Question

Fibonacci numbers have become a popular introduction to recursion for Computer Science students and there’s a strong argument that they persist within nature. For these reasons, many of us are familiar with them.

They also exist within Computer Science elsewhere too; in surprisingly efficient data structures and algorithms based upon the sequence.

There are two main examples that come to mind:

• Fibonacci heaps which have better
  amortized running time than binomial
  heaps.
• Fibonacci search which shares
  O(log N) running time with binary
  search on an ordered array.

Is there some special property of these numbers that gives them an advantage over other numerical sequences? Is it a spatial quality? What other possible applications could they have?

It seems strange to me as there are many natural number sequences that occur in other recursive problems, but I’ve never seen a Catalan heap.

Answer 1

The Fibonacci numbers have all sorts of really nice mathematical properties that make them excellent in computer science. Here’s a few:

1. They grow exponentially fast. One interesting data structure in which the Fibonacci series comes up is the AVL tree, a form of self-balancing binary tree. The intuition behind this tree is that each node maintains a balance factor so that the heights of the left and right subtree differ by at most one. Because of this, you can think of the minimum number of nodes necessary to get an AVL tree of height h is defined by a recurrence that looks like N(h + 2) ~= N(h) + N(h + 1), which looks a lot like the Fibonacci series. If you work out the math, you can show that the number of nodes necessary to get an AVL tree of height h is F(h + 2) – 1. Because the Fibonacci series grows exponentially fast, this means that the height of an AVL tree is at most logarithmic in the number of nodes, giving you the O(lg n) lookup time we know and love about balanced binary trees. In fact, if you can bound the size of some structure with a Fibonacci number, you’re likely to get an O(lg n) runtime on some operation. This is the real reason that Fibonacci heaps are called Fibonacci heaps – the proof that the number of heaps after a dequeue min involves bounding the number of nodes you can have in a certain depth with a Fibonacci number.
2. Any number can be written as the sum of unique Fibonacci numbers. This property of the Fibonacci numbers is critical to getting Fibonacci search working at all; if you couldn’t add together unique Fibonacci numbers into any possible number, this search wouldn’t work. Contrast this with a lot of other series, like 3ⁿ or the Catalan numbers. This is also partially why a lot of algorithms like powers of two, I think.
3. The Fibonacci numbers are efficiently computable. The fact that the series can be generated extremely efficiently (you can get the first n terms in O(n) or any arbitrary term in O(lg n)), then a lot of the algorithms that use them wouldn’t be practical. Generating Catalan numbers is pretty computationally tricky, IIRC. On top of this, the Fibonacci numbers have a nice property where, given any two consecutive Fibonacci numbers, let’s say F(k) and F(k + 1), we can easily compute the next or previous Fibonacci number by adding the two values (F(k) + F(k + 1) = F(k + 2)) or subtracting them (F(k + 1) – F(k) = F(k – 1)). This property is exploited in several algorithms, in conjunction with property (2), to break apart numbers into the sum of Fibonacci numbers. For example, Fibonacci search uses this to locate values in memory, while a similar algorithm can be used to quickly and efficiently compute logarithms.
4. They’re pedagogically useful. Teaching recursion is tricky, and the Fibonacci series is a great way to introduce it. You can talk about straight recursion, about memoization, or about dynamic programming when introducing the series. Additionally, the amazing closed-form for the Fibonacci numbers is often taught as an exercise in induction or in the analysis of infinite series, and the related matrix equation for Fibonacci numbers is commonly introduced in linear algebra as a motivation behind eigenvectors and eigenvalues. I think that this is one of the reasons that they’re so high-profile in introductory classes.

I’m sure there are more reasons than just this, but I’m sure that some of these reasons are the main factors. Hope this helps!