Iterated logarithm

For the theorem in probability theory, see Law of the iterated logarithm.

In computer science, the iterated logarithm of ${\displaystyle n}$, written log* ${\displaystyle n}$ (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 ${\displaystyle 1}$.^[1] The simplest formal definition is the result of this recurrence relation:

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

Figure 1. Demonstrating log* 4 = 2 for the base-e iterated logarithm. The value of the iterated logarithm can be found by "zig-zagging" on the curve y = log_b(x) from the input n, to the interval [0,1]. In this case, b = e. The zig-zagging entails starting from the point (n, 0) and iteratively moving to (n, log_b(n) ), to (0, log_b(n) ), to (log_b(n), 0 ).

In computer science, lg* is often used to indicate the binary iterated logarithm, which iterates the binary logarithm (with base ${\displaystyle 2}$) instead of the natural logarithm (with base e). Mathematically, the iterated logarithm is well defined for any base greater than ${\displaystyle e^{1/e}\approx 1.444667}$, not only for base ${\displaystyle 2}$ and base e. The "super-logarithm" function ${\displaystyle \mathrm {slog} _{b}(n)}$ is "essentially equivalent" to the base ${\displaystyle b}$ iterated logarithm (although differing in minor details of rounding) and forms an inverse to the operation of tetration.^[2]

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 log* n) time.^[3]
• Fürer's algorithm for integer multiplication: O(n log n 2^O(lg* n)).
• Finding an approximate maximum (element at least as large as the median): lg* n − 1 ± 3 parallel operations.^[4]
• Richard Cole and Uzi Vishkin's distributed algorithm for 3-coloring an n-cycle: O(log* n) synchronous communication rounds.^[5]

The iterated logarithm grows at an extremely slow rate, much slower than the logarithm itself, or repeats of it. This is because the tetration grows much faster than iterated exponential:

$${\displaystyle {^{y}b}=\underbrace {b^{b^{\cdot ^{\cdot ^{b}}}}} _{y}\gg \underbrace {b^{b^{\cdot ^{\cdot ^{b^{y}}}}}} _{n}}$$

the inverse grows much slower: ${\displaystyle \log _{b}^{*}x\ll \log _{b}^{n}x}$.

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 estimated number of atoms in the known universe), the iterated logarithm with base 2 has a value no more than 5.

The base-2 iterated logarithm

x	lg* x
(−∞, 1]	0
(1, 2]	1
(2, 4]	2
(4, 16]	3
(16, 65536]	4
(65536, 265536]	5

Higher bases give smaller iterated logarithms.

Other applications

The iterated logarithm is closely related to the generalized logarithm function used in symmetric level-index arithmetic. The additive persistence of a number, the number of times someone must replace the number by the sum of its digits before reaching its digital root, is ${\displaystyle O(\log ^{*}n)}$.

In computational complexity theory, Santhanam^[6] shows that the computational resources DTIME — computation time for a deterministic Turing machine — and NTIME — computation time for a non-deterministic Turing machine — are distinct up to ${\displaystyle n{\sqrt {\log ^{*}n}}.}$

See also

• Inverse Ackermann function, an even more slowly growing function also used in computational complexity theory

References

1. Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2009) [1990]. "The iterated logarithm function, in Section 3.2: Standard notations and common functions". Introduction to Algorithms (3rd ed.). MIT Press and McGraw-Hill. pp. 58–59. ISBN 0-262-03384-4.
2. Furuya, Isamu; Kida, Takuya (2019). "Compaction of Church numerals". Algorithms. 12 (8) 159: 159. doi:10.3390/a12080159. MR 3998658.
3. Devillers, Olivier (March 1992). "Randomization yields simple ${\displaystyle O(n\log ^{\ast }n)}$ algorithms for difficult ${\displaystyle \Omega (n)}$ problems" (PDF). International Journal of Computational Geometry & Applications. 2 (1): 97–111. arXiv:cs/9810007. doi:10.1142/S021819599200007X. MR 1159844. S2CID 60203.
4. Alon, Noga; Azar, Yossi (April 1989). "Finding an approximate maximum" (PDF). SIAM Journal on Computing. 18 (2): 258–267. doi:10.1137/0218017. MR 0986665.
5. Cole, Richard; Vishkin, Uzi (July 1986). "Deterministic coin tossing with applications to optimal parallel list ranking" (PDF). Information and Control. 70 (1): 32–53. doi:10.1016/S0019-9958(86)80023-7. MR 0853994.
6. Santhanam, Rahul (2001). "On separators, segregators and time versus space" (PDF). Proceedings of the 16th Annual IEEE Conference on Computational Complexity, Chicago, Illinois, USA, June 18-21, 2001. IEEE Computer Society. pp. 286–294. doi:10.1109/CCC.2001.933895. ISBN 0-7695-1053-1.