eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-07-04
64:1
64:13
10.4230/LIPIcs.ICALP.2019.64
article
A Composition Theorem for Randomized Query Complexity via Max-Conflict Complexity
Gavinsky, Dmitry
1
Lee, Troy
2
Santha, Miklos
3
4
5
Sanyal, Swagato
6
Institute of Mathematics, Czech Academy of Sciences, 115 67 Žitna 25, Praha 1, Czech Republic
Centre for Quantum Software and Information, Faculty of Engineering and Information Technology, University of Technology Sydney, Australia
CNRS, IRIF, Université de Paris, 75205 Paris, France
Centre for Quantum Technologies, National University of Singapore, Singapore 117543
MajuLab, UMI 3654, Singapore
Indian Institute of Technology Kharagpur, India
For any relation f subseteq {0,1}^n x S and any partial Boolean function g:{0,1}^m -> {0,1,*}, we show that R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * sqrt{R_{1/3}(g)}) , where R_epsilon(*) stands for the bounded-error randomized query complexity with error at most epsilon, and f o g^n subseteq ({0,1}^m)^n x S denotes the composition of f with n instances of g.
The new composition theorem is optimal, at least, for the general case of relational problems: A relation f_0 and a partial Boolean function g_0 are constructed, such that R_{4/9}(f_0) in Theta(sqrt n), R_{1/3}(g_0)in Theta(n) and R_{1/3}(f_0 o g_0^n) in Theta(n).
The theorem is proved via introducing a new complexity measure, max-conflict complexity, denoted by bar{chi}(*). Its investigation shows that bar{chi}(g) in Omega(sqrt{R_{1/3}(g)}) for any partial Boolean function g and R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * bar{chi}(g)) for any relation f, which readily implies the composition statement. It is further shown that bar{chi}(g) is always at least as large as the sabotage complexity of g.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol132-icalp2019/LIPIcs.ICALP.2019.64/LIPIcs.ICALP.2019.64.pdf
query complexity
lower bounds