Iterated logarithm

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

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

In computer science, lg* is often used to indicate the binary iterated logarithm, which iterates the binary logarithm (with base $2$ ) instead of the natural logarithm (with base e). Mathematically, the iterated logarithm is well defined for any base greater than $e^{1/e}\approx 1.444667$ , not only for base $2$ and base e. The "super-logarithm" function $\mathrm {slog} _{b}(n)$ is "essentially equivalent" to the base $b$ 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:

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

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

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
$(-\infty, 1]$	0
$(1, 2]$	1
$(2, 4]$	2
$(4, 16]$	3
$(16, 65536]$	4
$(65536, 2 65536]$	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 $O(\log ^{*}n)$ .

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 $n{\sqrt {\log ^{*}n}}.$

References

^ Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2009) . "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.
^ Furuya, Isamu; Kida, Takuya (2019). "Compaction of Church numerals". Algorithms. 12 (8) 159: 159. doi:10.3390/a12080159. MR 3998658.
^ Devillers, Olivier (March 1992). "Randomization yields simple $O(n\log ^{\ast }n)$ algorithms for difficult $\Omega (n)$ problems" (PDF). International Journal of Computational Geometry & Applications. 2 (1): 97–111. arXiv:cs/9810007. doi:10.1142/S021819599200007X. MR 1159844. S2CID 60203.
^ 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.
^ 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.
^ 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.

[1] Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2009) . "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 $O(n\log ^{\ast }n)$ algorithms for difficult $\Omega (n)$ 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.

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Iterated logarithm

Analysis of algorithms

Other applications

See also

References