Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

The Log-Star and Inverse Ackermann Functions
A surprising property of the tree-based partition data structure, when implemented
using the union-by-size and path-compression heuristics, is that performing a
series of nunion and find operations takes 0(nlog*n) time, where log*n is the
log-star function, which is the inverse of the tower-of-twos function. Intuitively,
log*n is the number of times that one can iteratively take the logarithm (base 2) of
a number before getting a number smaller than 2. Table 11.1 shows a few sample
values.
Table 11.1: Some values of log* n and critical values
for its inverse.
As is demonstrated in Table 11.1, for all practical purposes, log* n ≤ 5. It is an
amazingly slow-growing function (but one that is growing nonetheless).
In fact, the running time of a series of n partition operations implemented as
above can actually be shown to be O(nα(n)), where α(n) is the inverse of the
Ackermann function,A, which grows asymptotically even slower than log* n.
Although we will not prove this fact, let us define the Ackermann function here,
so as to appreciate just how quickly it grows; hence, how slowly its inverse
grows. We first define an indexed Ackermann function, Ai, as follows:
A 0 (n)= 2n
for n≥0
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