TY - JOUR
T1 - Query-to-communication lifting using low-discrepancy gadgets
AU - Chattopadhyay, Arkadev
AU - Filmus, Yuval
AU - Koroth, Sajin
AU - Meir,
AU - Pitassi, Toniann
N1 - Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics.
PY - 2021
Y1 - 2021
N2 - Lifting theorems are theorems that relate the query complexity of a function f : {0, 1}n → {0, 1} to the communication complexity of the composed function f ○ gn for some "gadget"g : {0, 1}b × {0, 1}b → {0,1}. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in recent years. In addition, such theorems can be viewed as a strong generalization of a direct-sum theorem for the gadget g. We prove a new lifting theorem that works for all gadgets g that have logarithmic length and exponentially-small discrepancy, for both deterministic and randomized communication complexity. Thus, we significantly increase the range of gadgets for which such lifting theorems hold. Our result has two main motivations: first, allowing a larger variety of gadgets may support more applications. In particular, our work is the first to prove a randomized lifting theorem for logarithmic-size gadgets, thus improving some applications of the theorem. Second, our result can be seen as a strong generalization of a direct-sum theorem for functions with low discrepancy.
AB - Lifting theorems are theorems that relate the query complexity of a function f : {0, 1}n → {0, 1} to the communication complexity of the composed function f ○ gn for some "gadget"g : {0, 1}b × {0, 1}b → {0,1}. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in recent years. In addition, such theorems can be viewed as a strong generalization of a direct-sum theorem for the gadget g. We prove a new lifting theorem that works for all gadgets g that have logarithmic length and exponentially-small discrepancy, for both deterministic and randomized communication complexity. Thus, we significantly increase the range of gadgets for which such lifting theorems hold. Our result has two main motivations: first, allowing a larger variety of gadgets may support more applications. In particular, our work is the first to prove a randomized lifting theorem for logarithmic-size gadgets, thus improving some applications of the theorem. Second, our result can be seen as a strong generalization of a direct-sum theorem for functions with low discrepancy.
KW - Communication complexity
KW - Lifting
KW - Query complexity
UR - http://www.scopus.com/inward/record.url?scp=85103792881&partnerID=8YFLogxK
U2 - 10.1137/19M1310153
DO - 10.1137/19M1310153
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AN - SCOPUS:85103792881
SN - 0097-5397
VL - 50
SP - 171
EP - 210
JO - SIAM Journal on Computing
JF - SIAM Journal on Computing
IS - 1
ER -