Polylogarithmic function

In mathematics, a polylogarithmic function in $n$ is a polynomial in the logarithm of $n$ ,^[1]

a_{k}(\log n)^{k}+a_{k-1}(\log n)^{k-1}+\cdots +a_{1}(\log n)+a_{0}.

The notation $log k n$ is often used as a shorthand for $(log n) k$ , analogous to $sin 2 θ$ for $(sin θ) 2$ .

In computer science, polylogarithmic functions occur as the order of time for some data structure operations. Additionally, the exponential function of a polylogarithmic function produces a function with quasi-polynomial growth, and algorithms with this as their time complexity are said to take quasi-polynomial time.^[2]

All polylogarithmic functions of $n$ are $o(n ε)$ for every exponent $ε > 0$ (for the meaning of this symbol, see small o notation), that is, a polylogarithmic function grows more slowly than any positive exponent. This observation is the basis for the soft O notation $Õ(n)$ .^[3]

References

^ Black, Paul E. (2004-12-17). "polylogarithmic". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved 2010-01-10.
^ Complexity Zoo: Class QP: Quasipolynomial-Time
^ Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2022). Introduction to Algorithms (4th ed.). Cambridge, Mass.: The MIT Press. pp. 74–75. ISBN 9780262046305.

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[1] Black, Paul E. (2004-12-17). "polylogarithmic". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved 2010-01-10.

[2] Complexity Zoo: Class QP: Quasipolynomial-Time

[3] Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2022). Introduction to Algorithms (4th ed.). Cambridge, Mass.: The MIT Press. pp. 74–75. ISBN 9780262046305.

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