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Randomized Lifting to Semi-Structured Communication Complexity via Linear Diversity

Authors Vladimir Podolskii , Alexander Shekhovtsov

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Vladimir Podolskii
  • Tufts University, Medford, MA, USA
Alexander Shekhovtsov
  • Moscow Institute of Physics and Technology, Russia

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Vladimir Podolskii and Alexander Shekhovtsov. Randomized Lifting to Semi-Structured Communication Complexity via Linear Diversity. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 78:1-78:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.78

Abstract

We study query-to-communication lifting. The major open problem in this area is to prove a lifting theorem for gadgets of constant size. The recent paper [Paul Beame and Sajin Koroth, 2023] introduces semi-structured communication complexity, in which one of the players can only send parities of their input bits. They have shown that for any m ≥ 4 deterministic decision tree complexity of a function f can be lifted to the so called semi-structured communication complexity of f∘Ind_m, where Ind_m is the Indexing gadget. 
As our main contribution we extend these results to randomized setting. Our results also apply to a substantially larger set of gadgets. More specifically, we introduce a new complexity measure of gadgets, linear diversity. For all gadgets g with non-trivial linear diversity we show that randomized decision tree complexity of f lifts to randomized semi-structured communication complexity of f∘g. In particular, this gives tight lifting results for Indexing gadget Ind_m, Inner Product gadget IP_m for all m ≥ 2, and for Majority gadget MAJ_m for all m ≥ 4. We prove the same results for deterministic case.
From our result it immediately follows that deterministic/randomized decision tree complexity lifts to deterministic/randomized parity decision tree complexity. For randomized case this is the first result of this type. For deterministic case, our result improves the bound in [Arkadev Chattopadhyay et al., 2023] for Inner Product gadget.
To obtain our results we introduce a new secret sets approach to simulation of semi-structured communication protocols by decision trees. It allows us to simulate (restricted classes of) communication protocols on truly uniform distribution of inputs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Oracles and decision trees
Keywords
  • communication complexity
  • decision trees
  • lifting

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References

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