ECCC-Report TR19-103https://eccc.weizmann.ac.il/report/2019/103Comments and Revisions published for TR19-103en-usTue, 24 Nov 2020 22:20:48 +0200
Revision 2
| Query-to-Communication Lifting Using Low-Discrepancy Gadgets |
Or Meir,
Sajin Koroth,
Arkadev Chattopadhyay,
Toniann Pitassi,
Yuval Filmus
https://eccc.weizmann.ac.il/report/2019/103#revision2Lifting theorems are theorems that relate the query complexity of a function $f:\left\{ 0,1 \right\}^n\to \left\{ 0,1 \right\}$ to the communication complexity of the composed function $f\circ g^n$, for some “gadget” $g:\left\{ 0,1 \right\}^b\times \left\{ 0,1 \right\}^b\to \left\{ 0,1 \right\}$. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in the 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.Tue, 24 Nov 2020 22:20:48 +0200https://eccc.weizmann.ac.il/report/2019/103#revision2
Revision 1
| Query-to-Communication Lifting Using Low-Discrepancy Gadgets |
Or Meir,
Sajin Koroth,
Arkadev Chattopadhyay,
Toniann Pitassi,
Yuval Filmus
https://eccc.weizmann.ac.il/report/2019/103#revision1Lifting theorems are theorems that relate the query complexity of a function $f:\left\{ 0,1 \right\}^n\to \left\{ 0,1 \right\}$ to the communication complexity of the composed function $f\circ g^n$, for some “gadget” $g:\left\{ 0,1 \right\}^b\times \left\{ 0,1 \right\}^b\to \left\{ 0,1 \right\}$. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in the 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.Tue, 31 Dec 2019 18:26:33 +0200https://eccc.weizmann.ac.il/report/2019/103#revision1
Paper TR19-103
| Query-to-Communication Lifting Using Low-Discrepancy Gadgets |
Or Meir,
Sajin Koroth,
Arkadev Chattopadhyay,
Toniann Pitassi,
Yuval Filmus
https://eccc.weizmann.ac.il/report/2019/103Lifting theorems are theorems that relate the query complexity of a function $f:\left\{ 0,1 \right\}^n\to \left\{ 0,1 \right\}$ to the communication complexity of the composed function $f\circ g^n$, for some “gadget” $g:\left\{ 0,1 \right\}^b\times \left\{ 0,1 \right\}^b\to \left\{ 0,1 \right\}$. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in the 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.Sun, 11 Aug 2019 07:04:36 +0300https://eccc.weizmann.ac.il/report/2019/103