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Revision #2 to TR19-103 | 24th November 2020 22:20

#### Query-to-Communication Lifting Using Low-Discrepancy Gadgets

Revision #2
Accepted on: 24th November 2020 22:20
Keywords:

Abstract:

Lifting 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.

Changes to previous version:

Minor fixes.

Revision #1 to TR19-103 | 31st December 2019 18:26

#### Query-to-Communication Lifting Using Low-Discrepancy Gadgets

Revision #1
Accepted on: 31st December 2019 18:26
Keywords:

Abstract:

Lifting 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.

Changes to previous version:

Fixed minor typos.

### Paper:

TR19-103 | 7th August 2019 14:41

#### Query-to-Communication Lifting Using Low-Discrepancy Gadgets

TR19-103
Publication: 11th August 2019 07:04
Keywords:

Abstract:

Lifting 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.

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