{
  "tag": "44165W",
  "kind": "definition",
  "slug": "iterated-logarithm",
  "name": "Iterated logarithm",
  "aliases": [
    "Iterated logarithm (log*)",
    "iterated logarithm",
    "log*",
    "log-star",
    "log-star function"
  ],
  "area": "foundations/asymptotic-analysis",
  "level": "core",
  "summary": "The number of times the logarithm must be applied before the value drops to at most one; it grows extremely slowly.",
  "prereqs": [
    "logarithms",
    "asymptotic-notation"
  ],
  "sources": [
    {
      "book": "isbn:9780262046305",
      "title": "Cormen, Leiserson, Rivest and Stein, Introduction to Algorithms, fourth edition",
      "locator": "Chapter 3, standard notations and common functions (iterated logarithm)"
    },
    {
      "paper": "Hopcroft and Ullman, Set merging algorithms, SIAM Journal on Computing 2 (1973)",
      "locator": "the O(m lg* n) bound for union by rank with path compression"
    },
    {
      "paper": "Cole and Vishkin, Deterministic coin tossing with applications to optimal parallel list ranking, Information and Control 70 (1986)",
      "locator": "the coloring of a cycle in O(lg* n) rounds"
    },
    {
      "paper": "Tarjan, Efficiency of a good but not linear set union algorithm, Journal of the ACM 22 (1975)",
      "locator": "the bound with the inverse Ackermann function"
    },
    {
      "paper": "Nathan Linial, Locality in distributed graph algorithms, SIAM Journal on Computing 21 (1992)",
      "locator": "the lower bound for coloring a cycle"
    },
    {
      "paper": "Martin Fürer, Faster integer multiplication, STOC 2007",
      "locator": "the main theorem"
    },
    {
      "paper": "David Harvey and Joris van der Hoeven, Integer multiplication in time O(n log n), Annals of Mathematics 193 (2021)",
      "locator": "the main theorem"
    }
  ],
  "see_also": [
    "inverse-ackermann-function",
    "ackermann-function",
    "disjoint-set-forest",
    "union-by-rank",
    "union-find-amortized-bound"
  ],
  "url": "https://algorithm.tamnd.com/iterated-logarithm/",
  "path": "nodes/foundations/asymptotic-analysis/iterated-logarithm",
  "status": {
    "slug": "iterated-logarithm",
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}