$} &\text{None}\\ \hline 2&\text{Fib-Heap-Insert}&O(1)&O(1)&\text{Asymptotic}&\Delta\Phi=1 \text{ ; $\widehat{c_i}=c_i=O(1)+1=O(1)$} &\text{None}\\ \hline 3&\text{Fib-Heap-Min}&O(1)&O(1)&\text{Asymptotic}&\Delta\Phi=0;\text{ ; $\widehat{c_i}=c_i=O(1)$} &\text{None}\\ \hline 4&\text{Fib-Heap-Union}&O(1)&O(1)&\text{Asymptotic}&\Delta\Phi=0;\text{ ; $\widehat{c_i}=c_i=O(1)$} &\text{None}\\ \hline 5&\text{Fib-Extract-Min}&O(D(n))&O(D(n)+t(n))&\text{Asymptotic}&\Delta\Phi=D(n)-t(n)+1 &\text{$\dagger$}\\ \hline 6&\text{Fib-Heap-Decrease-Key}&O(1)&O(c)&\text{Asymptotic}&\Delta\Phi=4-c &\text{$\ddagger$}\\ \hline \end{array}$$


$\dagger$ - The cost of performing each link is paid for by the reduction in potential due to the link's reducing the number of roots by one.

$\ddagger$ - Why the potential function was defined to include a term that is twice the number of marked nodes. When a marked node $у$ is cut by a cascading cut, its mark bit is cleared, so the potential is reduced by $2$. One unit of potential pays for the cut and the clearing of the mark bit, and the other unit compensates for the unit increase in potential due to node $у$ becoming a root.


Note: It is quite a difficult question in the sense that it involves the description the problem which I am facing to understand the intuition behind the concept of Fibonacci Heap operations which is in fact related to an entire chapter in the CLRS text. If it demands too much in a single then please do tell me then I shall split it accordingly into parts. I have made my utmost attempt to make the question clear. If at places the meaning is not clear, then please do tell me then I shall rectify it.(Even the authors say that it is a difficult data structure, having only theoretical importance.)

The entire corresponding portion of the text can be found here.

$\endgroup$
  • 1
    $\begingroup$ I haven't read CLRS, but question: Have you looked at binomial heaps first? This is why I ask: cs.stackexchange.com/questions/49710/… $\endgroup$ – Pseudonym Jul 16 at 7:42
  • $\begingroup$ Yes I am reading the text CLRS (2nd edition) and they use the concepts of binomial heaps to go into the concepts fibonacci heaps $\endgroup$ – Abhishek Ghosh Jul 16 at 7:44
  • $\begingroup$ @Pseudonym thanks for your help $\endgroup$ – Abhishek Ghosh Jul 16 at 8:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.