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In mathematics, a polylogarithmic function in n is a polynomial in the logarithm of n, a k ( log ⁡ n ) k + a k − 1 ( log ⁡ n ) k − 1 + ⋯ + a 1 ( log ⁡...

algorithm gets closer to the target word. An algorithm is said to run in polylogarithmic time if its time T ( n ) {\displaystyle T(n)} is O ( ( log ⁡ n ) k...

by a deterministic Turing machine in polynomial time (class P) and polylogarithmic space (class PolyL) (that is, O((log n)k) space for some constant k)...

the Lerch transcendent. Polylogarithms should not be confused with polylogarithmic functions, nor with the offset logarithmic integral Li(z), which has...

In mathematical analysis, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's...

Turing machine by an algorithm whose space complexity is bounded by a polylogarithmic function in the size of the input. In other words, polyL = DSPACE((log n)O(1))...

science is O ~ {\displaystyle {\tilde {O}}} (read soft-O), which hides polylogarithmic factors. There are two definitions in use: some authors use f ( n )...

(for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors...

integers could only exceed the square root of its largest value by a polylogarithmic factor, disproved by András Sárközy in 1978. The Erdős–Lovász conjecture...

Plouffe, Simon (April 1997). "On the Rapid Computation of Various Polylogarithmic Constants" (PDF). Mathematics of Computation. 66 (218): 903–913. Bibcode:1997MaCom...

If we use NC reductions, that is, reductions that can operate in polylogarithmic time on a parallel computer with a polynomial number of processors...

Babai, L. Fortnow, L. Levin, and M. Szegedy. "Checking computations in polylogarithmic time", in Proceedings of the 23rd ACM Symposium on the Theory of Computing...

solvable in polynomial time and polylogarithmic space on a deterministic Turing machine; in other words, given polylogarithmic space, a deterministic machine...

in parallel (see speedup). If a decision problem can be solved in polylogarithmic time by using a polynomial number of processors, then the problem is...

(help) Andoni, Alexandr; Krauthgamer, Robert; Onak, Krzysztof (2010). Polylogarithmic approximation for edit distance and the asymmetric query complexity...

Peter B.; Plouffe, Simon (1997). "On the Rapid Computation of Various Polylogarithmic Constants". Mathematics of Computation. 66 (218): 903–913. doi:10...

algorithms. The best known quantum algorithm for matrix completion runs in polylogarithmic time by using the HHL algorithm as a subroutine, and had been introduced...

of RATMs has been advanced through the exploration of deterministic polylogarithmic time and space and two-sorted logic, a concept explored in depth by...

PiHex web site David Bailey, Peter Borwein, and Simon Plouffe's BBP formula (On the rapid computation of various polylogarithmic constants) (PDF) v t e...

each node processes and sends a number of bits per round that are polylogarithmic in n, the network size. The Google Toolbar long had a PageRank feature...

are restricted not only to having polynomial-size but also to having polylogarithmic depth. The class AC is defined similarly to NC, however gates are allowed...

1999 paper titled Computationally Private Information Retrieval with Polylogarithmic Communication, where it was used in a private information retrieval...

based on rip-and-tent, which is practical and highly parallelized with polylogarithmic span. A divide and conquer algorithm for triangulations in two dimensions...

time, the minimum feedback arc set can be approximated to within a polylogarithmic approximation ratio, and maximum acyclic subgraphs can be approximated...

Fisher-Yates shuffle to get a parallel algorithm with linear work and polylogarithmic depth. The asymptotic time and space complexity of the Fisher–Yates...

contained in {1,2,...,n}, matching the classical lower bound up to a polylogarithmic factor. Jiří Matoušek and Joel Spencer later succeeded in getting rid...

Leonid A.; Szegedy, Mario (1991-01-01). "Checking computations in polylogarithmic time". Proceedings of the twenty-third annual ACM symposium on Theory...

NP-Complete problems such as SAT are known to be complete even under polylogarithmic time projections. It is known, however, that AC0 reductions define...

Lance; Levin, Leonid; Szegedy, Mario (1991), "Checking computations in polylogarithmic time", STOC '91: Proceedings of the twenty-third annual ACM symposium...

Stephen Cook after Nick Pippenger for his research on circuits with polylogarithmic depth and polynomial size. Pippenger became one of the most recent...

twelfth power of the number of digits in n times a factor that is polylogarithmic in the number of digits. However, this upper bound was rather loose;...

particularly in relation to the evaluation of many classes of logarithmic and polylogarithmic integrals, both definite and indefinite. They also have numerous applications...

protection against quantum computers, (2) quasi-linear proof time, (3) polylogarithmic verification time, and (4) transparency, which means no dependence...

2020. Andoni, Alexandr; Krauthgamer, Robert; Onak, Krzysztof (2010). Polylogarithmic approximation for edit distance and the asymmetric query complexity...

polynomial time and polylogarithmic space on a deterministic Turing machine; in other words, this result shows that, given polylogarithmic space, a deterministic...

have codewords polynomial in the size of the original message and polylogarithmic query complexity. Locally decodable codes have applications to data...

high-dimensional Euclidean space using polynomial preprocessing and polylogarithmic search time. The simplest solution to the NNS problem is to compute...

2 ) , {\displaystyle g_{n}=O\!\left((\log p_{n})^{2}\right)\!,} a polylogarithmic growth rate slower than any exponent θ > 0. Cramér's model, under which...

2023-09-16 CV (PDF) Peluse, Sarah; Prendiville, Sean (April 9, 2022). "A Polylogarithmic Bound in the Nonlinear Roth Theorem" (PDF). International Mathematics...

for their work constructing "an explicit two-source extractor with polylogarithmic min-entropy, resolving a central problem in the theory of computation...

Plouffe, Simon (April 1997). "On the Rapid Computation of Various Polylogarithmic Constants" (PDF). Mathematics of Computation. 66 (218): 903–913. Bibcode:1997MaCom...

group. NC (complexity), the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors...

time is polylogarithmic, leading to a total time for the parametric search that is slower than the decision algorithm by only a polylogarithmic factor...

1007/s11139-013-9528-5, S2CID 120943474. Broadhurst, D.J. (1998), "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ (...

{\displaystyle AC_{0}} or N C 0 {\displaystyle NC_{0}} circuits, or polylogarithmic projections where each subsequent reduction notion is weaker than the...

product of the reciprocal function and the natural logarithm is a polylogarithmic integral. Fubini's theorem allows this to be evaluated by translating...

Ben; Tao, Terence (2017). "New bounds for Szemerédi's theorem, III: A polylogarithmic bound for r4(N)". Mathematika. 63 (3): 944–1040. arXiv:1705.01703....

better than the trivial O ( n 2 ) {\displaystyle O(n^{2})} by only a polylogarithmic factor. Researchers in this area have also investigated the computational...

problem in ACC0 can be solved by circuits of depth 2, with AND gates of polylogarithmic fan-in at the inputs, connected to a single gate computing some symmetric...

four measures of performance. We say a quantity is small if it is a polylogarithmic function of n {\displaystyle n} , or is O ( n ϵ ) {\displaystyle O(n^{\epsilon...

Markus (1999). "Computationally Private Information Retrieval with Polylogarithmic Communication". Advances in Cryptology – EUROCRYPT '99. Prague, Czech...

diameters of Cayley graphs of finite simple groups of bounded rank are polylogarithmic in the size of the group, partially resolving a well-known conjecture...

kangaroo algorithm Pollard's rho algorithm for logarithms Polylogarithm Polylogarithmic function Prime number theorem Richter magnitude scale Grégoire de Saint-Vincent...

M. R.; King, V. (1995). "Randomized dynamic graph algorithms with polylogarithmic time per operation". Proceedings of the twenty-seventh annual ACM symposium...

R_{0}(f)=O(R_{2}(f)^{2}\log R_{2}(f))} . This relationship is optimal up to polylogarithmic factors. As for quantum decision tree complexities, D ( f ) = O ( Q...

Goyanes Guillera (2007). "Construction of binomial sums for π and polylogarithmic constants inspired by BBP formulas" (PDF). Applied Math. E-Notes. 7:...

polynomial time by a Merlin–Arthur protocol NC Solvable efficiently (in polylogarithmic time) on parallel computers NE Solvable by a non-deterministic machine...

Nikhil; Buchbinder, Niv; Madry, Aleksander; Naor, Joseph (2015). "A polylogarithmic-competitive algorithm for the k-server problem" (PDF). Journal of the...

logarithmic update-time lower bound by Pătrașcu and Demaine, and a polylogarithmic update-time algorithm by Holm and Rotenberg, improving on sub-linear...

a variety of problems, with the goal of designing highly parallel (polylogarithmic depth), work-efficient (linear in the sequential running time) algorithms...

differently from the rest of computer science. Standard Õ(·) usually hides polylogarithmic factors, but in fine-grained complexity it hides everything below polynomial:...

mathworld.wolfram.com. Retrieved 2024-10-02. Broadhurst, D. J. (1998). "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3)...

O notation. That is, it is bounded by an exponential function of a polylogarithmic function. This generalizes the polynomials and the functions of polynomial...

Theoretically, all algorithms in PAM are work-efficient and have polylogarithmic depth. PAM uses underlying persistent tree structure such that multi-versioning...

Peter; Plouffe, Simon (1997). "On the rapid computation of various polylogarithmic constants". Mathematics of Computation. 66 (1): 903–913. Bibcode:1997MaCom...

approximated in polynomial time by a solution whose length is within a polylogarithmic factor of optimal. Art gallery problem, which similarly involves viewing...

over the prime numbers up to n {\displaystyle n} ) is upper bounded polylogarithmically in n {\displaystyle n} . For some other Diophantine problems, the...

possible to find succinct greedy embeddings into a Euclidean space of polylogarithmic dimension, with the additional property that distances in the graph...

are restricted not only to having polynomial-size but also to having polylogarithmic depth. The class AC is defined similarly to NC, however gates are allowed...

implication is that "fast" parallel computers (i.e. those that run in polylogarithmic time) recognize exactly the languages in polyL. It was proven in 1978...

(informally, whether every polynomially-solvable problem can be solved by a polylogarithmic-time parallel algorithm) and Ketan Mulmuley has suggested an approach...

O(m\log n)} . One possible parallelisation of this algorithm yields a polylogarithmic time complexity, i.e. T ( m , n , p ) ⋅ p ∈ O ( m log ⁡ n ) {\displaystyle...

L_{n}[\alpha ,c]=L_{n}[0,c]=e^{(c+o(1))\ln \ln n}=(\ln n)^{c+o(1)}\,} is a polylogarithmic function (a polynomial function of ln n); When α {\displaystyle \alpha...

be gainfully employed in the design of deterministic log-space and polylogarithmic-space algorithms. In particular, we have a new set of tools to use...

Robert; Stauffer, Allan (2019). "Definite Integral of Arctangent and Polylogarithmic Functions Expressed as a Series". Mathematics. 7 (1099): 1099. doi:10...

the parameters" can produce a bound that is less than quadratic by a polylogarithmic factor, he does not supply details and his proof that the number is...

possible to construct a Danzer set of growth rate that is within a polylogarithmic factor of O ( r d ) {\displaystyle O(r^{d})} . For instance, overlaying...

Markus (1999). "Computationally Private Information Retrieval with Polylogarithmic Communication". In Stern, Jacques (ed.). Advances in Cryptology — EUROCRYPT...

the number of substructures and its condition number grows only polylogarithmically with the number of elements per substructure. The coarse space in...

M.; King, V.; Mountjoy, B. (2013). Dynamic graph connectivity in polylogarithmic worst case time. Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium...

algorithm, which performs a linear number of violation tests and a polylogarithmic number of basis evaluations. The basis evaluations for g may be performed...

{\displaystyle c\in (0,1)} . The next nearly linear goal is linear up to a polylogarithmic factor; n = poly ( log ⁡ k ) ∗ k {\displaystyle n={\text{poly}}(\log...

factor, can reduce the given problem to subproblems whose size is a polylogarithmic function of n {\displaystyle n} , in time O ( n 4 / 3 ) {\displaystyle...

Tao. They achieve bounds that differ from the optimal bounds only by polylogarithmic factors by strengthening the assumptions. Instead of the incoherence...

\Omega (N)} worst-case access overheads. Some ORAM constructions with polylogarithmic worst-case computational overheads are. The constructions of were in...

hidden-line problem is in the complexity class NC, i.e., it can be solved in polylogarithmic time by using a polynomial number of processors. Hidden-surface algorithms...

P. B. Borwein and S. Plouffe, On the rapid computation of various polylogarithmic constants. Math. Comp., Vol. 66 (1997). R. P. Brent and E. M. McMillan...

ISSN 0895-4801. Halperin, Eran; Krauthgamer, Robert (June 9, 2003). "Polylogarithmic inapproximability". Proceedings of the thirty-fifth annual ACM symposium...