Question

Can anybody give me an intuitive explanation of why the Ackermann function http://en.wikipedia.org/wiki/Ackermann_function is related to the amortized complexity of union-find algorithm used for disjoint sets http://en.wikipedia.org/wiki/Disjoint-set_data_structure?

The analysis in Tarjan’s data structure book isn’t very intuitive.

I also looked it up in Introduction to Algorithms, but it also seems too rigorous and non-intuitive.

Thanks for your help!

Answer 1

Applied to Disjoint-set forests

from Wikipedia

(about find and union) These two
techniques complement each other;
applied together, the amortized time
per operation is only O(α(n)), where
α(n) is the inverse of the function
f(n) = A(n,n), and A is the extremely
quickly-growing Ackermann function.
Since α(n) is the inverse of this
function, α(n) is less than 5 for all
remotely practical values of n. Thus,
the amortized running time per
operation is effectively a small
constant.

So why Ackerman?

from Kruskal algoritm

The Function lg*n

Note that lg*n is a very slow growing
function, much slower than lg n. In
fact is slower than lg lg n, or any
finite composition of lg n. It is the
inverse of the function f(n) = 2
^2^2^…^2, n times. For n >= 5, f(n)
is greater than the number of atoms in
the universe. Hence for all intents
and purposes, the inverse of f(n) for
any real life value of n, is constant.
From an engineer’s point of view,
Kruskal’s algorithm runs in O(e).
Note of course that from a
theoretician’s point of view, a true
result of O(e) would still be a
significant breakthrough. The whole
picture is not complete because the
actual best result shows that lg*n can
be replaced by the inverse of A(p,n)
where A is Ackermann’s function, a
function that grows explosively. The
inverse of Ackermann’s function is
related to lg*n, and is a nicer
result, but the proof is even
harder.