低通信叶子门的德摩根公式的算法与下界
计算复杂性
2020-02-21 v1
摘要
类 F O R M U L A [ s ] ∘ G FORMULA[s] \circ \mathcal{G} F O R M U L A [ s ] ∘ G s s s G \mathcal{G} G F O R M U L A [ n 1.99 ] ∘ G FORMULA[n^{1.99}]\circ \mathcal{G} F O R M U L A [ n 1.99 ] ∘ G G \mathcal{G} G R ( k ) ( G ) R^{(k)}(\mathcal{G}) R ( k ) ( G ) G \mathcal{G} G k k k G I P n k GIP^k_n G I P n k 1 / 2 + ε 1/2+\varepsilon 1/2 + ε F O R M U L A [ s ] ∘ G FORMULA[s]\circ \mathcal{G} F O R M U L A [ s ] ∘ G s = o ( n 2 ( k ⋅ 4 k ⋅ R ( k ) ( G ) ⋅ log ( n / ε ) ⋅ log ( 1 / ε ) ) 2 ) . s = o \! \left ( \frac{n^2}{ \left(k \cdot 4^k \cdot {R}^{(k)}(\mathcal{G}) \cdot \log (n/\varepsilon) \cdot \log(1/\varepsilon) \right)^{2}} \right). s = o ( ( k ⋅ 4 k ⋅ R ( k ) ( G ) ⋅ log ( n / ε ) ⋅ log ( 1/ ε ) ) 2 n 2 ) . G I P n k GIP^k_n G I P n k F O R M U L A [ n 1.99 ] ∘ P T F k − 1 FORMULA[n^{1.99}]\circ PTF^{k-1} F O R M U L A [ n 1.99 ] ∘ P T F k − 1 n / 2 + O ( s ⋅ R ( 2 ) ( G ) ⋅ log ( s / ε ) ⋅ log ( 1 / ε ) ) n/2 + O\left(\sqrt{s} \cdot R^{(2)}(\mathcal{G}) \cdot\log(s/\varepsilon) \cdot \log (1/\varepsilon) \right) n /2 + O ( s ⋅ R ( 2 ) ( G ) ⋅ log ( s / ε ) ⋅ log ( 1/ ε ) ) ε \varepsilon ε F O R M U L A [ s ] ∘ G FORMULA[s] \circ \mathcal{G} F O R M U L A [ s ] ∘ G F O R M U L A [ s ] ∘ L T F FORMULA[s] \circ LTF F O R M U L A [ s ] ∘ L T F O ( n 1 / 2 ⋅ s 1 / 4 ⋅ log ( n ) ⋅ log ( n / ε ) ) O\left(n^{1/2}\cdot s^{1/4}\cdot \log(n)\cdot \log(n/\varepsilon)\right) O ( n 1/2 ⋅ s 1/4 ⋅ log ( n ) ⋅ log ( n / ε ) ) ε ≤ 1 / n \varepsilon \leq 1/n ε ≤ 1/ n n n n o ( n ) o(n) o ( n ) 2 n − t 2^{n-t} 2 n − t # \# # F O R M U L A [ s ] ∘ G FORMULA[s] \circ \mathcal{G} F O R M U L A [ s ] ∘ G t = Ω ( n s ⋅ log 2 ( s ) ⋅ R ( 2 ) ( G ) ) 1 / 2 . t=\Omega\left(\frac{n}{\sqrt{s}\cdot\log^2(s)\cdot R^{(2)}(\mathcal{G})}\right)^{1/2}. t = Ω ( s ⋅ log 2 ( s ) ⋅ R ( 2 ) ( G ) n ) 1/2 . F O R M U L A [ n 1.99 ] ∘ L T F FORMULA[n^{1.99}]\circ LTF F O R M U L A [ n 1.99 ] ∘ L T F F O R M U L A [ n 1.99 ] ∘ X O R FORMULA[n^{1.99}]\circ XOR F O R M U L A [ n 1.99 ] ∘ X O R F O R M U L A [ n 1.99 ] ∘ X O R FORMULA[n^{1.99}] \circ XOR F O R M U L A [ n 1.99 ] ∘ X O R 2 O ( n / log n ) 2^{O(n/\log n)} 2 O ( n / l o g n )
引用
@article{arxiv.2002.08533,
title = {Algorithms and Lower Bounds for de Morgan Formulas of Low-Communication Leaf Gates},
author = {Valentine Kabanets and Sajin Koroth and Zhenjian Lu and Dimitrios Myrisiotis and Igor Oliveira},
journal= {arXiv preprint arXiv:2002.08533},
year = {2020}
}