Disjoint-set data structure

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In computer science, the iterated logarithm of n, written Template:Log-star n (usually read "log star"), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1. The simplest formal definition is the result of this recursive function:

$\log ^{*}n:={\begin{cases}0&{\mbox{if }}n\leq 1;\\1+\log ^{*}(\log n)&{\mbox{if }}n>1\end{cases}}$

On the positive real numbers, the continuous super-logarithm (inverse tetration) is essentially equivalent:

$\log ^{*}n=\lceil {\text{slog}}_{e}(n)\rceil$

but on the negative real numbers, log-star is 0, whereas $\lceil {\text{slog}}_{e}(-x)\rceil =-1$ for positive x, so the two functions differ for negative arguments.

Figure 1. Demonstrating lg* 4 = 2

In computer science, Template:Lg-star is often used to indicate the binary iterated logarithm, which iterates the binary logarithm instead. The iterated logarithm accepts any positive real number and yields an integer. Graphically, it can be understood as the number of "zig-zags" needed in Figure 1 to reach the interval [0, 1] on the x-axis.

Mathematically, the iterated logarithm is well-defined not only for base 2 and base e, but for any base greater than $e^{1/e}\approx 1.444667$.

Analysis of algorithms

The iterated logarithm is useful in analysis of algorithms and computational complexity, appearing in the time and space complexity bounds of some algorithms such as:

• Finding the Delaunay triangulation of a set of points knowing the Euclidean minimum spanning tree: randomized O(n Template:Log-star n) time
• Fürer's algorithm for integer multiplication: O(n log n 2^{Template:Lg-star n})
• Finding an approximate maximum (element at least as large as the median): Template:Lg-star n − 4 to Template:Lg-star n + 2 parallel operations
• Richard Cole and Uzi Vishkin's distributed algorithm for 3-coloring an n-cycle: O(Template:Log-star n) synchronous communication rounds.

The iterated logarithm grows at an extremely slow rate, much slower than the logarithm itself. For all values of n relevant to counting the running times of algorithms implemented in practice (i.e., n ≤ 2⁶⁵⁵³⁶, which is far more than the atoms in the known universe), the iterated logarithm with base 2 has a value no more than 5.

x	Template:Lg-star x
(−∞, 1]	0
(1, 2]	1
(2, 4]	2
(4, 16]	3
(16, 65536]	4
(65536, 265536]	5

Higher bases give smaller iterated logarithms. Indeed, the only function commonly used in complexity theory that grows more slowly is the inverse Ackermann function.

Other applications

The iterated logarithm is closely related to the generalized logarithm function used in symmetric level-index arithmetic. It is also proportional to the additive persistence of a number, the number of times one must replace the number by the sum of its digits before reaching its digital root.

Santhanam shows that DTIME and NTIME are distinct up to $n{\sqrt {\log ^{*}n}}.$

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• Template:Introduction to Algorithms