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In computer science, the iterated logarithm of ''n'', written ''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: : On the positive real numbers, the continuous super-logarithm (inverse tetration) is essentially equivalent: : but on the negative real numbers, log-star is 0, whereas for positive ''x'', so the two functions differ for negative arguments. In computer science, 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 (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 . ==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'' ''n'') time〔Olivier Devillers, "Randomization yields simple O(n log * n) algorithms for difficult ω(n) problems.". ''International Journal of Computational Geometry & Applications'' 2:01 (1992), pp. 97–111.〕 * Fürer's algorithm for integer multiplication: O(''n'' log ''n'' 2''O''( ''n'')) * Finding an approximate maximum (element at least as large as the median): ''n'' − 4 to ''n'' + 2 parallel operations〔Noga Alon and Yossi Azar, "Finding an Approximate Maximum". ''SIAM Journal of Computing'' 18:2 (1989), pp. 258–267.〕 * Richard Cole and Uzi Vishkin's distributed algorithm for 3-coloring an ''n''-cycle: ''O''( ''n'') synchronous communication rounds.〔Richard Cole and Uzi Vishkin: "Deterministic coin tossing with applications to optimal parallel list ranking", Information and Control 70:1(1986), pp. 32–53.〕〔 Section 30.5.〕 * Performing weighted quick-union with path compression 〔https://www.cs.princeton.edu/~rs/AlgsDS07/01UnionFind.pdf〕 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 estimated number of atoms in the known universe), the iterated logarithm with base 2 has a value no more than 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「iterated logarithm」の詳細全文を読む スポンサード リンク
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