Iterated Logarithm - Analysis of Algorithms

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, Olivier Devillers
• Fürer's algorithm for integer multiplication: O(n log n 2lg* n)
• Finding an approximate maximum (element at least as large as the median): lg* n − 4 to lg* n + 2 parallel operations
• Richard Cole and Uzi Vishkin's distributed algorithm for 3-coloring an n-cycle: O(log* 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.

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