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Communication Memento: Memoryless Communication Complexity

Authors Srinivasan Arunachalam, Supartha Podder



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Srinivasan Arunachalam
  • IBM T. J. Watson Research Center, Yorktown Heights, NY, USA
Supartha Podder
  • University of Ottawa, Canada

Acknowledgements

We thank Florian Speelman for the proof of the first inequality of Theorem 21 and letting us include it in our paper. We would like to thank Mika Göös, Robert Robere, Dave Touchette and Henry Yuen for helpful pointers. SP also thanks Anne Broadbent and Alexander Kerzner for discussions during the early phase of this project as a part of discussing gardenhose model. We also thank anonymous reviewers for their helpful comments and positive feedback. Part of this work was done when SA was visiting University of Ottawa (hosted by Anne Broadbent) and University of Toronto (hosted by Henry Yuen) and thank them for their hospitality.

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Srinivasan Arunachalam and Supartha Podder. Communication Memento: Memoryless Communication Complexity. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 61:1-61:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ITCS.2021.61

Abstract

We study the communication complexity of computing functions F: {0,1}ⁿ × {0,1}ⁿ → {0,1} in the memoryless communication model. Here, Alice is given x ∈ {0,1}ⁿ, Bob is given y ∈ {0,1}ⁿ and their goal is to compute F(x,y) subject to the following constraint: at every round, Alice receives a message from Bob and her reply to Bob solely depends on the message received and her input x (in particular, her reply is independent of the information from the previous rounds); the same applies to Bob. The cost of computing F in this model is the maximum number of bits exchanged in any round between Alice and Bob (on the worst case input x,y). In this paper, we also consider variants of our memoryless model wherein one party is allowed to have memory, the parties are allowed to communicate quantum bits, only one player is allowed to send messages. We show that some of these different variants of our memoryless communication model capture the garden-hose model of computation by Buhrman et al. (ITCS'13), space-bounded communication complexity by Brody et al. (ITCS'13) and the overlay communication complexity by Papakonstantinou et al. (CCC'14). Thus the memoryless communication complexity model provides a unified framework to study all these space-bounded communication complexity models. We establish the following main results: (1) We show that the memoryless communication complexity of F equals the logarithm of the size of the smallest bipartite branching program computing F (up to a factor 2); (2) We show that memoryless communication complexity equals garden-hose model of computation; (3) We exhibit various exponential separations between these memoryless communication models. We end with an intriguing open question: can we find an explicit function F and universal constant c > 1 for which the memoryless communication complexity is at least c log n? Note that c ≥ 2+ε would imply a Ω(n^{2+ε}) lower bound for general formula size, improving upon the best lower bound by [Nečiporuk, 1966].

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
Keywords
  • Communication complexity
  • space complexity
  • branching programs
  • garden-hose model
  • quantum computing

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